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UNIVERSIDADE DE LISBOA

INSTITUTO SUPERIOR TECNICO

Improving Information Security in Ranking, Recommender and Control

Systems

Guilherme Henrique Cacador Ramos

Supervisor: Doctor Carlos Manuel Costa Lourenco Caleiro

Thesis specifically prepared to obtain the PhD Degree in

Information Security

Draft

September 13, 2018

To my family and friends.

“We can only see a short distance ahead, but we can see plenty there that needs to be done.”

Alan Turing

Resumo

Nesta tese, propomos varios desenvolvimentos em seguranca de informacao nas areas de

comercio eletronico, sistemas de ranking, sistemas de recomendacao, e de controlo de sis-

temas, utilizando ideias provenientes da area de teoria de informacao.

Primeiro, propusemos um sistema de ranking que agrupa utilizadores usando metricas de

semelhanca, com base nas suas preferencias. O sistema apresenta rankings, possivelmente

distintos, para o mesmo produto a grupos de utilizadores diferentes. Alem da vantagem

de apresentar um ranking mais personalizado aos utilizadores, o sistema e mais resistente a

ataques e ruıdo do que o estado-da-arte, como avaliado pelos dados reais. Depois, exploramos

o efeito de suborno (bribing) em sistemas de ranking baseados em reputacao, no cenario de

um ranking para cada produto e no cenario que propusemos. Esta ideia provem da area de

controlo, pois o objectivo e controlar utilizadores de forma a levar o ranking de produtos

para valores desejados. Encontramos as estrategias de suborno optimas e avaliamos a nossa

metodologia com dados reais, verificando que o sistema proposto e mais robusto. Em sistemas

de recomendacao, propusemos o uso, em dois contextos, das metricas de semelhanca intro-

duzidas. Introduzimos um sistema de recomendacao eficiente e que apresenta resultados que

competem com o estado-da-arte, por vezes melhores, em dados reais e artificiais. Propusemos

um sistema de recomendacao para grupos usando uma das metricas introduzidas, com melhor

complexidade computacional que a metrica padrao, tendo um ganho de horas em dados reais.

Em controlo de sistemas, apresentamos metodos para encontrar o posicionamento do

numero mınimo de actuadores em sistemas LTI e em sistemas LTI alternados no cenario em

que um conjunto de controladores falhe, por exemplo devido a um ciberataque. Mostramos

que o primeiro problema e NP-completo e desenhamos algoritmos para o resolver explicita-

mente e para aproximar a solucao em tempo polinomial. Criamos, ainda, algoritmos para

resolver e aproximar a solucao de dois cenarios em sistemas LTI alternados. Por fim, explo-

rando uma decomposicao de grafos da area de controlo, apresentamos um novo limitara o

ındice de convergiria de matrizes Booleanas com propriedades adequadas.

Palavras-chave: Sistemas de Ranking, Sistemas de Recomendacao, Sistemas de Controlo,

Seguranca de Informacao, Matrizes Booleanas.

i

Abstract

In this thesis, we propose to advance in the field of information security in the areas of e-

commerce, ranking systems, recommender systems and control systems, using ideas from the

area of information theory.

First, we propose a ranking system that groups users based on their preferences, introduc-

ing similarity measures. The system presents possibly distinct rankings for the same product

in different user groups. Besides the advantage of presenting more personalized rankings to

users, it is a system more resistant to attacks and noise than the state-of-the-art, as evaluated

from real data. We then explore the effect of bribing in reputation-based ranking systems,

in the usual scenario (a ranking for each product) and in the scenario we proposed. This

idea is inspired by the area of control because the goal is to control users, in order to drive

the ranking of products to desired values. We find the optimal bribing strategies, and we

evaluate our methodology with real data, with the proposed ranking system being more ro-

bust to bribery. In recommender systems, we propose the use of the introduced similarity

measures in two contexts. In the first, we introduce an efficient recommender system that

presents results that compete with the state-of-the-art, being sometimes better, in both real

and synthetic data. In the second, we propose a group recommender system utilizing one of

the introduced measures. We achieve better computational complexity compared with the

standard measures, obtaining a gain of hours in real data.

In control systems, we present methods to find the placement of the minimum number

of inputs in LTI systems and switched LTI systems, in the eventual scenario where a set of

controllers may fail, e.g., due to a cyberattack. In the first case, we prove that the problem

is NP-complete. We design an algorithm to solve it explicitly, and also one to approximate

the solution in polynomial time. In the second case, we design algorithms to solve and to

approximate the solution for two scenarios in switched LTI systems.

Finally, exploring a decomposition of digraphs from the area of control, we present a more

general bound for the index of convergence of Boolean matrices.

Keywords: Ranking Systems, Recommender Systems, Control Systems, Information Secu-

rity, Boolean Matrices.

iii

Acknowledgments

The work presented in this thesis would not have been possible without the support, influence

and encouragement of several people, namely, professors, colleagues, friends and family.

First, and foremost, I would like to express my sincere gratitude to the support of

the DP-PMI and Fundacao para a Ciencia e a Tecnologia (Portugal), through scholarship

SFRH/BD/52242/2013. Second, a special thanks to the support of Instituto de Telecomu-

nicacoes, Lisboa, through Research Grant - BIM/No154 - 16/11/2017 - UID/EEA/50008/2017.

Further, I acknowledge that this work was developed under the scope of R&D Unit 50008,

financed by the applicable financial framework (FCT/MEC through national funds and when

applicable co-funded by FEDER - PT2020 partnership agreement).

Next, I would like to express my deepest gratitude to my supervisor, Professor Car-

los Caleiro, for constant and generous support, opportunities, encouragement and guidance

through the inevitable bumpy road that makes part of any PhD endeavor. The fruitful discus-

sions, comments and criticisms from his part have truly changed my way of thinking, which

is reflected in this thesis exposition. For all of the mentioned I give him a big thank you.

Moreover, I would like to express my gratitude to my colleagues and co-authors Joao

Saude, Sergio Pequito, Ludovico Boratto, Jaime Ramos and Soummya Kar for all the profuse

discussions and hard working hours spent together.

Finally, I would like to thank to my PhD colleagues, and also friends, Andreia Mordido,

Iolanda Velho and Filipe Casal and to my family: Henrique, Isabel, Joana, Ricardo, Clara,

Laura, Susana, Josue and specially to Tiago, for their great support.

Thank you,

Obrigado!

v

Contents

1 Introduction 1

1.1 Thesis Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Main Contributions of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 8

I Ranking and Recommender Systems 11

2 Preliminaries and Notation 13

3 A Robust Reputation- and Cluster-based Ranking System 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Reputation-based ranking algorithms . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Bipartite graph algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2 Similarity Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.3 Multipartite graph algorithms . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Computational complexity analysis . . . . . . . . . . . . . . . . . . . . . 26

3.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.1 Evaluation metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.2 Spamming and Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5.1 Robustness against random spamming (noise) . . . . . . . . . . . . . . . 29

3.5.2 Robustness against attacks . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5.3 Sensivity to parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

vii

4 Reputation-based Ranking Systems and their Resistance to Bribery 37

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Bribing in ranking systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Properties of strategies and its profit in the bipartite ranking systems . 42

4.3.2 Optimal Strategies in the Bipartite Ranking Systems . . . . . . . . . . . 44

4.3.3 Properties of strategies and its profit in Multipartite Ranking Systems . 46

4.3.4 Optimal Strategies in Multipartite Ranking Systems . . . . . . . . . . . 47

4.3.5 Bipartite vs. Multipartite Ranking Systems . . . . . . . . . . . . . . . . 49

4.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


5 Recommendation via Matrix Completion Using Kolmogorov Complexity 53

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.1 Setup specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.2 Complexity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4.2 Evaluation metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


6 A Novel Similarity Measure for Group Recommender Systems with Opti-

mal Time Complexity 63

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Background and Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.3 The Kolmogorov-based similarity . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.4 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.5 The group recommender system . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.6 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.6.1 Evaluation metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.6.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71


viii

II Control of dynamical systems 75

7 Preliminaries and Notation 77

8 The Robust Minimal Controllability Problem 79

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.2 Problems Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3 Preliminaries and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.4 Robust Minimum Controllability Problem . . . . . . . . . . . . . . . . . . . . . 84

8.4.1 Numerical and Computational Remarks . . . . . . . . . . . . . . . . . . 92

8.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


9 The robust minimal controllability problem for switched linear continuous-

time systems 97

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

9.2 Problems Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9.3 The Robust Minimal Controllability Problem . . . . . . . . . . . . . . . . . . . 101


9.4.1 Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.4.2 Example II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108


10 On the index of convergence of Boolean matrices with commutative SD-

decomposition 111

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

10.2 Preliminaries & Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

10.2.1 Boolean Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

10.2.2 Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

10.2.3 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

10.3 Index of Boolean matrices with commutative SD-decom-position . . . . . . . . 114



11 Conclusions and Future Work 123

Bibliography 141

ix

List of Algorithms

1 Clustering reputation-based ranking algorithm. . . . . . . . . . . . . . . . . . . 23

2 Matrix completion algorithm: KolMaC . . . . . . . . . . . . . . . . . . . . . . . 58

3 Group Recommender System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Polynomial reduction of the structural optimization problem (8.3) to a set-

covering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Approximate Solution to the rMCP . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Polynomial reduction of the structural optimization problem (9.5), to a set-

covering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7 Find a minimal set of state variables of a problem (9.4) that need to be actuated105

8 Merging procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

xi

List of Figures

1.1 Organization of the dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Bipartite graph representing n users, m items, and the ratings given by user u

to item i, Rui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Multipartite graph representing n users, m items, and the ratings given by user

u to item i, Rui. The lines represent the connection, through ratings, from the

users to the items. The dashed lines represent links between users, through

their similarities, suv with u, v ∈ U . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Evolution of the τ for random spamming with the proportion of spammers. . . 30

3.2 Evolution of the τ for the love/hate attack with proportion of spammers. . . . 30

3.3 Evolution of the ranking of the targeted item, rtarget, for love/hate attack with

proportion of spammers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Evolution of the ranking of the targeted item, rtarget, for reputation attack with

proportion of spammers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Evolution of the τ for the reputation attack with proportion of spammers. . . . 31

3.6 Evolution of r with proportion of attackers, for reputation attack, in the largest

cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.7 Variation of rtarget with the affinitity parameter, α, for different proportions of

attackers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.8 Variation of τ with the affinitity parameter, α, for different proportions of

attackers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 Bipartite/multipartite graph representation of users and items with edges in-

terconnecting them weighted by the users’ ratings for items, not consider-

ing/considering the dashed links. . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Profit of bribing strategies of the most rated item’s sellers in (a) bipartite

ranking system (σ1 – σ4), and (b) multipartite ranking system (σ1 and σ2). . . 50

xiii

4.3 Profit of bribing strategy σ2 in the bipartite ranking system, fixed users’ repu-

tations versus reputations recomputed after each user being bribed. . . . . . . . 51

5.1 Graph representing n users, u1, . . . , un and m items, i1, . . . , in. The filled

edges between users and items represent the products each user rated weighted

by the rating. The top dashed edges (between users) represent the weights com-

puted in the matrix SU . The dashed bottom edges (between items) represent

the weights computed in the matrix SI . . . . . . . . . . . . . . . . . . . . . . . 56

6.1 RMSE evolution with the number of users’ groups of a 5-fold-cross-validation

method, using Algorithm 3 with SVD for its step 2, with Pearson similarity

(blue points) and KS (yellow points) for the ML-100K. . . . . . . . . . . . . . . 71


method, using Algorithm 3 with SVD for its step 2, with Pearson similarity

(blue points) and KS (yellow points) for the ML-1M. . . . . . . . . . . . . . . . 72


method, using Algorithm 3 with KNN for its step 2, with Pearson similarity

(blue points) and KS (yellow points) for the ML-100K. . . . . . . . . . . . . . . 73


method, using Algorithm 3 with KNN for its step 2, with Pearson similarity

(blue points) and KS (yellow points) for the ML-1M. . . . . . . . . . . . . . . . 74

10.1 Two digraphs and their respective SD-decompositions. . . . . . . . . . . . . . 116

10.2 A family of digraphs G(Ai1)∞i=1 in (a); digraph G(A2) in (b); and digraph

G(A3) in (c). The SCCs are represented by the red edges in (a), each cycle is

an SCC, and they are represented by different (not black) colors, one per SCC,

in (b) and (c). The DAG of each digraph is represented by the black edges. . . 119

10.3 Graph representation of a 5-bus power system. . . . . . . . . . . . . . . . . . . 120

11.1 Future work directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xiv

List of Tables

3.1 Details of the datasets A and B. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Details of datasets MovieLens 100k and 1M. . . . . . . . . . . . . . . . . . . . . 59

5.2 RMSE of a 5-fold-cross-validation in four synthetic random and full rank 20×30

matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3 RMSE for the datasets ML–100k and ML–1M. . . . . . . . . . . . . . . . . . . 61

6.1 Time and space complexity of the similarities. . . . . . . . . . . . . . . . . . . . 68

6.2 Details of datasets MovieLens 100k and 1M. . . . . . . . . . . . . . . . . . . . . 70

6.3 Average and standard deviation of the computation time of the similarities

between every pair of users in a 5-fold cross validation. . . . . . . . . . . . . . . 72

10.1 Number of size n CSDD Boolean matrices vs. size n Boolean matrices. . . . . . 116

xv

Chapter 1

Introduction

Information Security is a ubiquitous concern of this century. It finds applications in a wide

range of areas such as military, political, social and economic domains. In fact, Information

Security has been a long time and primary concern for diplomats and military comman-

ders. They soon understood not only the need for protecting the confidentiality of exchanged

messages, but also that the access to critical information should be private (restricted to

authorized people). Another important concern of the diplomats and military commanders

was the need for detecting when someone tampered with their messages content, to keep the

messages’ integrity, maintaining the consistency, accuracy, and trustworthiness. For similar

purposes, in 50 B.C., Julius Caesar is considered the inventor of the Caesar cipher, created

to prevent his secret messages from being read if they fall into the wrong hands, see [Singh,

2000].

Two other relevant aspects in Information Security are the availability of the informa-

tion, it must be available when needed, and also non-repudiation of information, that is the

information must be “signed”, so that author’s ownership cannot be denied.

In the First World War, both the cryptology, the science of designing codes to cipher

and decipher messages, and the cryptanalysis, the art of cracking codes without holding the

secret key, were very important. But, the cryptograms designed for the Great War were

easily breakable by the cryptanalysis of that time. It was in the Second World War that

the areas of cryptology and cryptanalysis, more than important, became essential. The use

of machines to cipher messages was the ultimate weapon. In fact, the concern of keeping

information’s confidentiality translated in the Enigma Machine, employed by the Germans

to encrypt warfare data. The need to keep information’s integrity led to the creation of

levels of access to information and the design of complex and safe storages to garner it. Alan

Turing was an English computer scientist, mathematician, logician, cryptanalyst, philosopher,

and theoretical biologist, that is considered as the father of theoretical computer science

1

and artificial intelligence [Hodges, 2012]. He is the one who led the team that successfully

decrypted the Enigma machine.

The end of the last century and the beginning of this century have faced a pick of advances

in the field of telecommunications. The development of small and affordable devices with

computational power brought data processing power from military and powerful businesses

to single home users. This computational power leveraged the development of the Internet,

which, in its turn, enlarged the power of these small devices.

In 2017, the estimated world population number was 7, 519, 028, 970 and the estimated

number of Internet users was 3, 885, 567, 619 (51.7% of the population), the last value rep-

resents a growth of 976.4% from 2000 to 20171. Everyone with a computer and a network

connection can reach, using worldwide groups of connected networks, any other point on

the Internet, without borders or timezones restrictions. Consequently, we face a continually

growing interest in electronic communications and commerce. However, all the convenience

we acquire with the World Wide Web (WWW) to get information, services, and goods also

comes with its own risks. This comfortable mean of communicating and buying services and

goods can also be explored by malicious users to steal or tamper with valuable information.

Hence, it is of utmost importance to keep valuable information on the Internet confidential,

only known to who it is intended to, to keep its integrity, without being corrupted, and

available, reachable in normal and abnormal situations.

With the fast growth of Internet users and electronic devices, the need to extract and ex-

plore meaning from the data deluge they produce has a paramount role to electronic commerce

(e-commerce), military, advertisement and marketing strategies, and politics, to name a few.

In tandem, there is a growth of interest from malicious entities in obtaining and exploiting in-

formation from this amount of data. These entities may tamper with the systems/companies,

do espionage or sabotage of users/companies, or even, in an extreme scenario, they may trigger

a cyberwar.

In e-commerce, a traditional way of collecting information from users is to allow them to

rate and comment on products/services. In a subsequent step, the sellers/service providers

process this information to generate rankings for the products/services or to recommend more

products/services to users that may be of their interest. These simple methods of collecting

and processing users’ information help us improve our experience on the web [Forman et al.,

2008,Sparks and Browning, 2011], and sellers rely on this information to evaluate the viability

of the products, to predict sales and to target advertise campaigns [Chevalier and Mayzlin,

2006, Dellarocas et al., 2007, De Maeyer, 2012]. However, they are susceptible to attacks

from malicious users or even sellers/service providers [Hu et al., 2012]. A set of users may

1http://www.internetworldstats.com/stats.htm

2

http://www.internetworldstats.com/stats.htm

target a set of products/services and control, push or nuke, their rankings or tamper with

the recommendation list produced for determined users. Further, a seller may bribe a set of

users to control their ratings and increase the ranking of his/her own products or degrade

the ranking of competitors products. Hence, it is desirable that the ranking systems and

recommender systems, while essential tools, are robust to such kind of behaviors, trying to

keep information as secure as possible [Cialdini and Garde, 1987, Li et al., 2012, Apt and

Markakis, 2014,Simon and Apt, 2015,Grandi and Turrini, 2016].

In the cyberwar extreme scenario, the federal government of the United States of America,

in [Shiels, 2009], admitted that the electrical power grid is a cyberwarfare target [Singer and

Friedman, 2014]. In 2010, security experts from Kaspersky Lab found a malicious software

program named Stuxnet, in which the controllers’ input response to a tampered measured

output drives the system away from its usual operating conditions, see [Langner, 2011]. The

New York times said it was “the first attack on critical industrial infrastructure that sits

at the foundation of modern economies”. This software was not only infiltrated in factory

computers but also spread to electrical power grids around the world. Attacking the electrical

power grid of the target countries is considered to be the first tactical move in a technological

war, and this was successfully applied in [Case, 2016]. Hence, ensuring the security of the

electrical power grid is one of the main tasks of governments, see [Flick and Morehouse,

2010], for instance. Also, the United States Department of Homeland Security is working

together with industries to identify vulnerabilities and help them to enhance the security of

control system networks. Notwithstanding, several applications, not only power systems but

also control processes, multi-agents networks, control of large flexible structures and systems

biology [Egerstedt, 2011, Siljak, 2007, Skogestad, 2004] rely on the concept of controllability

to ensure their proper functioning.

The research of this thesis aims to contribute to information security aspects in the two

described scenarios. In the ranking and recommender systems scenario, the main goals are to

devise systems that are more robust to attacks, to provide a better-personalized experience,

and to design systems that have good computational complexity. In the cyberwar scenario, we

also focus on how the control system networks cope with attacks (robustness to attacks). We

design systems that can be controlled up to a specified number of controllers failures (possibly

due to attacks), while also focusing on the computational complexity of the design process.

1.1 Thesis Statement

The research on this dissertation aims at studying information security aspects in ranking,

recommender, and control systems and develop systems in these classes that are more robust

3

to attacks. Our claim is that the areas of information theory and control systems theory

may provide a valuable tool to approach the fields of ranking and recommender systems,

to improve the state-of-the-art in these two fields in terms of information security, to design

systems more robust to attacks and noise, and also to design systems that are more efficient in

terms of computational complexity. Hence, we seek to use ideas from control systems theory

and apply them in the context of ranking and recommender systems. In particular, we plan

to study optimal strategies to attack ranking systems and evaluate their effectiveness. That

is, what are the costs of an attack versus its gain.

1.2 Outline

The main topics we explore in the dissertation can be presented as a tree with the connections

represented as edges. In Figure 1.1, we map the main chapters of this thesis to a schema. We

omit Chapter 10 which is, in fact, related with the topic of control systems.

§

Information

Theory and Security

E-commerce ControlSystems

RecommenderSystems

RankingSystems

LTIsystems

SwitchedLTI systems

Ch. 5,6 Ch. 3,4Ch. 4 Ch. 8 Ch. 9

Part I Part II

Figure 1.1: Organization of the dissertation.

More specifically, the dissertation is organized in two parts and 11 chapters, as follows:

Introduction: The introduction chapter renders the context of the work presented in this

dissertation and its motivation. It summarizes the main contributions of this manuscript, and

it presents the structure of the document.

4

Part I – Ranking and Recommender Systems: In Part I, we use information theory,

in particular ideas from Kolmogorov complexity theory, from clustering, from graph theory,

and from collaborative filtering, to create a reputation-based ranking system for multipartite

networks that is more robust to noise and attacks than the state-of-the-art ranking systems.

This system is also more tailored to each user, because it presents (possible) different rankings

of the same item for different groups of users, see Chapter 3. We compare the robustness to

bribery of bipartite ranking systems versus multipartite ranking systems, showing that the

last case is more robust to bribery, in Chapter 4, and we explore the results with the ranking

system proposed in Chapter 3. We explore the similarity measures, based on Kolmogorov

complexity, that we introduced in Chapter 3, to develop a recommender system, in Chapter 5,

and a group recommender system, in Chapter 6.

Preliminaries and Notation: We introduce the preliminary results and notation used in

Part I of the manuscript in Chapter 2.

A Robust Reputation- and Cluster-based Ranking System: In Chapter 3, we pro-

pose a new reputation-based ranking system, utilizing multipartite rating subnetworks, which

clusters users by their similarities using three similarity measures, two of them based on Kol-

mogorov complexity. Our system is novel in that it reflects the diversity of preferences by

(possibly) assigning distinct rankings to the same item, for different groups of users. We

prove the convergence and efficiency of the system. By testing it, we see that it copes better

with spamming/spurious users, being more robust to attacks than state-of-the-art approaches.

This work was developed with Joao Saude, Ludovico Boratto, Carlos Caleiro and Soummya

Kar and was submitted for publication, see [Saude et al., 2018].

Reputation-based Ranking Systems and their Resistance to Bribery: In Chapter 4,

we study bribery resistance properties in the two classes of reputation-based ranking systems

presented in Chapter 3, where the rankings are computed by weighting the rates given by

users with their reputations. In the first class, the rankings are the result of the aggregation

of all the ratings, and we provide all users with the same ranking for each item. In the second

class, there is a first step that clusters users by their rating pattern similarities, and then

the rankings are computed cluster-wise. We study the setting where the seller of each item

can bribe users to rate the item, if they did not rate it before, or to increase their previous

rating on the item. We model bribing strategies under these ranking scenarios and explore

under which conditions it is profitable to bribe a user, presenting, in several cases, the optimal

bribing strategies. By computing dedicated rankings to each cluster, we show that bribing,

in general, is not as profitable as in the simpler scenario, without clustering. Finally, we

5

illustrate our results with experiments using real data. This work was developed with Joao

Saude, Carlos Caleiro and Soummya Kar and published in [Saude et al., 2017].

Recommendation via Matrix Completion Using Kolmogorov Complexity: In Chap-

ter 5, we explore the Kolmogorov-based similarity measures, introduced in Chapter 3, in the

context of recommender systems. Usually, recommender systems perform a completion of the

rating matrix via collaborative filtering algorithms. However, the choice of the algorithm to

employ is not trivial and usually depends on a set of factors and parameters. Indeed, when

choosing a neighborhood-based algorithm, we need to make assumptions on the ratio between

the number of users and items in the system, in order to choose between a user-based and

an item-based algorithm. Plus, the number of neighbors strongly impacts the effectiveness of

this class of algorithms. Memory-based algorithms, instead, assume that the matrix is either

low rank or that there are a small number of latent variables that encode the full problem.

However, it is hard to make such strong assumptions, especially for a recommender system

that will grow over time. To overcome these issues, in this chapter we propose a novel matrix

completion algorithm, without assumptions on the matrix rank. Also, it is model-free, i.e.,

the entries are not assumed to be a function of some latent variables. Instead, we use a

technique akin to information theory. Our method performs hybrid neighborhood-based col-

laborative filtering, using Kolmogorov complexity. Each component of the hybrid approach

is also parameter-free. It decouples the matrix completion into a vector completion problem

for each user. The recommendation for one user is, thus, independent of the recommenda-

tion for other users. For this reason, the algorithm is scalable, because the computations

are highly parallelizable. A large evaluation of our approach on 9 datasets shows that our

results outperform or compete with the state-of-the-art approaches. This work was developed

with Ludovico Boratto, Joao Saude and Carlos Caleiro and was submitted for publication,

see [Ramos et al., 2018c].

A Novel Similarity Measure for Group Recommender Systems with Optimal Time

Complexity: In Chapter 6, we again explore the novel Kolmogorov-based similarity mea-

sures introduced in Chapter 3 in the context of group recommender systems. Once we sub-

scribe to an e-commerce portal, or to a social media website, we interact with multiple brands

and with content from numerous providers. However, a unique user profile is created, contain-

ing all our preferences. Suppose that a company wants to understand who are its customers.

It wants to treat costumers as a target, and understand what campaigns the company should

run on them. On the one hand, an approach that clusters the users and performs group rec-

ommendations would be useful, while on the other hand, a generic user profile would not be

helpful, since the preferences in it are not specific for a brand. Hence, we have to determine

6

multiple user clusterings (one for each brand). This task makes the problem of producing

group recommendation challenging, since little and very sparse information about the users

is available, and for each pair of users we have to detect as many similarities as the brands

existing in the system. To tackle this problem, in this chapter, we introduce, in this context,

a novel and optimal measure to compute the similarity between users, based on Kolmogorov

complexity. Further, we test it in the group recommendation scenario. The results show that

our similarity measure can provide similar accuracy when compared to classical measures, but

with significant performance gains, having a strictly lower time complexity than the state-

of-the-art similarity measure. This work was developed with Ludovico Boratto and Carlos

Caleiro and was submitted for publication, see [Ramos et al., 2018a].

Part II – Control of dynamical systems: In Part II, we solve two controllability prob-

lems, that consist in selecting the minimal set of state variables that need to be actuated

such that the underlying system is controllable under the scenario where a set of actuators

may fail along the time (robust to attacks/actuators failures). This problem is solved for

linear time-invariant systems, in Chapter 8, and for switched linear continuous-time systems,

in Chapter 9. In Chapter 10, we make use of a digraph decomposition commonly utilized in

the areas of control systems (in particular, in structural control systems) to present a new

bound on the index of convergence of Boolean matrices.

Preliminaries and Notation: In Chapter 7, we introduce the preliminary results and

notation needed for the chapters of the Part II of the manuscript.

The Robust Minimal Controllability Problem: In Section 8, we address the robust

minimal controllability problem, where the goal is, given a linear time-invariant system, to

determine a minimal subset of state variables to be actuated to ensure controllability under

additional constraints. We study the problem of characterizing the sparsest input matrices

that assure controllability when the autonomous dynamics’ matrix is simple when a specified

number of inputs fail. We show that this problem is NP-hard, and under the assumption that

the dynamics’ matrix is simple, we show that it is possible to reduce the problem to a set

multi-covering problem. Additionally, under this assumption, we prove that this problem is

NP-complete, and polynomial algorithms to approximate the solutions of a set multi-covering

problem can be leveraged to obtain close-to-optimal solutions. This work was developed with

Sergio Pequito, Soummya Kar, Antonio Pedro Aguiar and Jaime Ramos and is published

in [Pequito et al., 2016b].

7

The robust minimal controllability problem for switched linear continuous-time

systems: In Chapter 9, we address the robust minimal controllability problem for switched

linear continuous-time systems, an extension of Chapter 8. The problem is to determine the

minimal subset of state variables to actuate such that the switching linear system is control-

lable, under the scenario where a set of actuators may fail along the time. Two variations

of this problem are considered, depending on whether we want to design an input matrix

for each mode (i.e., a different set of actuators that may fail in each mode), or if we want

to design an input matrix common across all the modes, when a set of actuators may fail.

In both cases, we want to ensure that, given an initial condition, we can drive the system

towards any desired state. For both problems, we characterize the sparsest input matrices

which ensure that the system is controllable, whenever the autonomous dynamics’ matrix of

each mode is simple, and for which a left-eigenbasis is available. We reduce these problems to

set multi-covering problems, showing that using a sufficient condition for controllability, the

first problem, i.e. to design an input matrix for each mode, is NP-complete. These allow us

to deploy known, close-to-optimal, (polynomial for the first case) algorithms approximating

the solutions of the problems we study. This work was developed with Sergio Pequito and

Carlos Caleiro and is published in [Ramos et al., 2018b].

On the index of convergence of Boolean matrices with commutative SD-decompo-

sition: In Chapter 10, we present a new bound for the index of convergence of Boolean matri-

ces that correspond to digraphs with commutative SD-decomposition, and we revisit the previ-

ously known bounds for the index of convergence of Boolean matrices, by Wielandt, Dulmage-

Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman. Boolean matrices emerge in con-

trol systems theory, more specifically in structural control systems theory and they are used

to ensure structural controllability of a system, a necessary condition to have a controllable

system. We use ideas from structural control to achieve this new bound, and we illustrate it

with examples. This work was developed with Carlos Caleiro and is submitted for publication,

see [Ramos and Caleiro, 2018].

1.3 Main Contributions of the Dissertation

Here, we summarize the main contributions of this thesis:

• we developed a multipartite reputation-based ranking system more robust to noise and

to attacks than the state-of-the-art ranking systems, that, by clustering users with

similarity measures based on their preferences, presents to users a personalized ranking

for each item, and we proved its convergence and efficiency;

8

• we studied the effect of bribing in ranking systems, we computed the optimal bribing

strategies, and we compared the resistance to bribery of bipartite ranking systems with

multipartite ranking systems, being the former less resistant than the latter;

• we explored the similarity measures proposed to cluster users (from the first bullet point)

to design a recommender system that, when evaluated with real-world data produces

similar results to the benchmark results;

• we explored the use of one of the proposed similarity measures in the context of group

recommender systems, where we use it to detect groups of users, and we obtained similar

results as the benchmark similarity measure but with a significant gain in terms of time

complexity and also in space complexity;

• we characterized the exact solutions to the robust minimal controllability problem for

continuous linear-time invariant systems, we showed that it is an NP-complete problem,

and we provided efficient approximated solutions to the problem;

• we extended the previous bullet point to characterize the solutions of two versions of the

robust minimal controllability problem for switched and continuous linear-time invariant

systems, also providing approximated algorithms, which only in one of the version may

have polynomial time complexity;

• we used ideas from control and structural control theory to present a new bound for the

index of convergence (transient) of a Boolean matrix.

9

Part I

Ranking and Recommender

Systems

11

Chapter 2

Preliminaries and Notation

In this part of the dissertation, we resort to information theory to contribute in the areas of

ranking systems and recommender systems. In particular, we use concepts from Kolmogorov

complexity theory, see [Ming and Vitanyi, 1990], from graph theory, see [Cormen et al.,

2001, Bollobas, 2013], from data mining (clustering), see [Jain and Dubes, 1988], and from

collaborative filtering1 (CF), see [Masthoff, 2015]. The outline of this part is as follows:

In Chapter 3, we present a reputation-based ranking system for multipartite networks. In

Chapter 4, we study the bribery effect in reputation-based bipartite and multipartite ranking

systems, including the one introduced in Chapter 3. In Chapter 5, we explore the similarity

measures that we introduce in Chapter 3 to design a recommender system. In the last chapter

of this part, Chapter 6, we use one of these similarity measures to design a group recommender

system.

We now introduce some notation and definitions that we use in the subsequent chapters.

Let U be a set of users, I a set of items, R⊥, R> ∈ Z+ the minimum and maximum ratings,

respectively. We denote by R = [R⊥, R>] ∩ Z+ the set of strictly positive integers, by ∆R =

R>−R⊥ the allowed ratings, and by R ⊆ U ×I×R the set of ratings given by users to items.

Note that we can consider continuous ratings of the form [R⊥, R>] ∩ R, with R⊥, R

> ∈ R,

and the results also apply (we just need, in the case of the ranking systems, to normalize

the ratings to be in ]0, 1]). For instance, if user u rates item i with rating Rui, then we

write it as (u, i, Rui) ∈ U × I × R, or simply Rui ∈ R. We denote the set of items rated by

user u as Iu = i | ∃Rui ∈ R s.t. (u, i, Rui) ∈ R, and the set of users who rated item i as

Ui = u | ∃Rui ∈ R s.t. (u, i, Rui) ∈ R.

A reputation-based ranking system assigns a reputation, cu ∈ R+, to each user, u ∈ U ,

and then utilizes it to weigh their ratings on products to compute the products’ rankings.

1Collaborative filtering is a technique to make automatic predictions (filtering) about the interests of a user,

by gathering preferences information from many users (collaborating).

13

Figure 2.1: Bipartite graph representing n users, m items, and the ratings given by user u to

item i, Rui.

From now on, for ranking systems, we consider normalized ratings (dividing by R>) and

reputations, Rui, cu ∈]0, 1] (i.e., the rankings and reputations take values in ]0, 1]).

To model a ranking system, we first consider a weighted bipartite graph, B = (U, I,R),

like the one presented in Figure 2.1, with two sets of vertices, U and I, representing the

users and the items. If a user rated an item, then there is an edge with the weight of the

rating connecting the two in B. From B, we build a graph G = (U,E), with the set of users

as vertices and where two users are connected if they are similar. Merging B and G gives

rise to a multipartite graph, M (Figure 2.2). A multipartite graph is a graph such that two

vertices that are connected by an edge have different colors [Bollobas, 2013]. Here, we need

one color for the items and at least two more whenever there is a cluster with more than one

user. This multipartite graph can either model users’ network generated by users themselves,

for instance, social networks, see [Symeonidis et al., 2011], or, as we propose, automatically

generated by the ranking system, based on ratings of items given by users. Note that a

partition on subnetworks may also be done on the items’ side, although we do not explore

this possibility in the context of ranking systems.

If |U | = n and |I| = m, n,m ∈ Z+, then we denote the n ×m matrix of ratings also by

R, because the matrix R is isomorphic to the set of ratings R. Again, Rui denotes the rating

that user u gave to item i. The entries take values on the allowed ratings together with a

special number denoting the absence of rating, ⊥ (in the following chapters, ⊥ is either 0

or 99, depending on the dataset used). Therefore, the set of given ratings corresponds to

the entries of the matrix R different from the special number. We denote by |R| the same

amount in both the set and matrix versions of the ratings, which is the number of ratings.

In other words, |R| is the number of elements of the set R or number of elements different

from ⊥ in the matrix R. We use the set version of R in the context of ranking systems and

the matrix version in the context of recommender systems. Note that, in this case, we have

14

Figure 2.2: Multipartite graph representing n users, m items, and the ratings given by user

u to item i, Rui. The lines represent the connection, through ratings, from the users to the

items. The dashed lines represent links between users, through their similarities, suv with

u, v ∈ U .

that the set Ui = u : Rui 6= ⊥ and u ∈ U. Similarly, the set Iu can now be written as

Iu = i : Rui 6= ⊥ and i ∈ I. We adopt standard notation to denote matrices and vectors.

For a matrix R, we denote the uth row of R by Ru, the ith column of R by Rᵀi , and the

ith column of the uth row by Rui. Further, we denote by Ru the average the ratings that

user u gave to items, i.e., Ru =∑

i∈Iu Rui/|Iu|. Given a set of objects X , a similarity is a

function s : X × X → [0, 1] such that whenever x ∈ X , s(x, x) = 1. For a square matrix

representing similarities, we use the letter S indexed by U or I, if the similarity matrix

represents similarities between users or items, respectively. Further, given two vectors with

dimension n, x and y, we denote by x y the vector whose entries are the product of the

entries of x and y, i.e., x y = (x1y1, . . . , xnyn). We use the semi-norm ‖ · ‖0. Given a vector

x, ‖x‖0 is the number of non zero entries of x.

Given a vector X ∈ Rn, we denote by X the average of the vector, i.e.,

X =

∑ni=1Xi

n. (2.1)

Further, we denote by σX the standard deviation of the vector X, i.e.,

σX =

√∑ni=1(Xi − X)2

n− 1. (2.2)

Let Σ = w1, . . . , wk be a finite alphabet (a set of characters). A word is an element in

Σ∗, i.e., a sequence of characters, such that the empty word ε ∈ Σ∗, and if x, y ∈ Σ∗ are two

words, then its concatenation xy is also a word, i.e., xy ∈ Σ∗. The length of a word x ∈ Σ∗

is inductively defined as |x| = 0 if x = ε and |x| = 1 + |z| if x = wz, with w ∈ Σ and z ∈ Σ∗.

15

Chapter 3

A Robust Reputation- and

Cluster-based Ranking System

In this chapter, we propose a reputation-based ranking system for multipartite networks.

Our system clusters users by their similarities in terms of tastes. Inspired by information

theory, We propose three similarity measures for this end, two of them based on Kolmogorov

complexity. Not only the proposed system is more user personalized (we may present different

rankings for the same item to distinct clusters of users), but it is also more robust to attacks

and spam/noise than the state-of-the-art ranking systems. Therefore, we take a step forward

in the information security of ranking systems. This work was submitted for publication,

see [Saude et al., 2018].

3.1 Introduction

In our daily life, electronic commerce, streaming media, and collaborative economy are ubiq-

uitous. Moreover, people’s opinions can be as effective as an advertisement. These facts

promoted the development of crowdsourced ratings/reviews. Consumers started to use, and

rely, on this information to decide whether or not to buy a product/service, have a meal

at a restaurant, or attend an event [Sparks and Browning, 2011, Forman et al., 2008]. The

sellers, aware of how the ratings of products/services impact sales [Chevalier and Mayzlin,

2006], started to rely on the ratings and reviews of their products to assess their commercial

viability as well as to predict sales [Dellarocas et al., 2007]. Further, they began to use this

information to improve their products and to target advertisement campaigns.

A domain in which these ratings and reviews can be employed efficiently is the systems

that rank the items for the users (e.g., Netflix and IMDB provide to the logged in users a

ranking of the available items). Given the relevance that ratings and reviews have for both

17

the users and the companies, it is of primary importance to detect and, automatically, correct

rating manipulations through fake users’ ratings.

Previous work. A simple way to collect and process ratings is to compute their arithmetic

average (AA). The main drawback of the AA is the indistinguishability of users, as it treats,

in the same way, the most relevant raters and spam. Therefore AA is prone to manipulation

of ratings through malicious attacks or spamming. Further, AA might be misleading, because

it does not capture the possible multimodal behavior of ratings [Hu et al., 2006]. For instance,

in a bimodal ratings’ distribution on the opposite extremes, the average is in the middle where

the density of votes is low.

Using weighted average algorithms allows the attribution of different importance to users.

This was explored in previous works, [Yu et al., 2006a,De Kerchove and Van Dooren, 2010].

The authors in [Li et al., 2012] used a modification of the weighted average. In [Mizzaro,

2003], the author used an additional time-dependent quantity to weigh the ratings of users.

These methods are more robust to spamming and attacks than the AA.

The mentioned methods have a bipartite graph structure because there are two types

of nodes, users, and items, with weighted edges (ratings) linking the two; see Figure 2.1 in

Chapter 2.

In prior work [Symeonidis et al., 2011], its authors extended the bipartite graph ap-

proaches. The authors considered implicit social networks, known as online Social Rating

Networks (SRN) (that emerge from different users commenting similarly on a given set of

products), and explicit social networks (built by users, through friendship or working rela-

tionship). They used the SRN to recommend products by a weighted combination of users’

similarities although not clustering users.

Open issues. By using AA or weighted average to rank the items, we are not taking into

account possible relations between users or users’ preferences. Furthermore, these approaches

do not incorporate the (possible) multimodal behavior of ratings of items, subjugating all

users to the average, hindering the rise of a multitude of preferences. There are two natural

negative consequences to this. On one hand, a ranking system does not take the most out

of the efforts made by the users to rate the items. Indeed, the current solutions do not offer

any form of personalization to the users, while it would be desirable that, if they belonged

to a segment with specific preferences, these should be reflected in the ranking (i.e., the

users should be presented first with items they might be interested in). On the other hand,

spurious users or spamming/malicious attacks might significantly affect the quality of the

ranking. Consequently, the ranking system would not reflect users’ preferences. Hence, there

would be negative consequences for the platform, regarding trust of their users.

Our contributions. In this chapter, we propose a generic class of iterative reputation-

18

based ranking systems on multipartite graphs. A reputation-based ranking system is a ranking

system in which each user has a reputation, and the ranking is the weighted average of users

ratings by their reputation. We prove, for the first time in the literature, that the algorithms

in that class converge and are efficient, extending the results in previous work. To design the

system, we use similarities between users. These similarities allow us to clusters users based,

solely, on their ratings (see Figure 2.2 of Chapter 2, where two subnetworks of users are

depicted in dashed lines, i.e., u1, u2, u3 and uN−1, uN). To cluster users, we propose two

novel similarity measures, the linear similarity (LS) and the Kolmogorov similarity (KS), and

we test them against the normalized compression similarity (CS), derived from the distance

measure proposed in [Li et al., 2004]. For an extensive introduction to Kolmogorov complexity

and its applications we refer the reader to [Ming and Vitanyi, 1990].

After, we compute (possible) different rankings for the same item, on different subnet-

works/clusters. Therefore, our method enables us to present, custom-built, items’ rankings

to each cluster.

Our approach adapts better to the preferences of similar users and also improves robust-

ness against both spurious users and spamming/malicious attackers. Further, it embeds the

multimodal behavior of ratings’ distribution. These aspects contrast with the algorithms that

we overviewed since they neglect the smaller subgroups that do not identify with the major-

ity because they are averaged out. Our proposal overcomes this issue, since we perform the

ranking on the base of a user clustering, while the other approaches consider the whole set of

users.

Our proposed similarities not only perform better but also have smaller computational

complexity than using CS. When comparing LS with KS, the former responds better to noisy

spam, and the latter is more robust to targeted attacks to a set of items. Both carry the same

order of computational complexity, although in our implementation KS is slightly faster than

LS. Finally, by using LS, we obtain better robustness results than state-of-the-art approaches.

Chapter structure. In Section 2, we introduce a generic class of reputation-based rank-

ing iterative algorithms and prove their convergence and efficiency. In Section 3.3, we design

a new reputation-based ranking system, we prove its convergence and explain its implemen-

tation. The experimental setup is described in Section 3.4 and we discuss our results in

Section 3.5. In Section 3.6, we conclude the chapter.

3.2 Reputation-based ranking algorithms

Next, we introduce the two classes of reputation-based ranking systems that we will study in

this chapter, and that will also be the subject of study of Chapter 4, but in the context of

19

bribery.

3.2.1 Bipartite graph algorithms

Here, we generalize the iterative reputation-based ranking methods, discussed above, as:rk+1 = gR(ck)

ck+1 = hR(rk+1), (3.1)

where k denotes the iteration index and c0 the vector of initial reputation of users, with

c0u ∈]0, 1]. Here, r = (r1, . . . , r|I|), where ri denotes the ranking of item i, computed by

gR : [0, 1]|U | → [0, 1]|I|, with the set of ratings, R, as a parameter. The users’ reputations,

c = (c1, . . . , c|U |), where cu denotes the reputation of user u, are determined by hR : [0, 1]|I| →[0, 1]|U |.

In [Li et al., 2012], the authors prove that their reputation-based ranking system converges

with a certain convergence rate. In this section, we prove that a class of reputation-based

ranking systems (more abstract) converges and with what convergence rate. Hence, we present

more general results that subsume all the proofs of convergence and efficiency in [Li et al.,

2012]. This allows for designing a more extensive range of convergent and efficient reputation-

based ranking systems.

Consider a Banach space, X , with an induced distance d : X 2 → [0, 1]. Using the Lipschitz

condition [Kreyszig, 1989], we prove the following results.

Lemma 3.2.1. Consider the iterative scheme (3.1). Let gR and hR be ηg and ηh-Lipschitz

maps, respectively. Then gR hR is an η-Lipschitz map, with η = ηgηh. If η < 1, then (3.1)

is a contraction.

Proof. Since the domain of g contains the codomain of h and both are Lipschitz the compo-

sition, g h, is also Lipschitz. Let d be a distance, we prove the induction’s basis:

d(r2, r1) = d(gR(c1), gR(c0)

)= d

((gR hR)(r1), (gR hR)(r0)

)≤ ηd(r1, r0),

where η ∈ [0, 1[ is the Lipschitz constant for gR hR. The induction step then reads

d(rn, rn−1) = d(gR(cn−1), gR(cn−2)

)= d

((gR hR)(rn−1), (gR hR)(rn−2)

)≤ ηd

(rn−1, rn−2

)= ηd

(gR(cn−2), gR(cn−3)

)≤ ηn−1d(r1, r0),

and the last inequality holds by the induction hypothesis.

20

Because we are working in a Banach space, if the algorithm (3.1) converges, then it

converges to a unique value. Using the previous lemma, we prove the following result:

Theorem 3.2.2. The class of iterative reputation-based ranking algorithms (3.1) converges.

Proof. Let m,n ∈ N. For any ε > 0, there exists an order, N , from which ηN < (1 −η)ε/d(r1, r0). Using the triangle inequality we have

d(rn, rm) ≤n∑

k=m+1

d(rk, rk−1) ≤n∑

k=m+1

ηk−1d(r1, r0)

≤ ηmd(r1, r0)+∞∑k=0

ηk ≤ ηNd(r1, r0)

1− η< ε,

since 0 < η < 1, therefore the algorithm (3.1) converges.

Theorem 3.2.3. Let d : X → [0, 1] be a normalized distance. Then the algorithm (3.1) has

exponential rate of convergence.

Proof. The basis of the induction reads:

d(r∗, r1) = d(gR(c∗), gR(c0)

)= d

((gR hR)(r∗), (gR hR)(r0)

)≤ ηd

(r∗, r0

)≤ η.

Assume that the induction hypothesis holds, for k = n, then it follows that

d(r∗, rn+1) = d ((gR hR)(r∗), (gR hR)(rn))

≤ ηd (r∗, rn) ≤ ηn+1d(r∗, r0

)≤ ηn+1.

To attain, at most, an error of ε > 0, we need κ = logη ε iterations, with η the Lipschitz

constant of gR hR.

3.2.2 Similarity Measures

To group users according to their preferences, we need to quantify how similar they are. For

each pair of users that have, at least, one rated item in common, we compute a similarity,

based on the item-rating information. We specify three similarities: one linear and two non-

linear. In the following, let Iu,v = Iu ∩ Iv denote the set of items that both users u and v

rated. Further, for each user, u, we denote by u the string composed by the concatenation of

the pairs (item, rating) of his/her rated items.

21

Linear similarity. We define the linear similarity as: LS (u, v) = 0 if Iu,v = ∅, and

otherwise

LS (u, v) = `(|Iu,v|)

1− 1

|Iu,v|∑i∈Iu,v

|Rui −Rvi|∆R

,where the function ` : Z+ → [0, 1] penalizes on how confident we are in the users’ similarity.

LD is a linear function of the absolute value of the rating difference, linearly encoding the

similarity between users, based on ratings of common rated items. If two users rated an item

with the same rating, the rating difference is zero, hence the similarity is one, on the other

hand, if the absolute rating difference is ∆R then the similarity is zero.

Next, we propose two compression-similarities based on Kolmogorov complexity [Cover

and Thomas, 2012]. Given the description of a string, x, its Kolmogorov complexity, K(x),

is the length of the smallest computer program that outputs x. In other words, K(x) is

the length of the smallest compressor for x. Although the Kolmogorov complexity is non-

computable, there are efficient and computable approximations by compressors. Let C be a

compressor; we denote by C(x) the length of the output string resulting from the compression

of x using C.

Compression similarity. Based on the normalized compression distance [Li et al.,

2004], we define the compression similarity as 1 minus the distance, i.e., CS (u, v) = 0 if

Iu,v = ∅, and otherwise

CS (u, v) = 1− C(uv)−minC(u), C(v)maxC(u), C(v)

,

for the string uv, the concatenation of u and v. Intuitively, we are measuring the information

(rating pattern) that users u and v have in common and normalizing it, by subtracting the

minimum and dividing by the maximum. If u and v have the same rating pattern, then

C(uv) ≈ C(u) = C(v) and CS (u, v) ≈ 1, while if the rating patterns are completely different

(they have nothing in common) we have that C(uv) ≈ C(u)+C(v) and CS (u, v) ≈ 0. Trivially,

when distance is maximum, 1, the similarity is minimum, 0, and vice-versa.

The main drawback of CS is that we need to compute the compression of each possible pair

of users with common rated items. To overcome this, we propose to use a nonlinear function

of the absolute disparity of users descriptions’ compressions, with lower time complexity.

Kolmogorov similarity. We define the Kolmogorov similarity as: KS (u, v) = 0 if

Iu,v = ∅, and otherwise

KS (u, v) =1

1 + |C(u)− C(v)|.

When the size of the compression of user u and user v rating patterns is the same, KS(u, v) =

1, and KS(u, v) goes to 0 when the absolute value of the compression sizes difference goes to

infinity.

22

3.2.3 Multipartite graph algorithms

We group users in subnetworks using a similarity measure SM . For a specified affinity level

threshold, α, we set Su,v = 1 if SM (u, v) > α and 0 otherwise, where S is the (possible

sparse) adjacency matrix, that characterizes the undirected graph M ≡ M(S). A large α

means that users need to be more strongly related in order to be connected, translating to a

larger number of clusters (automatically computed). We compute the subnetworks ofM,Mj

for j ∈ J , which are the connected components of M. Subsequently, we apply a reputation-

based ranking algorithm to each subnetwork Mj , to compute the reputation of users and

the ranking of items. Let rep rank denote a reputation-based ranking algorithm (3.1). We

summarize the clustering reputation-based ranking algorithm in Algorithm 1. Each time a

new user enters the system or an existing user changes his/her rating behavior all have to

be recomputed, as in most of the approaches in the literature. However, the update may be

performed only from time to time, not assigning new users to clusters until either some period

of time has passed or there is a sizable amount of new information.

Algorithm 1 Clustering reputation-based ranking algorithm.

1: input: α, dataset

2: build S from dataset and apply threshold α

3: build G, computing its adjacency matrix M≡M(S)

4: find the connected components of M, Mjmj=1

5: output: weighted average of rep rank(Mj)mj=1

3.3 Implementation

Here, we show how to compute the ranking of an item and the reputation of a user, by

considering the algorithms defined in equations (3.2) and (3.3) below. We compute the ranking

of the item, ri, as a weighted average. That is, the rating of user u to item i is weighted by

the user reputation, cu, and therefore gR in (3.1) becomes:

rk+1i =

∑u∈Ui

Ruick+1u

/∑u∈Ui

ck+1u . (3.2)

It is worth highlighting that our formulation of ranking differs from the one presented in [Li

et al., 2012] that, instead of normalizing by the sum of the users’ reputations, divides by the

number of users that rated the item i, |Ui|. Our definition allows us to have a ranking that

is based on the reputation of the users (thus more robust), but it makes more challenging

to prove the convergence of the method, which we present in Section 3.3.1 (indeed, having a

23

sum, instead of a constant value, means that we cannot simply get the absolute value of the

constant |Ui| out of the norm).

For hR in (3.1), we tested three different functions, parametrized by fλ,s:

ck+1u = 1− fλ,s(Iu)eR,u(r), (3.3)

where

eR,u =

1|Iu|

∑i∈Iu

|Rui − rki |p

maxi∈Iu|Rui − rki |p

mini∈Iu|Rui − rki |p

.

The users’ reputation is chosen as a function of the average, maximum, or minimum disagree-

ment of individual user’s ratings, Rui, and the rankings of the rated items, ri. In order to

control the penalization a user incurs on, for not rating according to the ranking, we define a

decay function fλ,s. We consider four different decay functions:

i) f1λ,s(x) = λ,

ii) f2λ,s(x) = λ

(1− e−

x2

),

iii) f3λ,s(x) = λ

[1− (1− υ)(1 + es−x)−1

],

iv) f4λ,s(x) =

1 if x ≥ 10

1/2 otherwise,

where λ ∈ [0, 1[ determines the penalization a user occurs in for rating differently than the

ranking, υ ∈]0, 1[ is the lowest penalization an user can incur, and s ∈ N is a parameter based

on the number of rated items such that the penalization is decreased by a half. The role of

the decay function is to control the penalization a user u suffers if it does not rate the item,

Rui, close to its ranking ri. The first, constant, function f1λ,s above is proposed in [Li et al.,

2012], the second is an exponential decrease function, the third is a logistic function, while the

fourth is a threshold function. In the second and third cases the penalization increases and

decreases, respectively, with the number of rated products. In the remaining of the chapter we

fix for hR the average and for fλ,s the constant function, f1λ,s, denoting by bipartite weighted

average (BWA) the resulting iterative scheme in equations (3.2) and (3.3). The choice to fix

these two functions is because they are easy to compute and, considering the different fλ,s and

the datasets used to evaluate our proposal, there is not much difference between the functions

(more details are provided in Section 3.5). Hence, they represent a good trade-off between

efficiency and effectiveness.

24

3.3.1 Convergence

Here, we prove the convergence of the proposed method. In what follows, for a given vector

x ∈ Rn and p ∈ Z+, the p-norm of x is ‖x‖p = (∑n

j=1 |xj |p)1p , and the ∞-norm is ‖x‖∞ =

maxj∈1,...,n |xj |.

Lemma 3.3.1. For all λ ∈ [0, (1 + ∆R)−1[, the iterative method in (3.1) with functions gR

and hR defined as in (3.2) and (3.3) converges.

Proof. Between iterations, rk+1 and rk, we get

‖rk+1i − rki ‖∞ =

∥∥∥∥Ri · ck+1

‖ck+1‖1− Ri · ck

‖ck‖1

∥∥∥∥∞.

Here, Ri ∈ [0, 1]|U | denotes a vector that contains the rating Rui that each user u gave to item

i (the element corresponding to a user is 0 if s/he did not rate the item).

Without loss of generality, assume that ‖ck+1‖1 ≥ ‖ck‖1, then the above difference is equal

to ∥∥∥∥Ri · ck+1

‖ck+1‖1− Ri · ck

‖ck+1‖1+

Ri · ck

‖ck+1‖1− Ri · ck

‖ck‖1

∥∥∥∥∞

≤∥∥∥∥Ri · ck+1

‖ck+1‖1− Ri · ck

‖ck+1‖1+Ri · ck

‖ck‖1− Ri · ck

‖ck‖1

∥∥∥∥∞

≤ R>‖ck+1‖1

∣∣∣ck+1γ − ckγ

∣∣∣ ,where

∣∣ck+1γ − ckγ

∣∣ = maxu∈Ui∣∣ck+1u − cku

∣∣. The iteration step for the reputation, c, gives us

|ck+1u − cku| ≤

|fλ,s(Iu)||Iu|

∑i

∣∣∣∣∣∣Rui − rki ∣∣∣p − ∣∣∣Rui − rk−1i

∣∣∣p∣∣∣≤ λ|rkβ − rk−1

β |,

where∣∣∣rk+1β − rkβ

∣∣∣ = maxi∈Iu

∣∣∣rk+1i − rki

∣∣∣, and using the triangular inequality, the translation

invariance of norms and the fact that |fλ,s(Iu)| ≤ 1. Combining the previous inequalities we

get

|rk+1i − rki | ≤

λ

‖ck+1‖1|rkβ − rk−1

β |, (3.4)

which is a contraction for λ < (1 + ∆R)−1, since 1 − ∆Rλ ≤ ‖c‖1 ≤ 1. Therefore (3.1)

converges.

In this work, we consider ∆R = 1− 0.2. Therefore, if λ ≤ 59 then the algorithm converges.

However, we may ensure convergence for any λ ∈ [0, 1[ changing the denominator of (3.3) to

max‖ck+1‖1, 1.

25

3.3.2 Computational complexity analysis

The time complexity of Algorithm 1 is given by the sum of the complexities of each step. Let

G = (V,E) denote a graph, where V is a set of vertices and E a set of edges. Step 3 consists in

building G, where V = U , this is done computing its sparse adjacency matrix,M, where each

rating is used once. Henceforth, the time complexity is O(|C||R|), where |C| = O(1) for simi-

larities LS and KS, and where, for the CS, |C| is the worst case complexity of compressing the

concatenation of pairs of users. Step 4 can be performed using Tarjan’s Algorithm [Hopcroft

and Tarjan, 1973], with time complexity in the worst case of O(|V | + |E|). Step 5 has, in

the worst case, the same time complexity of [Li et al., 2012], i.e., O(κ|R|). In summary,

Algorithm 1 has worst case time complexity of O ((κ+ |C|)|R|+ |V |+ |E|). In theory |E|can be, in the worst case |U |2, leading to a time complexity of O

((κ+ |C|)|R|+ |U |2

). In

practice, since (often) the users are sparsely connected in G, |E| = O(|U |), resulting in a

time complexity of O ((κ+ |C|)|R|+ |U |). In all cases, the space complexity of Algorithm 1

is O(|R|).

3.4 Experimental setup

Next, we detail the metrics we use to evaluate the ranking systems we propose. Further, we

detail the type of attacks and spam that we consider and that we explore in two datasets, in

Section 3.5.

3.4.1 Evaluation metrics

To quantitatively assess the quality of the ranking systems, we compute the Kendall rank

correlation coefficient, a.k.a. Kendall’s tau1, τ [Kendall, 1938]. This statistic measures the

ordinal association between two quantities. Intuitively, the Kendall correlation between two

variables is higher when observations are identical and lower otherwise.

The effectiveness is given by the Kendall tau of the rankings’ vector, r, versus a ground

truth, r, that is τ(r, r). Usually the used ground truth is the AA, due to its simplicity and

its popularity among ranking systems, [De Kerchove and Van Dooren, 2010, Jurczyk and

Agichtein, 2007]. However, evaluating the discrepancy between the ranking vector, r, and the

AA might not be very informative, since it does not capture the possible multimodal behavior

of ratings, and therefore might not be very useful to evaluate the quality of a ranking system.

For this reason, we opt for the robustness metric. Notice that, in the multipartite case, the

effectiveness is helpful to check for homogeneity within the clusters. We generalize the Kendall

1Given two sets X and Y , let C and D denote the sets of concordant and discordant pairs of elements in

X × Y , respectively. The Kendall’s tau is defined as τ = (|C| − |D|)/(|C|+ |D|).

26

tau as

τ =1

|M|

N∑j=1

|Mj |τMj , M =

N⋃j=1

Mj , Mj

⋂Mq = ∅,

whereMj is a subgraph ofM. We denote the effectiveness of a cluster,Mj , by τ(rMj , rAA|Mj).

The robustness evaluates the ability of the system to cope with noise or spamming

attacks. A noisy user gives random ratings to a random set of products [Aggarwal, 2016].

A spamming attacker targets a set of items with the intent of increasing (Push Attack) or

decreasing (Nuke Attack) their rankings.

For the multipartite case, the robustness Kendall tau is τ = τ(r, rspam), where r is a vector

of ri’s given by

ri =1

|M|∑m

|Mm|ri,Mm , where M =⋃m

Mm (3.5)

is the union of subnetworks where users rated item i, and ri,Mm denotes the ranking of

item i for the subnetwork Mm (if any user in the subnetwork rated the item, otherwise it

is undefined). This measure is useful to assess the quality of the partition of the original

network, and it can be used to tune the affinity level, α, between users so that they are in the

same cluster. For items not ranked in a subnetwork, or for new users, we average the rankings

among subnetworks, r, using weights proportional to the size of the subnetworks. Because

the weighted average is not a sufficient statistic, this protects the system against attacks.

We do not present the analysis of the personalization perspective in this chapter. That

is the analysis of how much closer to the real user preferences our cluster-based ranking

system is, compared to the ranking systems that use just the AA or a weighted average of

the ratings. This option because it is trivial to notice that a ranking produced by considering

the preferences of highly similar users is more personalized than a global one. Hence, in this

work, we focus on the robustness perspective and leave the personalization aspect as future

work.

3.4.2 Spamming and Attacks

In the bipartite graph scenario, the information available to a new user is every products’

rankings. This information can be used by malicious users to tamper with the ranking of

an item in a malicious way (push or nuke it.) For instance, in a reputation-based system,

an attacker can give ratings matching the ranking of items to increase its reputation, before

attacking an item.

When allowing for subnetworks, either the user is already classified into a cluster and

he/she accesses the item’s ranking within that cluster, or he/she is a new user. In this case,

the displayed ranking, rj , of the item, j, is the weighted average of its ranking within each

27

subnetwork. Both of these scenarios mitigate the spamming effect. Since the information

made available is not a sufficient statistic, a user cannot fully recover all the information to

efficiently attack the underlying ranking system.

Next, we discuss the robustness of the algorithm to different kinds of spamming/attacks:

• Random spamming : A set of spammers gives random ratings, uniformly distributed on

R, to a random number of items, following a Poisson distribution, starting at 1 with

parameter λP = 5. The rated items are randomly sampled from the initial dataset

distribution of ratings’ number per item.

• Love/hate attack : A set of spammers targets one item to push/nuke and selects another

set of items to nuke/push. In our simulations, each attacker nukes the most voted item

and pushes another random set of nine filler items.

• Reputation attack : In this case, a set of spammers targets one item to push/nuke its

ranking. They randomly select another fixed number of items, from the initial dataset,

typically the most popular ones, and give them the closest ratings to their rankings.

In all experiments we set λ = 0.3, α = 0.8, and for the LS method the confidence level

function `(|Iu,v|) = θ−1 if |Iu,v| ≤ θ and 1 otherwise. The parameter θ sets the number of

common rated items of users u and v from which we are confident that they can be similar.

We choose θ = 3. To evaluate the effect of the attacks/spamming, we compute the robustness

Kendall tau, τ(r, rspam).

3.5 Experimental results

We run all experiments on MATLAB 2016, using macOS 10.11 (2.8 GHz Intel Core 2 Duo

and 4 GB RAM).2

Datasets. In this work, we use two real world datasets obtained from the Stanford Large

Network Dataset Collection, [Leskovec and Krevl, 2014]. We use, as the first dataset, the 5-

core version of “Amazon Instant Video” dataset (Dataset A) that consists of users that rated

at least 5 items, as in [McAuley et al., 2015]. It has 5, 130 users, 1, 685 items and 37, 126

ratings, with R⊥ = 1 and R> = 5, see Table 3.1. We use, as the second dataset, the 5-core

version of “Tools and Home Improvement” (Dataset B), also in [McAuley et al., 2015], see

Table 3.1. This dataset has 16, 638 users, 10, 217 items and 134, 476 ratings, also with R⊥ = 1

2The code will be made available as soon as the anonymity restrictions are no longer required.

28

“Amazon Instant Video” “Tools and Home Improvement”

Users 5, 130 16, 638

Items 1, 685 10, 217

Ratings 37, 126 134, 476

Table 3.1: Details of the datasets A and B.

and R> = 5. Both datasets consist in tuples of the form (user, item, rating, timestamp), and

for both we normalize the ratings by dividing them by R>.

The choice to employ the 5-core version of the datasets is intrinsically related to the

scenario we consider in the evaluation, i.e., a ranking system where users are clustered based

on their similarity. Therefore, having information about the user preferences is key to measure

effective similarities (indeed, if two users did not rate any common items, their similarity would

be 0).

Benchmarks. We compare our results with the reputation-based ranking system in [Li et al.,

2012]. The authors already compared their algorithm with the state-of-the-art algorithms.

Namely, the HITS [Kleinberg, 1999], the Mizz [Mizzaro, 2003], the YZLM [Yu et al., 2006a]

and the dKVD [De Kerchove and Van Dooren, 2010] algorithms, showing that their algorithm

outperforms all the others, in the standard metrics.

Next, we test the robustness of the ranking systems against spamming (noise) and attacks

using the two real datasets. First, we analyze the behavior of the ranking system in the

presence of noise for the two datasets, in Section 3.5.1. Next, we evaluate the robustness of

the algorithms against Love/Hate and Reputation attacks, in Section 3.5.2. Finally, we discuss

how the robustness of the proposed ranking systems responds to changes in the parameters

of the system, namely the parameter α, in Section 3.5.3. We also test this response for the

different decay functions fλ,s and different parameters λ, but since the gains are small, we

omit the tests.

3.5.1 Robustness against random spamming (noise)

We test the random spamming (noise) by simulating a proportion of spammers ranging from

0 to 0.75 of the total number of ratings. The results are reported in Figure 3.1. Using the

multipartite ranking systems, we notice an increase of robustness for the LS, whereas for the

KS and CS we obtain similar robustness to the bipartite methods, because these similarities

29

××

×

××

××

× ×

++

+

++

++

+ +

0 0.15 0.35 0.55 0.75

0.6

0.7

0.8

0.9

1.0

Proportion of spammers

τ

(a) Dataset “Amazon Instant Video”.

××

×

×

××

××

×

++

+

+

++

++

+

0 0.15 0.35 0.55 0.75

0.6

0.7

0.8

0.9

1.0


τ

(b) Dataset “Tools and Home Improvement”.

Figure 3.1: Evolution of the τ for random spamming with the proportion of spammers.

××

××

× × × × ×

+ +

++

+ + + + +

0 0.15 0.35 0.55 0.75

0.80

0.85

0.90

0.95

1.00


τ


× × × × × × × × ×

+ + + + + + + + +

0 0.15 0.35 0.55 0.75

0.80

0.85

0.90

0.95

1.00


τ


Figure 3.2: Evolution of the τ for the love/hate attack with proportion of spammers.

××

××

××

××

×

++

++

++

++

+

0 0.15 0.35 0.55 0.750.5

0.6

0.7

0.8

0.9


r target


××

×

××

××

××

++

+

++

++

++

0 0.15 0.35 0.55 0.750.5

0.6

0.7

0.8

0.9


r target


Figure 3.3: Evolution of the ranking of the targeted item, rtarget, for love/hate attack with

proportion of spammers.

30

××

×

××

××

××

++

+

++

++

++

0 0.15 0.35 0.55 0.75

0.6

0.7

0.8

0.9


r target


××

×

××

××

××

++

+

++

++

++

0 0.15 0.35 0.55 0.75

0.6

0.7

0.8

0.9


r target


Figure 3.4: Evolution of the ranking of the targeted item, rtarget, for reputation attack with

proportion of spammers.

××

× × × × × × ×

++

+ + + + + + +

0 0.15 0.35 0.55 0.75

0.80

0.85

0.90

0.95

1.00


τ


× × × × × × × × ×

+ + + + + + + + +

0 0.15 0.35 0.55 0.75

0.80

0.85

0.90

0.95

1.00


τ


Figure 3.5: Evolution of the τ for the reputation attack with proportion of spammers.

accommodate new users by rearranging the clusters, and this degrades the τ .

3.5.2 Robustness against attacks

Now, we simulate two different attacks to the most voted item, rtarget, ranging the proportion

of attackers from 0 to 0.75 of the total number of voters, in this case, of the target item.

Love/Hate attack. For the love/hate attack, we obtain the results in Figures 3.2 and 3.3.

We can see that, using the multipartite ranking systems, the attack is less effective on both

datasets. In the case of the variation of τ , in Figure 3.2, the results are significantly better

when we perform the clustering with the LS. In both datasets, the effect of the attack on the

ranking of the target item, rtarget, in Figure 3.3, is more dimmed in the multipartite scenario,

thus less effective. The best similarity measure to avoid the effect of the attack on the target

31

item’s ranking is the KS.

A remarkable property is that while the KS is effective to deter the attack on the ranking of

the product, it has the most nefarious effect on τ . This is a consequence of the reorganization

of the subnetworks, to minimize the effect of the attack on rtarget, and our generalization of

the Kendall tau does not account for this repercussion. This indicates that the attackers are

not grouped with normal users and thus do not affect the rankings of items in the cluster.

The ranking is not nuked in the multipartite cases as when using the ranking system in [Li

et al., 2012] and BWA.

Reputation attack. In both datasets, using subnetworks, the effect of the reputation attack

on the ranking of the targeted item is dimmed, see Figure 3.4, because the intelligent attacker

chooses the closest rating to the ranking of the filler items, it should not affect drastically

the rankings of the filler items, while the attackers increase their reputation. In fact, in the

multipartite ranking systems, the ranking of the nuked item drops less than in the bipartite

ranking systems, and the best case is when using LS. The robustness τ , see Figure 3.5, has

a similar behavior as in the love/hate attack, and the best robustness is achieved in the

multipartite scenario when using the LS.

The organization of subnetworks changes with the increasing number of spammers, this is

a collateral effect of the system, that helps to cope with the attack. Thus, this effect produces

a bigger change in τ , because it reduces drastically the effect of the targeted attack. Since

the ranking of the filler items does not change drastically (the attackers rate those items with

their weighted average ranking), this is not an important side effect. Moreover, in the larger

cluster, containing users who rated the targeted item, the ranking of the item is almost kept

unchanged, when using LS. For the KS, it has a small variation and has a large variation

for the CS. Both variations reflect the opposite effect on the ranking of the targeted item as

what is intended by the attacker, see Figure 3.6. The clustering produced by the KS and the

CS aggregate attackers with legit users (that gave smaller ratings to the target item) on a

separated cluster, leaving raters who gave high ratings on the biggest cluster. Recall that for

new users, the displayed rankings are a weighted average of the ratings by user’s reputations,

whereas in each cluster they are the weighted average within the users of the cluster.

An important observation is that the experiments, in both datasets, are coherent in the

sense that we obtain similar results for the different attacks and spam. This indicates that

when we scale the size of the dataset, we expect to get similar robustness to the attacks for

the evaluated metrics, τ and Rtarget. As we pointed out, the multipartite scenario allows us

to present rankings of items to users that allow a multimodal behavior, and this can also

32

××

××

××

××

×

+

+ +

+ +

+

+ +

+

0 0.15 0.35 0.55 0.750.5

0.6

0.7

0.8

0.9

1.0


r targetinlargestcluster


××

××

××

××

×

+

+

+

+

++

+

++

0 0.15 0.35 0.55 0.750.5

0.6

0.7

0.8

0.9

1.0


r targetinlargestcluster


Figure 3.6: Evolution of r with proportion of attackers, for reputation attack, in the largest

cluster.

be explored for the item recommendation scenario, where we expect to get recommendations

more tailored to the users.

3.5.3 Sensivity to parameters

Here, we discuss the response of our reputation-based ranking system to the variation of its

parameters, using Dataset B (the results for Dataset A are almost the same, so they have not

been reported due to space constraints).

In Figure 3.7, we look for the affinity level, α, such that the variation of rtarget with the

increase of the number of attackers is smaller, in the case of the love/hate attack. For the LS

case, Figure 3.7a we have best results for α ∈ [0.4, 0.6]. For the CS, Figure 3.7b we obtain

best results α ∈ [0.5, 0.6]. Finally, for KS, Figure 3.7c the best level of the affinity level is

α ∈ [0.6, 0.9]. To chose a satisfactory level of affinity we need to analyze the effect of α, not

only on the ranking of the attacked item, rtarget , but also on the robustness, τ . In this case,

we look for values of τ close to one.

When using LS for clustering and comparing both Figures 3.7a and 3.8a, we see that the

best affinity level lays in the interval α ∈ [0.4, 0.6]. Choosing some α in this interval allows the

systems to protect the ranking of an item, rtarget, maintaining the robustness of the system, τ ,

close to one. In both the CS and KS cases, the affinity level that protects better the ranking

of the attacked item produces worst robustness to attacks, not only within the clustering

method, but also when compared to LS. These results are in line with the previous discussion

in Section 3.5.2. We suspect that the effect of parameter α on the τ metric is due to the

fact that CS and KS produce more clusters (without a bigger one) and the users tend to be

33

Proportion of Attackers× 0.05 0.15 0.25

× × × ××

×

×

× × ×

0.1 0.3 0.5 0.7 0.90.70

0.75

0.80

0.85

0.90

0.95

1.00

α

τ

× × ×

× × × × ×

×

0.1 0.3 0.5 0.7 0.9

0.75

0.80

0.85

0.90

α

r target

(a) rtarget versus α, using LS.

× × × ×

××

× × ×

0.1 0.3 0.5 0.7 0.9

0.75

0.80

0.85

0.90

α

r target

(b) rtarget versus α, using CS.

× × × × ×× × × ×

0.1 0.3 0.5 0.7 0.9

0.75

0.80

0.85

0.90

α

r target

(c) rtarget versus α, using KS.

Figure 3.7: Variation of rtarget with the affinitity parameter, α, for different proportions of

attackers.

regrouped as the proportion of attackers change, and this effect is not captured by τ .

3.6 Concluding Remarks

In this chapter, we advanced state of the art in ranking systems, both theoretically and

algorithmically. We developed a new multipartite ranking system that allows the coexistence

of multiple preferences by enabling different rankings for the same item for different users. This

is achieved by automatically clustering similar users, based on their given ratings. For each

cluster, we used a bipartite reputation-based ranking system, for which we proved convergence

and efficiency in a more general setting than previous results. Our method favors the creation

of bubbles, i.e., segregates users into groups, which we show that makes the ranking system

more robust to attacks and spamming.

As future work, we will analyze the impact of our approach regarding personalization. We

will also investigate the effect of bribing users to influence the ranking of items, as in [Grandi

and Turrini, 2016]. Also, in order to reduce the rate of change in the clusters, we will

34

Proportion of Attackers× 0.05 0.15 0.25

× × × ××

×

×

× × ×

0.1 0.3 0.5 0.7 0.90.70

0.75

0.80

0.85

0.90

0.95

1.00

α

τ

× × × × × × × × ×

0.1 0.3 0.5 0.7 0.90.70

0.75

0.80

0.85

0.90

0.95

1.00

α

τ

(a) τ versus α, using LS.

× ××

×

×

×

× × ×

0.1 0.3 0.5 0.7 0.90.70

0.75

0.80

0.85

0.90

0.95

1.00

α

τ

(b) τ versus α, using CS.

×× ×

× ×

× × × ×

0.1 0.3 0.5 0.7 0.90.70

0.75

0.80

0.85

0.90

0.95

1.00

α

τ

(c) τ versus α, using KS.

Figure 3.8: Variation of τ with the affinitity parameter, α, for different proportions of attack-

ers.

explore the use of steadiness functions, based on a timestamp, so that established clusters do

not change so easily. Another possible extension of the proposed algorithm is to use it for

recommendation systems.

35

Chapter 4

Reputation-based Ranking Systems

and their Resistance to Bribery

This chapter studies the effect of bribing in two classes of reputation-based ranking systems,

the bipartite and the multipartite reputation-based ranking systems. For both scenarios,

we define bribing strategies and the profit of playing a bribing strategy. We compute the

optimal bribing strategies, and we test the bipartite and multipartite reputation-based ranking

systems, introduced in Chapter 3. As expected, the latter is more robust to bribery than the

former. Hence, we endorse the step forward we gave in Chapter 3, when designing a ranking

system that assures more information security. This work is published in [Saude et al., 2017].

4.1 Introduction

The evolution of contemporary society towards an information economy boosted the devel-

opment of e-commerce. The fast pace of information spreading around the world has been

reconfiguring our social interactions. The social networks and online fora instigated the ex-

change of opinions and the online word of mouth (WOM), which in turn gained the potential

to drive e-commerce sales, see [Davis and Khazanchi, 2008, Kietzmann and Canhoto, 2013]

and [Maslowska et al., 2017], for instances.

Nowadays, the importance of reviews and rankings became paramount for sellers, as the

visibility and the sales numbers are related with them, [Chevalier and Mayzlin, 2006,Dellarocas

et al., 2007] and [De Maeyer, 2012]. Further, studies pointed out that, in several cases, online

reviews may be more influential than traditional marketing, see [Bickart and Schindler, 2001].

This strong influence increased the attempts to manipulate them, [Hu et al., 2012], and it

fostered the need for designing ranking systems robust to spam and attacks, as in [Li et al.,

2012, Saude et al., 2017] and references therein. The companies invest money to convince

37

users to vouch for their products/services, either by giving samples of them so that users can

comment on them or by directly paying users to provide positive feedback on their products

and negative on competitor ones, see [Cialdini and Garde, 1987]. Aware of the importance

and influence of bribing to manipulate rankings, we model this phenomenon and characterize

it quantitatively, so that its impact can be better understood. We mitigate the impact of such

behaviors, by showing that reputation-based ranking systems using clusters are, in general,

more robust to bribery.

Previous work. The influence of individual decisions on global properties in network-

based rating systems was studied in several works. In [Apt and Markakis, 2014], the authors

investigate how to turn a product into a tendency among users by changes on a social network.

In [Simon and Apt, 2015], the authors explore how to design an impartial mechanism for peer

review to mitigate the effect of a reviewer interfering with the likelihood of its work being

accepted.

In Chapter 3, we proposed a reputation-based ranking system that clusters users by their

ranking pattern similarities, an idea that we also explore in the context of recommender

systems in the subsequent chapters of the dissertation. We showed that, by doing so, our

approach is more robust to both spamming users and users trying to attack the ranking

system to change the ranking of a set of items.

The authors of [Grandi and Turrini, 2016] analyzed the resistance of two ranking systems,

one that simply averages the ratings of users (AA), another that takes into account the

influence network of a given user, using the AA to compute the ranking for each network.

They showed that the AA ranking system is bribable, and, in particular, bribing users who did

not rate is profitable. When considering social networks of users, they show that the bribery

effect is diminished. Their work assumes a fixed set of users with only one item to rate. The

AA does not capture a possible multimodal ratings’ behavior, as noticed in [Hu et al., 2006].

This motivated us to study bribing in reputation-based ranking systems and to explore the

case where users are clustered in groups which, intuitively, must lessen the bribing effect.

Our contribution. Here, we study the resistance to bribery of a class of ranking systems

that assign reputations to users. We show that a ranking system computing the items rank by

the weighted average of users ratings with their reputations is bribable since users that rated

the item with a reputation above the users’ average reputation are bribable. By clustering

users by their rating pattern and assigning possibly different rankings for the same item for

each cluster, we increase the bribery resistance of the ranking system. This makes the ranking

system that we propose in Chapter 3 much more robust to bribing. Further, a user is bribable

if its reputation is larger than the average reputation of the users that rate the item. This

bound increases in the clustering scenario, since within each cluster the number of users that

38

rated the item is smaller than the non-clustering scenario. Our model also applies to evaluate

marketing strategies, where a company is willing to invest money to augment its sales, either

increasing the users base or boosting positive reviews.

4.2 Definitions

Here, we set up the notational conventions and definitions to keep the chapter self-contained.

Recall that reputation-based ranking systems assign weights to users to aggregate their rank-

ings on given items, see Section 2. Further, we discuss two classes of ranking systems, namely,

the bipartite reputation-based ranking systems and the multipartite reputation-based ranking

systems, that include the instances presented in Chapter 3.

Schematically, as we stated in Chapter 3, we represent these ranking systems as a bipartite

graph, in which a set of vertices corresponds to users and the other to items. The edges

connecting vertices are weighted by the ratings that users gave to items and only connect

users to items, recall Figures 2.1 and 2.2 from the Chapter 2 that we condensate in Figure 4.1.

The bipartite graph corresponds to Figure 4.1 when not considering the dashed lines.

Recall that a multiparite ranking takes two steps. In the first step, the users are clustered

by their rating pattern similarities. In the second step, the ranking of each item is iteratively

computed, as in the bipartite ranking system case, only using information from each cluster,

producing (possibly) different rankings for different clusters. Since the ranking of an item may

differ from cluster to cluster, it is worth recalling that every user on the same cluster accesses

the local ranking of a given item, in the case where the item was rated by, at least, one user

belonging to that cluster. If for a given item, within a given cluster, no user rated that item,

the available ranking for that item is the weighted average of that item’s rankings among

clusters where the item was rated. Each user can link to several items, by edges weighted by

its ratings, as before. Now, we allow for edges between users (encoding similarities between

them), with clusters of connected users, forming a multipartite graph, see Figure 4.1. In the

bipartite ranking, recalling (3.2) and writing it without the iteration index, the ranking of

the item i is computed by

ri =1

α

∑u∈Ui

cuRui, where α =∑u∈Ui

cu,

and Ui denotes the set of users that rated item i ∈ I, where I is the set of all users.

Next, we recall the definition of rankings for the multipartite ranking systems, introduced

in (3.5). In the multipartite scenario, let M1, . . . ,MN be a partition of the set of users U

39

Figure 4.1: Bipartite/multipartite graph representation of users and items with edges inter-

connecting them weighted by the users’ ratings for items, not considering/considering the

dashed links.

into N disjoint groups of users, that is,

U =N⋃n=1

Mn and, for m 6= n, Mm

⋂Mn = ∅.

We denote the set of items rated by users from cluster Mn by IMn , with IMn =⋃u∈Mn

Iu,

where Iu is the set of items rated by user u. The set of users in the clusterMn that rated item

i is denoted by UMni , where UMn

i = Ui ∩Mn. Now, the ranking is computed independently

for each cluster as

rMni =

1

α

∑u∈UMn

i

cuRui,

where α =∑

u∈UMni

cu. As described in Chapter 3, note that for users belonging to a cluster

Mn, the displayed ranking of item i can be one of the following two possibilities: (i) the

ranking of the item for that cluster rMni , whenever there are users in the cluster that rated

item i; (ii) otherwise, the ranking of item i is the weighted average of the rankings of i for

the clusters with users that rated item i, that is,

ri =∑n∈Xi

|UMni |rMn

i

/ ∑n∈Xi

|UMni |,

where Xi = m : i ∈ IMm and m = 1, . . . , N. In what follows, for a set of users U ′ ⊆ U ,

cU ′ =∑

u∈U ′ cu/|U ′|.

Suppose the seller of item i has an initial wealth proportional to the item’s ranking and

to the number of customers that rated the item. To boost the sales of i, the seller may invest

his resources (wealth) to promote the item’s popularity so that users like it more, and/or to

expand his consumer base by making people buy it, like it, but not necessarily love it. Here,

we model this setting assuming that the popularity is an increasing function of the ranking

40

of the item, ri, and supposing that the number of consumers that bought the item is an

increasing function of the number of users that rated the product, |Ui|.We define the reward function or wealth, in the bipartite and multipartite ranking systems,

for the seller of item i as

Ji = |Ui| ri and Ji =∑n∈Xi

JMni ,

respectively, where JMni = |UMn

i |rMni .

We define the strategy of the seller of item i as a vector σi ∈ Si, with size |U |, where the

u-th entry is the value of the invested wealth to convince user u to increase his rating by ρu,

and Si ⊆ [0, 1]|U | \ 0, where 0 is the null strategy that does not bribe any user. If user u

rated item i with Rui, then ρu ≤ 1 − Rui. If ρu = 0 this means the seller does not try to

persuade user u to rate or to change his rating on item i.

For the seller of item i, we denote by

Ξi = σi ∈ Si : σi(u) = ρu = 0 for all u /∈ Ui

the set of strategies that consists, exclusively, in bribing users that already rated the item i.

Analogously, we denote by

Ξi = Si \ Ξi = σi ∈ Si : σi(u) = ρu = 0 for all u ∈ Ui,

the set of strategies of bribing users that did not rate item i. We say that a bribing strategy

σi is an elementary strategy if for some user u ∈ U we have that σiu > 0 and, for all v ∈ Uwith v 6= u, σiv = 0. To easy notation, instead of denoting by σi(u) the strategy of seller of

item i to bribe user u, we write σiu. Further, the wealth spent by playing strategy σi is given

by

‖σi‖1 =∑u∈U

σiu.

After strategy σi, the wealth of seller i becomes

Jσi = |Uσi | rσi −∑u∈Uσi

ρu, with Jσi =∑n∈Xi

∣∣∣UMn

σi

∣∣∣ rMn

σi−∑u∈Uσi

ρu,

respectively, for the bipartite and the multipartite ranking systems, where rσi is the new value

of ri after σi.

The profit of playing the strategy σi is

πσi = Jσi − Ji and πσi = Jσi − Ji,

respectively, for the bipartite and multipartite ranking systems.

41

4.3 Bribing in ranking systems

Here we study the resistance to bribery of reputation-based ranking systems. To simplify

the analysis, we assume a fixed assignment of reputations to users. First, we describe the

set of decomposable bribing strategies. After, we find the conditions for the strategies to be

profitable. Lastly, we compare bipartite ranking systems with multipartite ranking systems

and show that, by using clusters, the multipartite system is more robust to bribery.

4.3.1 Properties of strategies and its profit in the bipartite ranking systems

First, we investigate what particular conditions allow us to decompose a strategy into elemen-

tary ones. We start by considering the case where item i ∈ I sellers bribe users that already

rated the item, proving that all strategies bribing several users at once are decomposable into

several elementary ones.

Proposition 1. Let u, v ∈ Ui be two users that rated the item i ∈ I. If two strategies, σiu and

σiv, consist in bribing users to change their ratings from Rui and Rvi to Rui+ρu and Rvi+ρv,

respectively, then we have that πσiu+σiv= πσiu + πσiv .

Proof. When the seller of item i plays the strategy σiu, the ranking of item i changes according

to rσiu = ri + α−1cuρu. Thus, an elementary strategy’s profit is:

πσiu = |Ui| rσiu − ρu − |Ui| ri =

(cucUi− 1

)ρu. (4.1)

The profit of the sum of strategies is given by:

πσu+σv = |Ui| rσiu+σiv− (ρu + ρv)− |Ui|ri = πσu + πσv ,

the profits’ sum of elementary strategies, πσiu and πσiv .

Now, we consider the case when a seller opts to bribe users that did not rate the item i.

Proposition 2. Consider a user that did not rate the item i, i.e., u /∈ Ui, and any other user

v ∈ U . The strategy that is to bribe both users, u and v, does not carry the same profit as

the sum of the profits of bribing each user, i.e., πσiu+σiv6= πσiu + πσiv , unless both elementary

strategies have zero profit.

Proof. If both users did not rate the item i, their strategies change the ranking of the product

in the same way

rσiu =

∑v∈Ui cvRvi + cuρu

α+ cu=αri + cuρuα+ cu

.

Thus, we have

πσiu = (α− |Ui| cu) (ri − ρu)/(α+ cu). (4.2)

42

Hence, the profit for the sum of strategies, σiv + σiu, is

πσiu+σiv= |Uσiu+σiv

|rσiu+σiv− (ρu + ρv)− |Ui| ri

=α+ cuα

πσiu +α+ cvα

πσiv +1

α(ρu − ρv)(cu − cv),

where α = α+ cu + cv. To have a positive profit of the sum of strategies that is equal to the

sum of the profits of each elementary strategy, we need the following conditions to hold

α+ cuα

=α+ cvα

= 1 and (ρu − ρv)(cu − cv) = 0,

this implies cu = cv = α− α > 1, which contradicts the fact that cu, cv > 0. However, in the

case that cu = cv = cUi the sum of the strategies’ profit (each being zero) is zero.

The case where one of the users to be bribed did not rate the item, v /∈ Ui, but the other

user did, u ∈ Ui, yields a profit

πσiu+σiv=∣∣Uσiv ∣∣ rσiu+σiv

− (ρu + ρv)− |Ui| ri

=|Ui|α+ cv

cv(ρv − ri) +|Ui|α+ cv

cuρu

+α

α+ cv(ri − ρv) +

1

α+ cvρu(cu − cv)−

α

α+ cvρu

=α

α+ cvπσiu + πσiv +

1

α+ cvρu(cu − cv),

(4.3)

which carries the same conclusion as above.

As we noted in the previous proof, special conditions on the users’ reputation make the

profit zero, hence decomposable into elementary strategies. We discuss this in the next result.

Proposition 3. Pick an item i ∈ I. Consider the following case: The seller of i bribes

users that already rated the item i, u, v ∈ Ui, and all the users have the same reputation

cu = cv = cUi. In this case, the strategy is not profitable, and the sum of the elementary

strategies is zero, πσiu+σiv= πσiu + πσiv = 0.

Proof. For the strategies composition, the profit is given by (4.3), with cv = cw = cUi , thus

πσiv+σiw= 0. σiv has profit given by (4.1), with cv = cUi , thus πσiv = 0. σiw has profit given

by (4.2), where cw = cUi , hence πσiw = 0.

Next, we analyze strategies regarding the profit they carry, in order to classify users into

bribable and non-bribable ones, based on their reputation. We assume that all information

is publicly available to sellers, both users’ ratings and reputations. First, we analyze bribing

users that already rated the item.

Proposition 4. If user v rated item i, v ∈ Ui, a valid strategy, σi ∈ Ξi, s.t. ‖σi‖1 = σiv = ρv,

is profitable if cv > cUi.

43

Proof. Since, ρv > 0, the profit of such strategy, σiv is given by (4.1), which is positive whenever

cv >αUi

= cUi .

We obtain, as a corollary of Proposition 4, the result of Lemma 2 in [Grandi and Turrini,

2016], if v ∈ Ui, and cv = cu = cUi for all u ∈ Ui (the ranking is given by the arithmetic

average) then πσiv = 0.

Suppose item i seller wants to bribe a user who did not rate the item. We study what

conditions make this action profitable.

Proposition 5. Let v /∈ Ui, the strategy σiv is profitable whenever one of the following holds:

1) cv < cUi and ρv < ri, or

2) cv > cUi and ρv > ri.

Proof. The result follows from (4.2).

Note that this marks a difference from the work in [Grandi and Turrini, 2016], where, in

Example 1, the authors showed that a user that did not rate the item can be bribed and it

always increase the wealth.

Further, notice that the result of Proposition 5 means that, in case 1), if a seller bribes a

user (that did not rate item i) that has reputation below the average then the bribing value,

ρv, must be smaller than ri. This happens because, the effect of bringing a new rater to the

set of raters increases the wealth, as long we do not pay a high price, ρv, since the reputation

of the user is smaller, henceforth the effect on the rating is small. In the case where the bribed

user has a reputation above the average, case 2), its effect on the ranking of the item is large,

so bribing with a value below the ranking degrades it, thus lessening the wealth. Hence, if

the ranking is computed by the AA, it is not profitable to bribe a user that did not rate the

item.

4.3.2 Optimal Strategies in the Bipartite Ranking Systems

Here, we investigate what is the optimal investment strategy that the seller of item i should

use to increase his/her initial wealth, by influencing the opinion of customers. First, we

consider two simpler cases where a vendor either tries to change the opinion of users that

already rated item i or tries to persuade users that did not rate it. Then, we analyze the

more complex case when a seller influences both raters and non-raters. We obtain the optimal

bribing strategies in closed form.

To model these problems, we consider a common set up. The seller of item i has an initial

wealth of Ji, and we consider two reference customers, u and v, with reputations cu > cv. We

44

compute the profit per amount of invested wealth, πσ‖σ‖1 , so we can design the optimal bribing

strategy.

Bribing users that already rated item i. Let us consider the case where the seller wants

to bribe users that already rated item i, i.e. u ∈ Ui. We formulate this problem as

maximize: πσi , subject to: ‖σi‖1 ≤ Ji, σi ∈ Λi, (4.4)

where Λi = Ξi. As we show in Proposition 4, to have a positive profit πσiu , when bribing user

u, we need to have cu >α|Ui| = cUi . Therefore, we do not consider strategies that bribe users,

v, s.t. cv < cUi , since it would not increase the wealth, Ji.

Let cu > cv > cUi , we look into the profit per unit of invested resources, πσiu/ρu−πσiv/ρv =

(cu − cv)/cUi > 0. Hence, the profit per unit of invested wealth is larger for user u than for

user v. The optimal strategy is then: to bribe users by decreasing order of their reputation,

investing all the available wealth until either the exhaustion of available profitable users

(cu > cUi) or the depletion of funds.

Bribing users that did not rate the item i before. Suppose that the seller of item i

wants to bribe users that did not rate the item, i.e., u /∈ Ui. We formulate this problem as

(4.4) with Λi = Ξi. Let users u, v /∈ Ui be s.t. cu > cv, and let

α =∑w∈Ui

cw, γ =|Ui| cu − αcu + α

and δ =|Ui| cv − αcv + α

.

The profit is given by (4.2), hence, we have, for user u and v,

ρu − ricu + α

(|Ui| cu − α) andρv − ricv + α

(|Ui| cv − α) ,

respectively. The difference of profits is

(ρu − ri)γ − (ρv − ri)δ,

and hence for the same amount of wealth spent,

πσiu/(ρu − ri) > πσiv/(ρv − ri),

because γ > δ.

Again, the optimal strategy is to bribe users by decreasing order reputation, investing all

the available wealth until either the exhaustion of profitable users (cu > cUi) or funds.

45

General case. Again, under the same conditions for the seller of item i, we now consider

that all users, u ∈ U , are bribable. The problem of finding the best bribing strategy is (4.4)

with Λi = Si = Ξi ∪ Ξi. Next, we investigate when it is better to bribe a user u ∈ Ui or a

non-rater user v /∈ Ui. For this, we consider the profit change rate, which are

πσiu/ρu = δ and πσiv/(ρu − ri) = γ,

respectively. In the case, cu ≥ cv we always have δ ≥ γ. In the other case, cu < cv, we have

γ < δ whenever either cUi < 1/|Ui| and cu < α, or cUi ≥ 1/|Ui|. Again, the optimal strategy

consists in ordering bribable users by decreasing reputation for each of the sets Ui and U \Ui,and start allocating wealth first to Ui.

4.3.3 Properties of strategies and its profit in Multipartite Ranking Sys-

tems

Now, we explore the profit of bribing on the multipartite case. To simplify the analysis, we

assume that, when a user is bribed and changes his/her rating for an item, his/her reputation

keeps unchanged. This assumption is not unrealistic since not only whenever the user has

rated several items its reputation’s change is small if only one of his ratings changes, but also

because in real systems the re-computation of the reputations is often performed only from

time to time. We assume that the users’ ratings and reputations are publicly available, but

the network of users, i.e., the clusters’ partition is private.

Proposition 6 (Bribing a user in a cluster that already rated the item). Suppose that v ∈UMsi , for some cluster s ∈ 1, . . . , N. If cv > c

UMsi

, then any σv ∈ Ξv is profitable.

Proof. Following the same steps as in the proof of Proposition 4, replacing Ui by UMsi , we

have that

πσiv = Jσiv − Ji = ρv(cv/cUMsi− 1) > 0.

This result is Proposition 4 applied to Ms.

Proposition 7 (Bribing a user in a cluster to rate a non-rated item in the cluster). Suppose

that v ∈Ms, for a cluster s ∈ 1, . . . , N, and consider an item, i, that was not rated by any

member of the cluster, that is i /∈ IMs. In this case, any σv ∈ Ξv is non-profitable.

Proof. Since |UMsi | = 0, then

πσiv =∑m∈Xi

|UMmi |rMm

i + (|UMsi |+ 1)

cvρvcv− ρv −

∑m∈Xi

|UMmi |rMm

i = 0.

46

Proposition 8 (Bribing a user in a cluster to rate an item that he did not rate before, but

i ∈ IMs). Suppose that we want to bribe a user that did not rate item i and the user belongs

to a cluster where some user already rated item i, in other words, v ∈ Ms, v /∈ UMsi and

i ∈ IMs. The strategy σiv is profitable whenever one of the following holds:

1) cv < cUMsi

and ρv < rMsi ,

2) cv > cUMsi

and ρv > rMsi .

Proof. By an adaptation of (4.2), the profit of σiv is

πσiv = (|UMsi |+ 1)rMs

σiv− ρv − |UMs

i |rMsi = (α− |UMs

i |cv)rMsi − ρvα+ cv

,

where α =∑

u∈UMsi

cu. It is profitable if 1) or 2) holds.

4.3.4 Optimal Strategies in Multipartite Ranking Systems

Next, we study the optimal bribing strategies for the multipartite ranking system, as we did

in Section 4.3.2 for the bipartite ranking systems. Again, we consider three scenarios: (i)

bribing users that rated the item; (ii) bribing users that did not rate the item; (iii) bribing

users from the set of all users. We compute the close form of the optimal strategies for some

cases, for the others LP can be used.

To model these problems we assume that the seller of item i disposes of an initial wealth

given by Ji, and we consider two reference customers, u and v, with reputations s.t. cu > cv.

Bribing users that rated item i. Consider that the seller wants to bribe users that

already rated item i, u ∈ Ui, i.e.,

maximize: πσi , subject to: ‖σi‖1 ≤ Ji, σi ∈ Υi, (4.5)

where Υi = Ξi. There are two cases to explore: (i) both users are in the same cluster; (ii)

each user is in a different cluster.

(i) Suppose that u, v ∈ Ms are two users that already rated item i. By Proposition 6, to

have a positive profit πσiu , when bribing user u, we need to have cu > cUMsi

. Thus, we do

not consider strategies that bribe a user, v, s.t. cv < cUMsi

, because it would not increase the

wealth, Ji.

Let cu > cv > cUMsi

, we compute the profit per unit of invested resources,

πσiu/ρu − πσiv/ρv = (cu − cv)/cUMsi

> 0.

Thus, the profit per unit of invested wealth is larger for user u. Hence, as we obtained for the

bipartite ranking systems, the optimal strategy is: to bribe users by decreasing reputation,

47

investing all the wealth until either the lack of available profitable users (cu > cUMsi

) or the

exhaustion of funds to bribe profitable users.

(ii) When each reference user belongs to distinct clusters, u ∈Ms, v ∈Mt and s 6= t, we have

that if |UMsi | ≥ |UMt

i |, then the profit per unit of invested wealth (πσiu/ρu versus πσiv/ρv) is

larger for user u. If |UMsi | < |UMt

i | then the profit per unit of invested wealth is larger for

user u if |UMsi | > (cu− cv)−1 and |UMt

i | < (|UMsi |cu− 1)/cv, and larger for user v, otherwise.

Bribing users that did not rate the item i. Under the same conditions for item i seller,

suppose that he wants to bribe users that did not rate i, i.e. u /∈ Ui. We formulate this as

(4.5) with Υi = Ξi. Recalling Proposition 7, we only need to explore the case where the seller

of item i wants to bribe users belonging to clusters with users that already rated the item,

clusters m s.t. i ∈ IMm , otherwise the profit is zero. Let users u, v ∈ Ms and u, v /∈ Ui be

s.t. cu > cv, and let

α =∑

w∈UMsi

cw, γ =

∣∣∣UMsi

∣∣∣ cu − αcu + α

and δ =

∣∣∣UMsi

∣∣∣ cv − αcv + α

.

By Proposition 8, we have that the profits for bribing users u and v are

ρu − rMsi

cu + α(|UMs

i |cu − α) andρv − rMs

i

cv + α(|UMs

i |cv − α),

respectively. The difference of profits is

(ρu − rMsi )γ − (ρv − rMs

i )δ,

hence, for the same amount of spent wealth,

πσiu/(ρu − ri) > πσiv/(ρv − rMsi ),

because γ > δ.

Again, the optimal strategy is to bribe users by decreasing order reputation, investing all

the available wealth until either the exhaustion of profitable users (cu > cUMsi

) or funds.

In the case both users are in distinct clusters and did not rate item i, we cannot derive

simple conditions and we need to solve a LP for each instance.

General case. Under the same conditions for item i seller, we consider that all users,

u ∈ U , can be bribed. The problem of finding the best bribing strategy is written as (4.5)

with Υi = Si = Ξi ∪ Ξi. Next, we investigate when it is better to bribe a user u ∈ UMsi or a

non-rater user v /∈ UMsi . The result is the adaptation of the one for the general case in 4.3.2.

48

We consider the profit change rate, which areπσiuρu

= δ andπσiv

ρu−ri = γ, respectively. In the

case, cu ≥ cv we always have δ ≥ γ. In the other case, cu < cv, we have γ < δ whenever either

cUMsi

< 1/|UMsi | and cu < α, orcUi ≥ 1/|UMs

i |.

Again, the optimal strategy is to order bribable users by decreasing reputation for each of the

sets UMsi and U \ UMs

i , and start allocating wealth to UMsi and, afterward, to U \ UMs

i . If

the reference users are in different clusters, we cannot draw simple conditions, and we need

to solve the LP for each instance.

4.3.5 Bipartite vs. Multipartite Ranking Systems

Here, we compare the profits obtained in the multipartite case and bipartite case, for same

conditions. In the case where the user rated the item, we have the following result:

Proposition 9. Suppose that the seller of item i wants to bribe a user v that already rated

the item, i.e. v ∈ Ui. Let the user v be in cluster Ms, then the profit is larger in the bipartite

ranking systems, πσi < πσi, if and only if c(Ui\UMs

i )< c

UMsi

, the average of the reputations in

(Ui \ UMsi ) and UMs

i , respectively.

Proof. By definition, πσi < πσi is the same as∣∣∣UMs

i

∣∣∣ cv∑u∈UMs

icu− 1

ρv <

(|Ui| cv∑u∈Ui cu

− 1

)ρv,

which is equivalent to

|UMsi |

∑u∈Ui

cu < |Ui|∑

u∈UMsi

cu.

Noticing that

Ui = UMsi ∪ (Ui \ UMs

i ),

we can rewrite it as

|UMsi |

∑u∈UMs

i

cu +∑

u∈Ui\UMsi

cu

<(|UMsi |+ |Ui \ UMs

i |) ∑u∈UMs

i

cu.

This is

c(Ui\UMsi ) < c

UMsi

.

49

5 10 15 20 25 30 350

100

200

300

400

σ1 σ2 σ3 σ4

0 100 200 300 400403

404

405

406

407

(a)

0 100 200 300 400

396

398

400

402

404

(b)

Figure 4.2: Profit of bribing strategies of the most rated item’s sellers in (a) bipartite ranking

system (σ1 – σ4), and (b) multipartite ranking system (σ1 and σ2).

Hence, there are cases where bribing a user in the multipartite ranking system is more

profitable than in the bipartite ranking system. Since the clusters’ partition is assumed to be

unknown for the sellers, they cannot determine the users that verify the previous condition.

Unlike users’ reputations that are often public. Now, we compare the profit of bribing a user

that did not rate the item i in the case the bribed user v belongs to a network where no users

rated the item, v ∈ Ms and i /∈ IMs . In this case, bribing user v in the multipartite ranking

system yields zero profit, but in bipartite one the strategy can be profitable, as we showed in

Proposition 5. In the case that the bribed user did not rate the item, but he/she belongs to a

cluster where some user rated the item, we cannot draw simple conditions as in the previous

cases. We need to check for each concrete case which one is the most profitable.

4.4 Simulations

Here, we illustrate the main results of the chapter with real data, using the 5-core version of

“Amazon Instant Video” data set, [McAuley et al., 2015], as in Chapter 3, with 5130 users,

1685 items, 37126 ratings, and where each user rated, at least, 5 items.

We simulate bribing strategies for the seller of the most rated item (455 ratings). This

item allows us to have more data to explore (the results would be similar for other items).

We study the effect of four strategies in bipartite and multipartite ranking systems, which

are: σ1 – bribe users that rated the item, by a random order; σ2 – bribe users that rated the

item, by decreasing reputation; σ3 – bribe users uniformly at random, from all users (only

for bipartite ranking systems); σ4 – bribe users in decreasing order of reputation (only for

bipartite ranking systems). In Figure 4.2 (a) and (b), we show the results of different bribing

50

Fixed reputations Dynamic reputations

0 100 200 300 400

404.0404.5405.0405.5406.0406.5

Figure 4.3: Profit of bribing strategy σ2 in the bipartite ranking system, fixed users’ reputa-

tions versus reputations recomputed after each user being bribed.

strategies for bipartite and multipartite ranking systems, respectively. The steps where the

rewards are constant, in Figures 4.2 (a) and (b), represent choosing users that already rated

the item with the maximum allowed rating. For the bipartite ranking system, Figure 4.2 (a),

after bribing the same users in strategies σ1 and σ2, both strategies yield the same profit,

as stated in Proposition 1. Finally, the strategy σ3 of Figure 4.2 (a) is to bribe users, from

the set of all users, by decreasing reputation. As expected, bribing users among the ones

who rated the item and are more influential (have a larger reputation) results in a faster

increase of reward, whereas random bribing among the item’s raters has an expected profit

close to zero, and does not increase wealth. The strategy σ4 is the most profitable, but only

after a certain number of users in comparison to strategy σ2. For all strategies, the profit

is positive. In the multipartite ranking system scenario, Figure 4.2 (b), for σ1 and σ2, the

wealth is strictly smaller at the end of the bribing strategy. This illustrates the fact that

the multipartite rankings system is more robust to bribery than the bipartite one, which

meets the discussion in Section 4.3.5. Lastly, we apply strategy σ2 to the bipartite ranking

system, assuming that the users’ reputations are fixed, or without this restriction (each time

a user is bribed, both rankings and reputations updated) as in Chapter 3. The results of this

experiment are depicted in Figure 4.3. In Figure 4.3, we see that the reputations of the bribed

users decrease therefore the impact of their ratings is smaller as well their profits than when

the reputations are fixed.


We model bribing in two reputation-based ranking systems. The first ranking system does

not aggregate users, while the second one clusters users by their similarities and, therefore,

presents a dedicated ranking of items for each cluster. In both settings, we show which users

to bribe to get positive profit, and we show that clustering users decrease the number of

profitable bribing strategies. We illustrate our results with a real-world dataset. In future

51

work, we would like to study the interactions between bigger and smaller players, and the

scenario where sellers bribe users to decrease a competitor item’s ranking through a game

theory model with the sellers as players. Another aspect we want to explore and incorporate

into the bribery analysis is the impact on the profit of strategies of dynamic reputations that

changed when the ratings change. Therefore studying new conditions to design profitable

bribing strategies.

52

Chapter 5

Recommendation via Matrix

Completion Using Kolmogorov

Complexity

This chapter explores the two Kolmogorov complexity based similarity measures introduced

in Chapter 3, KS and CS, to design a recommender system. The proposed system performs

neighborhood-based collaborative filtering (CF), and it allows to compute individual user

recommendation lists incurring in low time and space complexity costs, faster than the state-

of-the-art approach to which we compare. Further, when tested, our algorithm presents

comparable results to the algorithms we test, sometimes even better. This work was submitted

for publication, see [Ramos et al., 2018c].

5.1 Introduction

Recommender systems suggest products that might be interesting for the users [Ricci et al.,

2015]. Collaborative Filtering (CF) is by far the most popular and effective recommendation

technique [Ning et al., 2015,Koren and Bell, 2015].

The CF approaches divide into two classes: model-based and neighborhood-based [Ricci

et al., 2015]. The first tries to model latent factors of both users and items, and it is widely

employed due to its success for movie recommendation in the Netflix prize [Bennett et al.,

2007]. The second class does recommendation based on users with similar preferences or

items that are similar to the users’ preferences. This class further divides into three main

approaches, user-based, item-based, and hybrid. User-based methods select a set of similar

users based on similarity among them to recommend items [Zhao and Shang, 2010]. Item-

based methods are analogous, but performed using similarities among the items [Sarwar et al.,

53

2001a]. The hybrid approaches combine the previous [Wang et al., 2006].

In general, in order to choose the CF algorithm that will implement the recommender

system, the authors need to do several assumptions. Moreover, in order to produce effective

suggestions, we have to set several parameters inside the algorithm. Model-based algorithms

assume that there is an underlying model that uncovers latent features and explains unob-

served ratings [Koren and Bell, 2015]; however, in order to employ this class of algorithms,

it is necessary to assume which model will fit the data (and thus will lead to effective pre-

dictions), and its parameters have to be learned. When a neighborhood-based approach is

chosen, it is necessary to make an assumption on the ratio between the number of users and

the number of items. Indeed, if there are more users, item-based CF leads to more accurate

predictions [Ning et al., 2015, Fouss et al., 2007, Sarwar et al., 2001b]. Moreover, both user-

and item-based approaches require to set the number of neighbors at predictions stage, which

is key to build accurate predictions [Ning et al., 2015].

It should be clear that it might be hard to make these assumptions in advance, especially

for a new business and for a system that will grow over time.

Contributions. In this chapter, we propose a hybrid (user and item) neighborhood-

based CF. We made the choice of employing a neighborhood-based approach, because they

are known to be simple, justifiable, and stable [Ning et al., 2015]. Moreover, with respect to

approaches in the literature, our solution is:

• efficient: the algorithm that we propose is modular, and the recommendation for each

user can be computed independently. Moreover, the computations of the algorithm can

be done in a distributed fashion, making it scalable;

• assumption-free: our algorithm works with a small number of data points, it works for

both low-rank and high-rank matrix completion, without the need for any initialization;

• model-free: the entries are not assumed to be a function of some latent variables;

• parameter-free: our algorithm does not need a parameter to select the neighbors, since

they are represented by the users who rated items in common with the target users

(more details on this will be provided in Sections 5.3 and 5.4).

While these problems might have been tackled individually, this is the first approach

that combines together efficiency, with the absence of assumptions on the data, models, and

parameters.

Our method explores Kolmogorov complexity to construct a similarity measure from in-

formation theory [Cover and Thomas, 2012], and to propose new similarity measure. A large

evaluation of our approach on 9 datasets shows that our approach outperforms or competes

with 10 state-of-the-art approaches chosen as baseline.

54

Chapter structure. We organized the rest of the chapter as follows. In Section 5.2, we

present related work. In Section 5.3, we introduce some notation and present our setup spec-

ification. In Section 5.4, we use our matrix completion algorithm to evaluate its performance,

with both synthetic data and real-world datasets. In Section 5.5, we conclude the chapter,

and we draw avenues for further research.

5.2 Related Work

Several approaches to the matrix completion problem reformulate it into an optimization

problem, assuming that the matrix to recover has low rank and that the observed entries’

positions are sampled according to a uniform distribution, see [Candes and Tao, 2010]. Al-

though the rank minimization problem is NP-hard, approaches following the ideas in [Candes

and Tao, 2010] are used with relative success. It consists of relaxing the problem so that it

becomes convex, and then in minimizing the nuclear norm of the matrix. These methods are

very used in practice. In other approaches, authors assume that the matrix to complete has a

high rank. These approaches also lead to deal with a NP-hard problem. Nonetheless, under

certain assumptions, we can complete some incomplete high-rank or even full-rank matrices,

as in [Balzano et al., 2012]. In their work, the authors assume that the columns of the matrix

to complete belong to a union of multiple low-rank subspaces. In a recent work by [Ganti

et al., 2015], the authors addressed the matrix completion problem without assuming that

the matrix is low rank, as it is usual. They considered recovering the entries of a low-rank

matrix through a Lipschitz monotonic function. In [Song et al., 2016], the authors address

the matrix completion problem using a novel framework for nonparametric regression over

latent variable models. They propose to model the unknown matrix entries as a Lipschitz

function of two latent variables, one for users and another for items. In [Wang et al., 2006], the

authors presented a generative probabilistic framework that considers the similarity between

users and between items. The prediction of each unknown matrix entry is the average of the

individual ratings weighted by the users’ confidence.

5.3 Setup

Next, we present our matrix completion algorithm and its computational complexity analysis.

5.3.1 Setup specification

We propose a recommender system, by making matrix completion as in hybrid neighborhood-

based CF approaches. Our approach computes two matrices of similarities, one between users,

55

...

...

... ...

un-1 unu1 u2

i1 i2 im-1 im

SU1n

SU12 SUn-1n

SIm-1m

SI1m

SI12

Ru1i1Ru2i1

Runim-1

Runim

Figure 5.1: Graph representing n users, u1, . . . , un and m items, i1, . . . , in. The filled

edges between users and items represent the products each user rated weighted by the rating.

The top dashed edges (between users) represent the weights computed in the matrix SU . The

dashed bottom edges (between items) represent the weights computed in the matrix SI .

SU , and another between items, SI .After, we complete each entry of user u and item i by

assigning a convex combination of two quantities, by a parameter α1. The first quantity is a

weighted average of the ratings that user u gave to other items by the similarities of the other

items with item i. The second is a weighted average of the ratings of item i given by other

users similar to user u, see Figure 5.1.

To build the matrices SU and SI , we propose the two compression similarities based on

Kolmogorov complexity [Cover and Thomas, 2012] that we introduced in Chapter 3. Recall

that, given the description of a string, x, its Kolmogorov complexity, K(x), is the length

of the smallest computer program that outputs x (i.e., K(x) is the length of the smallest

compressor for x). Although Kolmogorov complexity is non-computable, there are efficient

and computable approximations by compressors. Let C be a compressor and C(x) denote the

length of the output string resulting from the compression of x using C. The first measure

we proposed in Chapter 3 is the following.

1Note that parameter α is a way to weigh the importance that each component (i.e., user- and item-based)

should take, in classic hybrid fashion. As stated in the Introduction, the individual components can perform

matrix completion without parameters.

56

Compression similarity. Using the normalized compression distance, see [Li et al.,

2004], we define the compression similarity as:

CS (x, y) = 1− C(xy)−minC(x), C(y)maxC(x), C(y)

,

where string xy is the concatenation of x and y. We implement the description of users/items

as the string composed by the index of rated items/rating users and respective rating. For

instance, if user u rated the items i1u, i2u, . . . , i

lu, l ≤ M , then we write the description of user

u as the string “i1uRui1ui2uRui2u . . . i

luRuilu”.

Inspired by CS, in order to reduce the computational complexity, we proposed, also in

Chapter 3, another similarity measure.

Kolmogorov similarity. We define the Kolmogorov similarity as:

KS (x, y) = (1 + |C(x)− C(y)|)−1 .

Here, to compress the description strings, we use the standard compression tools from the

zlib library2. Intuitively, both similarities measures quantifies how related are the compactest

descriptions of a pair of users or a pair of items.

The compression similarity measures are used to compute the two similarity matrices, SU

and SI .

To complete the rating matrix R, we set each non-filled entry Rui in the completed one,

R, as a convex combination of two quantities by the parameter α. The first quantity is

based on users that are similar to the user for which we are estimating the rating. A known

phenomenon, as pointed in [Schafer et al., 2007b], is that users may vary the scale of the ratings

that they use. For example, an optimistic user may only give high ratings and a pessimistic one

only low ratings. To compensate for this effect, we do the following rating prediction, based

on the similarities between users, that accounts for each user’s rating average, as proposed

in [Schafer et al., 2007b].

predU (u, i) = Ru +

∑v∈Ui

|Iu,v|2SUuv(Rvi − Rv)∑v∈Ui

|Iu,v|2SUuv.

The second is the sum of the ratings of each item j 6= i, weighed by the square of the number

of user rating the item together with SIij , predI(u, i).

predI(u, i) =

∑j∈Iu

|Ui,j |2SIijRuj∑j∈Iu

|Ui,j |2SIij.

2https://tools.ietf.org/html/rfc1950

57

https://tools.ietf.org/html/rfc1950

Lastly, fixed the parameter 0 ≤ α ≤ 1, we estimate each non filled matrix entry as

Rui = α predU (u, i) + (1− α) predI(u, i).

Observe that if α = 1, it corresponds to user-based CF, and if α = 0, it corresponds to

item-based CF. The previous steps are summarized in Algorithm 2.

Our approach allows to decouple the problem into a set of independent user-by-user sub-

problems. Hence, to generate recommendations for a user, we do not need to complete the

entire rating matrix, but only the corresponding matrix row.

Algorithm 2 Matrix completion algorithm: KolMaC

1: input: α, training set R

2: compute SU from the training set

3: compute SI from the training set

4: set R = R

5: for each user u do

6: for each item i such that Rui = ⊥ do

7: set Rui = α predU (u, i) + (1− α) predI(u, i)

8: end for

9: end for

10: output: R

5.3.2 Complexity analysis

To build the user similarity matrix SU , we pre-compute the quantity C(u) for each user

u ∈ U . After, we do not need to build an n × n matrix where each entry SUuv = KS(u, v)

for each u, v ∈ U , because we can compute each of this entries in O(1) time by accessing the

pre-computed values. The pre-computed values consist in, at the end, compressing strings

that are a partition of the string with all the ratings, which takes O(|R|). Hence, to compute

the similarity of each pair of users, we need time complexity of O(maxn2, |R|) and space

complexity of O(n). Mutatis mutandis, for the time and space complexities of the items’

similarities, which are O(maxm2, |R|) and O(n), respectively. Note that |R| ≤ n×m, but

usually, |R| n2 and |R| m2.

For the CS measure, we perform the same pre-computations, but to build SU and SI ,

we further need to compute the compression of the concatenation of pairs of users and pairs

of items, respectively. Hence, the time complexity is O(n2m) and O(nm2), whilst the space

complexity is O(n2) and O(m2), respectively for SU and SI .

58

For the matrix completion problem, steps 4-9 of Algorithm 2, the time complexity is

O(maxn,m) (to compute the weighted averages in step 7) times the number of elements of

the matrix nm. This yields a time complexity of O(maxn2m,nm2). The space complexity

of those steps is O(nm).

Hence, the time complexity of Algorithm 2, when using KS and CS, is O(maxn2m,nm2).Note that, in fact, for the KS case it is O(maxn2m,nm2)+O(maxn2, |R|+maxm2, |R|),and O(maxn2m,nm2) > O(maxn2, |R| + maxm2, |R|). Hence, it is strictly less than

for the CS case, which is 2×O(n2m+nm2). The space complexity when using KS is O(nm),

and when using CS it is O(maxn2,m2).

5.4 Experimental setup

Next, we describe our experimental settings and analyze the results.

5.4.1 Datasets

We test Algorithm 2 on synthetic and real-world datasets. All experiments were done in a

3.33GHz Six-core Intel Xeon, with 6GB 1333MHz RAM, using Matlab 2016 and Python 3,

and with OS X 10.13. For the synthetic data, we generate randomly four full-rank matrices,

with dimension 20× 30, and with entries in [1, 5].

For the real-world datasets we use MovieLens 100k (ML–100k) and MovieLens 1M (ML–

1M)3, and both have ratings in [1, 5], with ⊥ = 0. Further, we use the Jester datasets

in “Dataset 1” of http://eigentaste.berkeley.edu/dataset/. The datasets consist in

ratings to a set of 100 jokes, with continuous ratings in ] − 10, 10[, with ⊥ = 99. Jester-1

has 24,983 users who rated 36 or more jokes. Jester-2 consists of 23,500 users that rated

36 or more jokes. Jester-3 has 24,938 users who rated between 15 and 35 jokes. Table 5.1

contains a compact description of the datasets.

ML–100k ML–1M Jester-1 Jester-2 Jester-3

|U | 983 6040 24,983 23,500 24,938

|I| 1682 3952 100 100 100

|R| 100,000 1,000,000 1,810,455 1,708,993 616,912

Table 5.1: Details of datasets MovieLens 100k and 1M.

3http://movielens.umn.edu

59

http://eigentaste.berkeley.edu/dataset/

http://movielens.umn.edu

5.4.2 Evaluation metric

To evaluate and compare the performance of the proposed algorithm, Algorithm 2, we use

the 5-fold-cross-validation method on both synthetic and real data. For the ML–100k, the

dataset already provides a set of 5 train and test files. For the ML–1M, we randomly split

the original dataset in a set of 5 train/test files. In the synthetic data, the four randomly

generated full rank matrices, with dimension 20× 30, were split as in the ML–1M case.

We use the root-mean-square error (RMSE) [Koren, 2008] to evaluate the accuracy of our

algorithm, by measuring the difference between the estimated and the original values. Let R

be the original matrix, R∗ equal to R except on the missing entries of the test set T , where

it has the value ⊥, and let R be the estimation of M by a matrix completion method when

applied to R∗. The RMSE is given by

RMSE(R, R) =

√√√√ 1

|T |∑

(u,i)∈T

(Rui − Rui)2.

5.4.3 Experimental results

We compare our algorithm, using both similarity measures KS and CS, against the following

algorithms: NormalPredictor, BaselineOnly [Koren, 2010], KNNBasic [Altman, 1992], KN-

NWithMeans [Altman, 1992], KNNBaseline [Koren, 2010], SVD [Salakhutdinov and Mnih,

2007], SVD++ [Koren, 2008], NMF [Lee and Seung, 2001], Slope One [Lemire and Maclachlan,

2005] and Co-clustering [George and Merugu, 2005]. The Python toolkit Surprise4 presents

an implementation for these algorithms. We summarize the results of the experiments in

Table 5.2, for the synthetic data, and in Table 5.3, for the real datasets. In Table 5.3, the “-”

in the competitive neighborhood-based algorithms means we could not get the results with

the available RAM memory (Jester case), and in the SVD++ case means we could not get

a result in reasonable time. Hence, these methods suffer from scalability problems. For the

synthetic data, the best result is obtained by our approach with the similarity CS. When using

similarity KS, the result is the third best. We obtain these results because the majority of

the compared methods assume that the matrix they are completing is low rank, which might

be the case in these datasets, but might not be the case in general.

With real data, using both KS and CS similarity measures, our algorithm does not present

the lowest RMSE, except for the Jester-1 dataset. The reason may be the fact that most

of the compared methods assume that the completed matrix is low rank. However, the

results are comparable and of the same order as the best-reported ones. In conclusion, our

proposal represents an effective and efficient solution, since it combines the intrinsic values of

4http://surpriselib.com/

60

http://surpriselib.com/

Method M1 M2 M3 M4

NormalPredictor 1.8692 1.8944 1.7140 1.9263

BaselineOnly 1.4667 1.4663 1.4306 1.4803

SVD 1.5155 1.5120 1.4660 1.5222

SVD++ 1.5205 1.5176 1.4698 1.5279

NMF 1.6999 1.6703 1.7052 1.7686

Slope One 1.5270 1.5287 1.4760 1.5310

Co-clustering 1.5808 1.5630 1.5461 1.6442

KNNBasic 1.4665 1.4840 1.4383 1.5049

KNNWithMeans 1.5107 1.5150 1.4721 1.5269

KNNBaseline 1.4838 1.4998 1.4549 1.5126

KolMaC KS 1.4663 1.4676 1.4303 1.4848

KolMaC CS 1.4530 1.4520 1.4260 1.4714

Table 5.2: RMSE of a 5-fold-cross-validation in four synthetic random and full rank 20× 30

matrices.

neighborhood-based CF (i.e., its simplicity, justifiability, and stability), with the advantages

offered by our algorithm (i.e., its efficiency, and being assumption-, model-, and parameter-

free).

Method ML–100k ML–1M Jester-1 Jester-2 Jester-3

NormalPredictor 1.5228 1.5037 7.4572 7.2695 7.4490

BaselineOnly 0.9445 0.9086 4.5877 4.3139 4.5971

SVD 0.9396 0.8936 4.4957 4.5594 4.4957

SVD++ 0.9200 – 4.5192 4.7277 4.5192

NMF 0.9634 0.9155 6.2372 7.1256 6.2768

Slope One 0.9454 0.9065 4.5187 4.2517 4.5187

Co-clustering 0.9678 0.9155 4.6634 4.3627 4.6693

KNNBasic 0.9789 0.9207 – – –

KNNWithMeans 0.9514 0.9292 – – –

KNNBaseline 0.9306 0.8949 – – –

KolMaC KS 0.9582 0.9330 4.4719 4.5164 4.7160

KolMaC CS 0.9465 0.9216 4.4582 4.5027 4.7107

Table 5.3: RMSE for the datasets ML–100k and ML–1M.


We presented a novel hybrid neighborhood-based CF recommender system. Our system makes

independent, user-by-user, matrix completion, utilizing Kolmogorov complexity. Our method

61

does not require assumptions about the rank of the matrix, we do not need to specify dimen-

sions of subspaces, and it is model-free. Therefore, it is more general. We present experimental

results on both synthetic and real dataset which show that our approach is comparable with

state-of-the-art approaches. The avenues for further research include exploring matrix com-

pletion under the presence of noise and, to extend this work, in an initial step, clustering by

similarities both users and items.

62

Chapter 6

A Novel Similarity Measure for

Group Recommender Systems with

Optimal Time Complexity

In line with the previous chapter, allied to the fact that KS is very fast to compute, in this

chapter, we explore applying this similarity in the context of group recommender systems. A

crucial phase of a group recommender system is to generate the groups of users. By using KS

in this step of a group recommender system, we obtain statistically the same results as when

employing the standardly utilized Pearson similarity. Moreover, we get a considerable gain in

terms of time complexity when using KS. This gain translates to, in the computations of the

users’ similarities phase, spending a few minutes against a few hours when using Pearson’s

similarity. This work was submitted for publication, see [Ramos et al., 2018a].

6.1 Introduction

Our online experience can tell much about our preferences. Indeed, from the analysis of

browsing sessions to the comments, likes, and ratings we leave, lots of implicit and explicit

traces are available on the Web. These preferences are usually stored in a user profile and

can be exploited by those running services, such as e-commerce websites or social media

platforms, to turn them into actionable knowledge, and provide us tailored services, like

recommendations.

Given that the experience of users on the Web is usually individual, these services are

single users’ tailored. Group recommendation operates in contexts in which more than one

person is involved in the recommendation process [Boratto and Carta, 2011]. This area usually

focuses on offline scenarios, in which people have to experience something together (e.g., a

63

group that goes to dinner, or watches a movie).

Suppose that a brand wants to run an advertising campaign specific to its customers.

The usual way of doing it would be targeting a group of users, and recommend them a set of

possibly interesting products. Usually, we base this targeting on users’ global preferences, i.e.,

on the whole user’s profile. However, it is not trivial to understand how the users interacted

with the items of a specific brand. Indeed, the preferences contained in the user profiles

are available only to persons who are running an e-commerce website. This website should

provide a personalized service to the brand, by extracting segments of users, based only on

how they interacted with that brand.

We can effectively employ group recommendation in this scenario, by first analyzing how

the users interacted with a brand and detecting groups of users, and by providing group

recommendation to these groups, treating them as a target.

Open problem and scientific contribution Group recommendation is naturally consid-

ered a challenging area [Jameson and Smyth, 2007,Ricci, 2014], due to the fact that we have

to take into account multiple preferences in the recommendation process. Hence, the problem

we are tackling is even more challenging. Indeed, the information about the user preferences

becomes even more sparse than the usual recommendation scenarios, and detecting brand-

specific groups is not trivial. This problem is due to the fact that the similarity for each pair

of users does not have to be detected just once, considering the whole profile, but once for

each brand. Hence, we need a fast similarity measure, able to deal both with the fact that

the group recommendation problem has to be solved multiple times, and with the continuous

evolution of the user preferences.

In this chapter, we propose a new similarity measure to group users. This similarity has

lower time and space complexity than the state-of-art Pearson correlation similarity measure

that presents statistically the same root mean-squared-error results (RMSE) when tested in

offline datasets. More specifically, our contributions are the following:

• we propose a novel similarity measure based on Kolmogorov complexity that detects

the similarity for the users in an efficient and effective way;

• we show that our measure has lower and optimal time complexity than the state-of-the-

art measure used to compute the similarities between users (Pearson’s correlation);

• we embed our similarity measure in a group recommender system, and test its effective-

ness on two real-world datasets;

• our group recommender system is the first in the literature in which the recommen-

64

dations are both provided and meant to be consumed online1. This contrasts with

classic group recommender systems, in which the users consume the items together in

real-world scenarios.

Chapter structure. The chapter is organized as follows. In Section 6.2, we present re-

lated work in group recommendation. In Section 6.3, we propose a novel similarity measure.

In Section 6.4, we analyze the computational complexity of the proposed similarity metric.

In Section 6.5, we describe the setup of our group recommender system, and we test it in

Section 6.6. In Section 6.7, we conclude the chapter and draw avenues for further research.

6.2 Background and Related Work

Group recommender systems provide suggestions in contexts in which the objective of the

recommendations is not an individual, but multiple users [Boratto and Carta, 2011,Masthoff,

2015].

Group recommender systems naturally adapt to any scenario that involves a group of users.

Indeed, approaches have been developed for people who perform activities together, such as

going to the cinema [O’Connor et al., 2001], planning a travel [Ardissono et al., 2003,McCarthy

et al., 2006,De Pessemier et al., 2015], watching TV [Goren-Bar and Glinansky, 2004,Yu et al.,

2006b], or working out in a gym [McCarthy and Anagnost, 1998] (to name a few).

Providing group recommendations in online scenarios is an approach that has been ex-

plored, mostly taking advantage of social networks [Sanchez et al., 2014]. However, as men-

tioned in the Introduction, the group is meant to consume the items offline (e.g., in the

previously mentioned paper, movies are recommended based on user preferences and social

interactions). Recent literature has also shown that the offline interaction between the users

can help moving from individual to group preferences [Delic et al., 2016].

It is also worth highlighting that Ntoutsi et. al, in [Ntoutsi et al., 2012], previously

introduced the concept of fast group recommendation. By “fast” the authors meant that the

users are clustered in order to speed up the computation when the neighbors are selected

at the prediction stage. However, the group recommender system is run once for the whole

dataset, so this would not solve the problem tackled in this chapter. However, a comparison

with this approach will be presented in Section 6.6.

As the analysis of the literature shows, no approach in the literature performed group

recommendation at subsets of a dataset, thus facing the efficiency and effectiveness problems

1It is worth noting that, even if the users do not consume the recommendation together, this is still a group

recommendation, since the same set of items is recommended to a group of users.

65

at the same time. Moreover, our approach is the first where recommendations may be provided

and consumed online.

6.3 The Kolmogorov-based similarity

First, we introduce a standard similarity measure to group users for group recommender

systems. The Pearson product-moment correlation coefficient [Lee Rodgers and Nicewander,

1988], or Pearson similarity, is the standard to measure the similarities between users [Schafer

et al., 2007a]. Given two vectors X,Y ∈ Rn, the Pearson similarity between the two vector is

given by

Pearson(X,Y ) =

∑ni=1(Xi − X) · (Yi − Y )

(n− 1) · σX · σY. (6.1)

To keep the chapter self-contained, in this section, we recall the similarity measure we

proposed in Chapter 3 and that we used in Chapter 5 to design a recommender system. The

similarity is inspired by the notion of Kolmogorov complexity [Cover and Thomas, 2012], from

information theory. Given the description of a string, x, its Kolmogorov complexity, K(x),

is the length of the smallest computer program that outputs x. In other words, K(x) is the

length of the smallest compressor for x. Although Kolmogorov complexity is non-computable,

there are efficient and computable approximations by compressors. Let C be a compressor

and C(x) denote the length of the output string resulting from the compression of x using C.

Kolmogorov-based similarity. We define the Kolmogorov-based similarity between strings

x and y as

KS (x, y) =1

1 + |C(x)− C(y)|. (6.2)

In the context of this work, different from the previous chapters, the string x is the

string with the pairs of items and ratings, given by a user, or the pairs of items and ratings

estimated/predicted for that user. To compress the description strings, we use the standard

Python function, from the numpy package, savez compressed. Intuitively, the Kolmogorov

similarity KS measures how related are the compactest descriptions of a pair of users or a

pair of items.

The presented Pearson similarity and the KS have, here, the same purpose. However, it

is not easy to compare them mathematically, although we are able to compare them in terms

of computational complexity.

66

6.4 Complexity Analysis

Now, we compare the theoretical time and space complexity that we need to compute the

Pearson and the KS similarities between every pair of users.

Lemma 6.4.1. Let U = u1, . . . , un be a set of users, I = i1, . . . , im a set of items and

R a set of ratings given by users to items. The time and space complexity of computing the

Pearson similarity of each pair of users are O(mn2) and O(n), respectively.

Proof. First, we compute and store the mean and standard deviation of the ratings for each

user. It takes O(m) time to compute expression (2.1) for each user. Therefore, we need

O(nm) time to compute the mean vector of the ratings of all users and O(n) space to store

it. Having the mean vector, we may compute the standard deviation, expression (2.2), also

in O(nm) times and store it in O(n) space. Now, we need to compute expression (6.1). For a

pair of users ratings, having the vector of means and the vector of standard deviation stored,

we need O(m) time to compute the Pearson similarity between the pair of users. To compute

the Pearson similarity of every pair of users, O(n× n) pairs, we need O(mn2) time.

Lemma 6.4.2. Let U = u1, . . . , un be a set of users, I = i1, . . . , im a set of items and

R a set of ratings given by users to items. The time and space complexity of computing

the Kolmogorov-based similarity (LS) of each pair of users are O(maxn2, nm) and O(n),

respectively.

Proof. First, we compute, and store, the compression size of each set of ratings of each user.

For each user, the set of ratings has O(m) size, and we can compute its compression in O(m)

times, using [Williams, 1991], for instance. The O(n) compressions for each user take O(nm)

time and O(n) space to store them. Now, we can compute the KS between a pair of users

in O(1) time, using expression (6.2) and the stored values of the previous step. Finally, to

compute the KS similarity for each pair of users, O(n2) pairs, hence we need O(maxn2, nm)time.

Notice that for the Pearson similarity we need 2n space to store the vector of means and

the vector of the standard deviation of the users’ ratings. For KS, we need only n space

to store the size of the compressions of the set of ratings that each user gave. The time

complexity of KS contrasts with the one of Pearson similarity, because we need less an order

of operations, as we summarize in Table 6.1.

Observe that KS has optimal time complexity whenever n ≥ m, since to compute the

similarity between each pair of users we always need to compute O(n2) values, which is the

67

Pearson KS

Time O(mn2) O(maxn2, nm)Space O(n) O(n)

Table 6.1: Time and space complexity of the similarities.

minimum possible time complexity and the complexity of computing these similarities with

KS.

6.5 The group recommender system

Here, we present the group recommender algorithm we use in this work, Algorithm 3.

Algorithm 3 Group Recommender System.

1: input: ratings’ matrix R, number of clusters k, a similarity measure s, and rating pre-

diction function Pred

2: compute P = Pred(R)

3: compute similarity of each pair of users with function s

4: compute G the k groups of users, clustered by similarities

5: group estimated ratings in P for each group as the average of predictions

6: output: Recommendation list for each group in G

In particular, in this work, we use as the prediction function Pred the benchmark SVD

algorithm for matrix completion, see [Mnih and Salakhutdinov, 2008]. It has time complexity

of O(minmn2,m2n) [Holmes et al., 2007]. Also, for a comparison, we test the k-nearest

neighbors algorithm (KNN) as the prediction function Pred, see [Koren, 2010], with time

complexity of O(m2n).

Further, we use a polynomial time approximation of the k-means clustering algorithm.

The k-means Algorithm [MacQueen et al., 1967] is, in general, NP-hard for a generic number

of clusters k, even in the plane, see [Mahajan et al., 2009]. In practice, k-means may be

approximated by Lloyd’s heuristic algorithm [Hartigan and Wong, 1979], which has time

complexity of O(nkdi), where k is the number of clusters, d is the dimension of the elements

that we are clustering, and i is the number of iterations needed until convergence, which is

usually small. In our case d = 1, and the number of clusters k is a parameter of Algorithm 3.

Theorem 6.5.1. Let i denote the number of iterations that Lloyd’s algorithm takes to compute

the users’ clusters. Let n be the number of users, m the number of items and k the number

68

of users’ groups. The time complexity of Algorithm 3, using SVD as the Pred algorithm of

step 2, is:

• O(mn2 + nki+ km logm), using Pearson similarity;

• O(minmn2,m2n+ maxn2,mn+ nki+ km logm), using KS.

Proof. The time complexity of Algorithm 3 is the sum of the time complexities of each

step. The SVD step, step 2, takes time of O(minmn2,m2n). The Lloyd’s algorithm,

the clustering step 4, takes O(nki), where i is the number of iterations needed until con-

vergence. If we user the Pearson similarity in step 3, by Lemma 6.4.1, we need O(mn2)

time to compute the similarities between every pair of users. This yields a total time of

O(n2m+nki)), because mn2 ≥ minmn2,m2n. If we use the KS in step 3, by Lemma 6.4.2,

we need O(maxn2,mn) time to compute the similarities between every pair of users, and

maxn2,mn < minmn2,m2n. Step 4 takes O(mn) to average the ratings’ estimations

for the users of each group. Step 5 takes O(km logm) to sort the ratings’ prediction for

each of the k groups. Hence, the total amount of time is O(n2m + nki + km logm) and

O(minmn2,m2n+ maxn2,mn+ n2i+ km logm) when using Pearson similarity and KS,

respectively.

In fact, Theorem 6.5.1 states that the complexity order of Algorithm 3 is strictly better

when using KS whenever n > m, otherwise it has the same order than when using the Pearson

similarity. However, we notice that if we compare not only the complexity order but also the

exact complexity, we have the following. When we use Pearson similarity, the total amount

of time is c1mn2 + c2 minmn2,m2n + c3nki + c4km logm, for some non zero constants

c1, c2, c3, c4 ∈ R+. When we use KS, the total amount of time is c2 minmn2,m2n+ c3nki+

c4km logm+c′mn+c′′maxn2, nm, for the same c2, c3, c4 ∈ R+ as in the Pearson’s scenario,

and c′, c′′ ∈ R+. Therefore, the time complexity is always strictly better in the case that we

use KS.

Observe that if, instead of using the SVD algorithm in step 2 of Algorithm 3, we use the

KNN algorithm then the time complexity is the following. Using the Pearson similarity, we

get O(mn2 + m2n + nki + km logm), and using KS we always obtain a better complexity

order of O(m2n+ nki+ km logm+ maxn2, nm).

6.6 Experimental Setup

In this section, we test our similarity measure in two real-world datasets. We use the Movie-

Lens 100k (ML–100k) and the MovieLens 1M (ML–1M), available in http://movielens.

umn.edu, and both datasets have ratings in 1, . . . , 5, with ⊥ = 0. Recall Table 5.1, which

69



we repeat in Table 6.2. The choice of two relatively-small and very sparse datasets was made

to simulate our scenario, in which group recommendations have to be computed for medium-

and large-sized companies.

All experiments were done in a 3.33GHz Six-core Intel Xeon, with 6GB 1333MHz RAM,

using Python 3, and with OS X 10.13. We use the Surprise scikit [Hug, 2017] to compute the

individual predictions with the SVD algorithm and the KNN algorithm, step 2 of Algorithm 3.

Further, to compute the Pearson similarity we use the pearsonr function from the Python

package scipy.stats.

ML–100k ML–1M

|U | 983 6040

|I| 1682 3952

|R| 100,000 1,000,000

Table 6.2: Details of datasets MovieLens 100k and 1M.

6.6.1 Evaluation metric

To evaluate and compare the performance of the proposed algorithm, Algorithm 3, we use,

again, the 5-fold-cross-validation method. For the ML–100k, the dataset already provides a

set of 5 train and test files. For the ML–1M we randomly split the original dataset in a set of

5 train/test files.

We use, as in the previous chapter, the root-mean-square error (RMSE) [Koren, 2008]

to evaluate the performance of the proposed group recommender algorithm. Recall that it

measures the difference between the estimated missing values and the original values as we

detail next. Let R be the original ratings matrix, and let R∗ be the train set, equal to R

except on the missing entries of the test set T , where it has the value ⊥. Let Rui denote the

estimated rating of the group where user u belongs for item i, and R the matrix with all the

estimated ratings. The RMSE is given by

RMSE(R, R) =

√√√√ 1

|T |∑

(u,i)∈T

(Rui − Rui)2. (6.3)

Observe that, here, the RMSE is not measuring the same as in Chapter 5. Here, it is measuring

the difference between the real rating that a user gave to an item and the estimated group

rating of that item, for the group that user belongs to.

70

6.6.2 Experimental Results

Now, we present the experimental results of Algorithm 3. We test the users’ clustering/grouping

phase using the Pearson similarity versus our proposed Kolmogorov-based similarity (KS). For

the ratings’ prediction phase, step 2 of Algorithm 3, we tested with the SVD and the KNN

algorithms2.

Figure 6.1 and Figure 6.2 depict the RMSE (6.3) evolution (yy axis) with the number of

users’ groups (xx axis) as the average of a 5-fold-cross-validation method. The blue points

correspond to using the Pearson similarity and the yellow points to using the KS.

0 200 400 600 800

0.94

0.96

0.98

1.00

1.02

Movielens 100K with SVD

Pearson KS

Figure 6.1: RMSE evolution with the number of users’ groups of a 5-fold-cross-validation

method, using Algorithm 3 with SVD for its step 2, with Pearson similarity (blue points) and

KS (yellow points) for the ML-100K.

Next, we test that the results depicted in Figure 6.1 and Figure 6.2 are not related with

the prediction algorithm (SVD), step 2 of Algorithm 3. For this purpose, we replace the SVD

by the KNN algorithm, and we obtain the results in Figures 6.3 and Figure 6.4.

We obtain better RMSE results when using SVD for the ratings’ prediction phase than

when using KNN, which is expected. More important, we get the same behavior in the

evolution of the RMSE with the number of groups for the KS and the Pearson similarity,

when using either the SVD or the KNN in the prediction step.

2In order to speed up the process furthermore and embrace the concept of fast group recommendation

proposed by Ntoutsi et al. [Ntoutsi et al., 2012], we also considered an alternative to the KNN approach, in

which the neighbors were only selected inside the cluster of the target user. However, results show that, in our

context, the effectiveness decreases. These results are not presented, to improve the readability of the chapter.

71

0 1000 2000 3000 4000 50000.90

0.92

0.94

0.96

0.98

Movielens 1M with SVD

Pearson KS


method, using Algorithm 3 with SVD for its step 2, with Pearson similarity (blue points) and

KS (yellow points) for the ML-1M.

We can see in Figures 6.1–6.4 that the RMSE results when using the Pearson similarity

or the KS are very close. Hence, we test the null hypothesis (H0) that the differences we

obtain in the results are due to randomness. We compare the two means for the 5-fold tests

of each different group size with the Student’s t-test [O’Mahony, 1986]. For both datasets,

we obtained p-values larger than 0.05 and, thus, we must accept the H0. In other words, the

RMSE results depicted in Figure 6.1 and Figure 6.2, and the ones depicted in Figure 6.3 and

Figure 6.4 are, essentially, the same.

Finally, in Table 6.3, we present average and standard deviation of the time that Algo-

rithm 3 spends in step 3, the computation of the similarities between each pair of users, also

using a 5-fold-cross-validation method. Table 6.3 compares, in practice, the time complexity

ML–100k ML–1M

Pearson 2’5.1710”±1.1862” 3h23’20.3505”±10’29.6983”

KS 4.5608”±0.0688” 1’13.9046”±3.1560”

Table 6.3: Average and standard deviation of the computation time of the similarities between

every pair of users in a 5-fold cross validation.

results of Lemma 6.4.1 and Lemma 6.4.2, which are part of Algorithm 3 and responsible for

the difference of the two cases of time complexity in Theorem 6.5.1. Recall that, to compute

the similarity of each pair of users, we need O(mn2) time using the Pearson similarity, and

O(n2) using KS. In practice, we notice that for the ML-100k the KS takes a few seconds

against the 2 minutes needed in the Pearson similarity case. Further, for the ML-1M the

72

0 200 400 600 800

0.94

0.96

0.98

1.00

1.02

Movielens 100K with KNN

Pearson KS


method, using Algorithm 3 with KNN for its step 2, with Pearson similarity (blue points) and

KS (yellow points) for the ML-100K.

gain is even more notorious because KS takes only about 1 minute versus more than 3 hours

needed for the Pearson similarity scenario.


In this chapter, we tackled the problem of producing brand-specific group recommendations,

i.e., recommendations to groups of users, considering only the preferences expressed for a

specific brand. Since we need to compute the similarity for a pair of users multiple times,

we devised a novel and fast to compute similarity measure, the Kolmogorov-based similarity

(KS). Our similarity measure has better (and optimal) theoretical computational complexity

than the state-of-the-art Pearson similarity, which is widely used in the group recommendation

community. We tested these similarity measures in the context of group recommendation in

two real-world datasets. The RMSE that we obtained for both similarities is statistically the

same, up to some randomness. For the larger dataset, the computation of users’ similarities,

took 1 minute, using the KS, while it took more than 3 hours when using the Pearson similarity.

In future work, we will analyze the obtained clusters. This analysis allows us to explain to

a brand what are the characterizes of each targeted group, in terms of users’ preferences.

Also as future work, we would like to study the relation of the KS with known Kolmogorov-

based distances, see [Li et al., 2004] and references therein, and also to explore using different

compressors to compute KS.

73

0 1000 2000 3000 4000 5000

0.92

0.93

0.94

0.95

0.96

0.97

0.98

Movielens 1M with KNN

Pearson KS


method, using Algorithm 3 with KNN for its step 2, with Pearson similarity (blue points) and

KS (yellow points) for the ML-1M.

74

Part II

Control of dynamical systems

75

Chapter 7

Preliminaries and Notation

In this part of the thesis, we solve the robust minimal controllability problem for both linear-

time invariant (LTI) systems and switched LTI systems, with some additional assumptions.

Under the scenario that a specified number of actuators may fail over the time, the problem

consists in determining a placement of the minimal number of actuators that ensures that

the system is controllable. Then, we use a digraph decomposition, used in structural control

theory, to present a more general bound for the index of convergence of Boolean matrices.

The outline of this part is the following: In Chapter 8, we solve the robust minimal

controllability problem for LTI systems. In Chapter 9, we extend the result of the previous

chapter for switched LTI systems. Finally, in Chapter 10, we present a more general bound

for the index of convergence of Boolean matrices.

Now, we introduce the notation used in the first two subsequent Chapters, to avoid repeat-

ing definitions and notation. The notation of Chapter 10 is introduced within the chapter, to

improve the readability of the manuscript.

We denote vectors by small font letters such as v, w, b and its corresponding entries by

subscripts. A collection of vectors is denoted by vjj∈J , where the superscript indicates an

enumeration of the vectors using indices from a set such as I,J ⊂ N. We use square brackets

in vectors or matrices, to separate an enumeration of those from their entries, e.g., [Bk]i,j

stands for the jth column of the ith row of the kth matrix of an enumeration Bk, or [vji ]k

stands for the kth entry of the vector vji . The number of elements of a set S is denoted by |S|.We denote by In the n-dimensional identity matrix. Given a matrix A, σ(A) denotes the set

of eigenvalues of A, the spectrum of A. Given two matrices M1 ∈ Cn×m1 and M2 ∈ Cn×m2 ,

the matrix [M1 M2] is the n × (m1 + m2) concatenated complex matrix. If I = i1, . . . , ikand B ∈ 0, 1n×m, with m,n ≥ k, B(I) is the matrix where [B]j,i = 1 for i ∈ I and the

remaining entries of B are equal to zero.

The structural pattern of a vector/matrix or a structural vector/matrix have their entries

77

in 0, ?, where ? denotes a non-zero entry, and they are denoted by a vector/matrix with a

bar on top of it. We denote by Aᵀ the transpose of A. The function · : Cn×Cn → C denotes

the usual inner product in Cn, i.e., v·w = v†w, where v† denotes the adjoint of v (the conjugate

of vᵀ). With some abuse of notation, · : 0, ?n×0, ?n → 0, ? also denotes the map where

v · w 6= 0, with v, w ∈ 0, ?n if and only if there exists i ∈ 1, . . . , n such that vi = wi = ?.

Similarly, given u, v ∈ 0, ?n, we extend the plus operation, + : 0, ?n × 0, ?n, to be

w = u + v, with wi = 0 if ui = vi = 0 and wi = ? otherwise, for i = 1, . . . , n. Additionally,

‖v‖0 denotes the number of non-zero entries of the vector v in either 0, ?n or Rn. Given a

subspace H ⊂ Cn we denote by Hc its complement with respect to C, i.e., Hc = Cn \H. With

abuse of notation, we will use inequalities involving structural vectors as well – for instance,

we say v ≥ w for two structural vectors v and w if and the only if the following two conditions

hold: (i) if wi = 0, then vi ∈ 0, ?, and (ii) if wi = ? then vi = ?.

A multiset is a set where each element may occur more than once. Consider multisets Xand Y. The multiset X t Y is such that, if a ∈ X or a ∈ Y then a ∈ X t Y. If a occurs n1

times in X and n2 times in Y, then a occurs maxn1, n2 times in X tY. Although not very

intuitive, this ‘unusual’ union will be useful to address (ii), in Section 9.3.

78

Chapter 8

The Robust Minimal Controllability

Problem

In this chapter, we solve the robust minimal controllability problem for LTI systems, with

some additional assumptions. The problem is to find the minimal number of actuators and

their placement, ensuring that the system is controllable in the scenario where a specified

number of actuators may fail. This may happen due to an external agent tampering with

the system or due to natural phenomenon reasons. We show that the problem in hands is

NP-complete, and we present an algorithm that solves it, explicitly. Further, we provide

polynomial time algorithms that approximately solve the problem.

Hence, we gain ground on the topic of information security applied to the are of control

systems. We published this work in [Pequito et al., 2016b].

8.1 Introduction

The problem of guaranteeing that a dynamical system can be driven toward the desired state

regardless of its initial position is a fundamental question studied in control systems and it is

referred to as controllability. Several applications, for instance, control processes, multi-agents

networks, control of large flexible structures, systems biology and power systems [Egerstedt,

2011, Siljak, 2007, Skogestad, 2004] rely on the notion of controllability to safeguard their

proper functioning. Subsequently, it is important to identify which subsets of state variables

need to be actuated, or what is the placement of actuators required, to ensure controllabil-

ity [van de Wal and de Jager, 2001,Olshevsky, 2014,Pequito et al., 2016a].

Moreover, actuators may malfunction over time due to the adverse nature of the environ-

ments where the actuators are deployed, e.g. due to the wear and tear of the materials, or

due to external (adversarial) influence of an agent aiming to disrupt the proper functioning

79

of the dynamical system. In fact, a classical example of such malicious attack is the Stuxnet

malware incident [Langner, 2011], in which the controller’s input response to a tempered mea-

sured output lead the system away from its normal operating conditions. Thus, the control

designer needs to consider such scenarios, while accounting for the actuator placement [Velde

and Carignan, 1984]. Additionally, as the systems become larger (i.e., the dimension of their

state space), we aim to identify a relatively small subset of state variables that ensure the

controllability of the system, for instance, due to economic constraints [Olshevsky, 2014].

Consequently, in this chapter we address the following natural design question:

Q1: What is the minimum number of actuated state variables, and what is the configuration

of actuators, we need to consider to ensure the controllability of a dynamical system if a specific

number of actuators failures occur?

To formally capture Q1, we introduce and study the robust minimal controllability prob-

lem (rMCP) that aims to determine the minimum number of state variables that need to be

actuated to ensure system’s controllability, under the possible failure of a specified number of

actuators. This is a generalization of the of the minimal controllability problem (MCP) [Ol-

shevsky, 2014], which can be obtained as a particular case of the rMCP when no actuator

fails. Therefore, the MCP is the first step to understand resilience and robustness properties

of dynamical systems since it unveils which variables need to be actuated.

Finally, it is important to mention that the rMCP can be stated regarding observability,

by invoking the duality between controllability and observability in LTI systems [Hespanha,

2009]. In particular, [Shoukry and Tabuada, 2014,Chen et al., 2015,Fawzi et al., 2012] provide

necessary and sufficient conditions concerning the sensor deployment to ensure that a reliable

estimate of the system is recovered. More importantly, those conditions can be achieved by

design, when solving the rMCP. Hence, guaranteeing the design of stable observers to proper

monitor the state evolution of an LTI system. Furthermore, the results presented in this

chapter are for discrete-time, but they readily applicable to continuous-time LTI systems.

Related Work: This chapter follows up and subsumes previous literature by consider-

ing the deployment of actuators to ensure controllability under possible actuation failures.

When no actuators fail, it extends the results available for the MCP, as we overview next.

In [Nabi-Abdolyousefi and Mesbahi, 2013] the controllability of circulant networks is ana-

lyzed by exploring the Popov-Belevitch-Hautus eigenvalue criterion, where the eigenvalues

are characterized using the Cauchy-Binet formula. The controllability in multi-agents with

Laplacian dynamics was initially explored in [Tanner, 2004]. Later, in [Rahmani et al., 2009]

and [Egerstedt et al., 2012], necessary and sufficient conditions are given in terms of partitions

of the Laplacian graph. In [Parlangeli and Notarstefano, 2012], the controllability is explored

for paths and cycles, and later extended by the same authors to the controllability of grid

80

graphs by means of reductions and symmetries of the graph [Notarstefano and Parlangeli,

2013], and considering dynamics that are scaled Laplacians. In [Kibangou and Commault,

2014] and [Zhang et al., 2011], the controllability is studied for strongly regular graphs and

distance-regular graphs. Recently, new insights on the controllability of Laplacian dynamics

are given regarding the uncontrollable subspace, in [Aguilar and Gharesifard, 2014] and [Chap-

man and Mesbahi, 2014]. In addition, in [Pasqualetti and Zampieri, 2014] the controllability

of isotropic and anisotropic networks is analyzed.

Furthermore, [Aguilar and Gharesifard, 2014] concludes by pointing out that further study

of non-symmetric dynamics and the controllability is required – which we address in the

present chapter. Therefore, we consider a much less restrictive assumption: A is a simple

matrix, i.e., all of its eigenvalues are distinct. Moreover, there are several applications where

A satisfies this assumption, for instance, all dynamical systems modeled as random networks

of the Erdos-Renyi type [Tao and Vu, 2014], as well as several known dynamical systems used

as benchmarks in control systems engineering [Ogata, 2001,Siljak, 1991,Siljak, 2007].

Observe that the MCP problem presents both continuous and discrete optimization prop-

erties, captured by the controllability property and the number of non-zero entries, respec-

tively. To avoid the nature of this problem, in [Olshevsky, 2014], the non-zero entries of the

input matrix were randomly generated. In the present chapter, we ‘decouple’ the continuous

and discrete optimization properties, and show that by first solving the discrete nature of

the problem, it is always possible to deterministically obtain a solution to MCP in a second

phase. Besides, the first step reduces the MCP to the set covering problem – well known to be

NP-hard. Nonetheless, the set covering problem is one of the most studied NP-hard problems

(probably second only to the SAT problem). Subsequently, although the set covering problem

is NP-hard, some subclasses of the problem are equipped with sufficient structure that can

be leveraged to invoke a polynomial algorithm that approximate the solution with ‘almost’

optimality guarantees [Bronnimann and Goodrich, 1995]. This contrasts with the approach

proposed in [Olshevsky, 2014], where an approximated solution particular to the MCP prob-

lem was provided. In addition, we study the rMCP which has not been previously addressed

in the literature. Similarly to the MCP, we show that the rMCP can be polynomially re-

duced to the set multi-covering problem, i.e., a set covering problem that allows the same

elements to be covered a predefined number of times. Furthermore, extensions of polynomial

approximation algorithms are also available with similar optimality guarantees.

Alternatively, when the parameters of the LTI system are not exactly known, and assumed

to be independent, structural systems theory [Dion et al., 2003] can be used to address the

MCP and rMCP while ensuring structural controllability, see [Pequito et al., 2016a] and [Liu

et al., 2015], respectively. Notwithstanding, the tools and conditions to ensure structural

81

controllability are quite different from those adopted in this chapter, and a solution to the

MSCP is not necessarily a solution to the MCP when the dynamics’ matrix is simple [Pequito

et al., 2016b].

Main Contributions of the present chapter are as follows: (i) we characterize the exact

solutions to the MCP; (ii) we show that for a given dynamics’ matrix almost all input vectors

satisfying a specified structure are solutions to the MCP; (iii) we show that the rMCP is an

NP-hard problem; (iv) we characterize the exact solutions to the rMCP; (v) we prove that

the decision version of both MCPs are NP-complete; (vi) we provide approximated solutions

to the rMCPs and discuss their optimality guarantees; and, finally, in (vii) we discuss the

limitations of the proposed methodology.

The remainder of this chapter is organized as follows. In Section 8.2, we formally state

the rMCP addressed in this chapter. Next, in Section 8.3, we review some concepts required

to prove the main results of this chapter. In Section 8.4, we present the main results of this

chapter, i.e., we characterize the solutions to the rMCP, its complexity, and a polynomial

algorithm that approximates the solutions. Finally, in Section 8.5, we provide some examples

that illustrate the main results of the chapter and discuss the limitations of the proposed

methodology.

8.2 Problems Statement

Under the adverse scenarios of failure or malicious temper of the actuators, the dynamics of

the system can be modeled by

x(k + 1) = Ax(k) +BM\Au(k), (8.1)

where x(k) ∈ Rn is the state of the system, u(k) ∈ Rp is the input signal exerted by the

actuators, and k ∈ N denotes the time instance. The matrix A ∈ Rn×n, which is referred to

as the system dynamics’ matrix, describes the coupling between state variables. In addition,

BM\A consists of the subset of columns with indices in M\A, the set M = 1, . . . , p is the

set of inputs’ labeling indices and A the set of indices of malfunctioning actuators. Therefore,

an extra set of actuators should be in place to ensure that it is still possible to control the

system if some inputs fail. By identifying the system (8.1) with the pair (A,BM\A), we aim

to ensure that this pair is controllable, so the rMCP can be posed as follows.

P: Given a dynamics’ matrix A ∈ Rn×n and the number of possible input failures s, determine

the matrix B∗ ∈ Rn×(s+1)n such that

82

B∗ = arg minB∈Rn×(s+1)n

‖B‖0 (8.2)

s.t. (A,BM\A) is controllable,

|A| ≤ s, A ⊂M,

where M ⊂ 1, . . . , n are the indices of the non-zero columns of the matrix B. Notice

that the dimension of B is n× (s+ 1)n, to ensure that a solution always exist. In particular,

in the worst case scenario the matrix B that concatenates s times the identity matrix is a

feasible solution. In practice, only the non-zero columns of B matter, which we refer to as

effective inputs. Notice that when s = 0, we recover the MCP problem, so we first provide

the solution to the MCP, which we later extend to provide the characterize the solution to

the rMCP.

The main assumptions in this chapter are as follows:

Assumption 1: The dynamics’ matrix is simple, i.e., all the eigenvalues of A are distinct.

Observe that Assumption 1 is not very restrictive since there are several applications where

A satisfy this assumption. For example, dynamical systems modeled as random networks

of the Erdos-Renyi type [Tao and Vu, 2014], as well as known dynamical systems used as

benchmarks in control systems engineering [Ogata, 2001,Siljak, 1991,Siljak, 2007].

Assumption 2: A left-eigenbasis of A is available, i.e., the eigenbasis consisting of left-

eigenvectors of A.

The second assumption is required by technical reasons, since an eigenbasis is determined

using numerical methods. Therefore, in practice, it may be composed of approximated eigen-

vectors to a given floating-point error – see Section 8.4.1 for further discussion.

8.3 Preliminaries and Terminology

In this section, we use introduce some basic concepts of computational complexity required

to characterize the rMCP using the following NP-hard problem.

Definition 8.3.1 ([Chekuri et al., 2012]). (Minimum Set Multi-covering Problem) Given a set

ofm elements U = 1, 2, . . . ,m referred to as universe, a collection of n sets S = S1, . . . ,Sn,with Sj ⊂ U , with j ∈ 1, . . . , n,

⋃nj=1 Sj = U , and a demand function d : U → N that

indicates the number of times an element i needs to be covered. In other words, d(i) is

the minimum number of sets in S that need to be consider such that i is member of all

of this sets. The minimum set multi-covering problem consists of finding a set of indices

J ∗ ⊆ 1, 2, . . . , n corresponding to the minimum number of sets covering U , where every

83

element i ∈ U is covered at least d(i) times, i.e.,

J ∗ = arg minJ⊆1,2,...,n

|J |

s.t. |j ∈ J : i ∈ Sj| ≥ d(i) .

In particular, if d(i) = 1 for all i ∈ 1, . . . , n, then we obtain the minimum set covering

problem.

The minimum set multi-covering problem plays a double role in this chapter: (i) we reduce

the rMCP to a minimum set multi-covering problem; and (ii) by polynomially reducing [Garey

and Johnson, 1979] it to the rMCP, we show the latter to be NP-hard. Such reduction is useful

to determine the qualitative complexity class a particular problem belongs to, see [Garey and

Johnson, 1979] for an introduction to the topic.

8.4 Robust Minimum Controllability Problem

In this section, we propound the main results of this chapter. First, notice that when there

are no input failures (i.e., s = 0) in the rMCP, we recover the MCP problem. Therefore, we

first provide the solution to the MCP, which we later extend to provide the characterize the

solution to the rMCP.

To obtain the solution to the MCP, we perform the following two steps: (i) we polynomial

reduce the structural optimization problem in (8.3) to a set-covering problem using Algo-

rithm 4, and (ii) we determine a numerical parametrization of an input matrix with a specific

input structure in a deterministic polynomial fashion, by solving (8.4). Simply speaking, by

performing these two steps, we are ‘decoupling’ the discrete and continuous properties of the

MCP without losing optimality. In fact, in Theorem 8.4.2, we provide a generic characteriza-

tion of the solutions to the MCP, and a particular instance can be found using Theorem 8.4.6.

Next, we design a similar procedure to that used to solve MCP to obtain the solution to

the rMCP, which we show to be NP-hard (Theorem 8.4.8). Specifically, we determine the

sparsity of an input matrix, by polynomially reducing the problem to a minimum set multi-

covering problem (see Theorem 8.4.9), which is later used to characterize the solutions to the

rMCP (Theorem 8.4.11).

Complementary to the solutions to the MCPs, in what follows, we show that (under

Assumption 1) the decision versions of the rMCP is NP-complete (Theorem 8.4.12). Subse-

quently, we provide a polynomial approximation algorithm (see Algorithm 5), which solution

is feasible (see Theorem 8.4.15) and has sub-optimality guarantees (see Theorem 8.4.16).

Finally, in Section 8.4.1, we explore numerical implications of waiving Assumption 2.

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Let us start by considering the MCP and only one input, i.e., instead of an input matrix

B, we only consider an input vector b. The first set of results provides necessary conditions

on the structure that an input vector b must satisfy to ensure that (A, b) is controllable, and

a polynomial complexity procedure (Algorithm 4) that reduces the problem of obtaining such

necessary structural patterns to a minimum set covering problem.

Lemma 8.4.1. Given a collection of non-zero vectors vjj∈J with vj ∈ 0, ?n, the procedure

of finding b∗ ∈ 0, ?n such that

b∗ = arg minb∈0,?n

‖b‖0

s.t. vj · b 6= 0, for all j ∈ J(8.3)

is polynomially (in |J | and n) reducible to a minimum set covering problem with universe

U and a collection S of sets by applying Algorithm 4.

Proof. Consider the sets S and U obtained in Algorithm 4. The following equivalences hold:

let I ⊂ 1, · · · , n be a set of indices and bI the structural vector whose i-th component is

non-zero if and only if i ∈ I. Then, the collection of sets Sii∈I in S covers U if and only

if ∀j ∈ J , ∃k ∈ I such that j ∈ Sk, which is the same as ∀j ∈ J , ∃k ∈ I such that vjk 6=0 and bk 6= 0 , this can be rewritten as ∀j ∈ J , ∃k ∈ I such that vjk bk 6= 0 and therefore

∀j ∈ J vj · b 6= 0. In summary, bI is a feasible solution to the problem in (8.3). In addition,

it can be seen that by such reduction, the optimal solution b∗ of (8.3) corresponds to the

structural vector bI∗ , where Sii∈I∗ is the minimal collection of sets that cover U , i.e., I∗

solves the minimum set covering problem associated with S and U . Hence, the result follows

by observing that Algorithm 4 has polynomial complexity, namely O(max|J |, n3).

Next, we show that given the structure obtained in Lemma 8.4.1, almost all possible real

numerical realizations lead to a vector b ∈ Rn that is a solution to the MCP.

Theorem 8.4.2. Let vii∈J to be the set of left-eigenvectors of A, and b a solution to (8.3).

Then, almost all numerical realizations b of b are solutions to the MCP.

Proof. The proof follows by showing that if vii∈J with countable J such that vi 6= 0 for

all i ∈ J and b a solution to (8.3), then the set Ω = b ∈ Rn : vi · b = 0 for some i ∈J , and b is a numerical instance of b has zero Lebesgue measure. The proof follows similar

steps to those proposed in [Wonham, 1985], but due to the additional sparsity constraint we

devise an independent proof. Let vii∈J , with countable J , be given and let b be a solution to

problem (8.5). For b ∈ Rn, the equation vi · b = 0 represents a hyperplane Hi ⊂ Cn (provided

85

Algorithm 4 Polynomial reduction of the structural optimization problem (8.3) to a set-

covering problem

Input: vjj∈J , a collection of |J | vectors in 0, ?n.

Output: S = Sii∈1,...,n and U , a set of n sets and the universe of the sets, respectively.

Step 1. set Si = for i = 1, . . . , n

Step 2. for j = 1, . . . , |J |for i = 1, . . . , n

if vji 6= 0 then

Si = Si ∪ j;end if

end for

end for

Step 3. set S = S1, . . . ,Sn and U =⋃n

i=1 Si.

vi 6= 0 for all i), thus the equation vi · b 6= 0 defines the space Cn \Hi. Therefore, the set of b

that satisfies vi ·b 6= 0 for all i ∈ J , is given by⋂i∈J

(Cn \ Hi

)= Cn\

( ⋃i∈JHi)

and the set Ω of

values which does not verify the equations is the complement, i.e.,

(Cn \

⋃i∈JHi)c

=⋃i∈JHi,

which is a set with zero Lebesgue measure in Cn, since |J | is countable.

Now, if vii∈J is taken to be the set of left-eigenvectors of A and b the corresponding

solution to problem (8.5), each member of the set Ω constitutes a solution to (8.5) and hence

the MCP. Since, by the preceding arguments, Ω has Lebesgue measure zero in Cn, it follows

readily that almost all numerical instances of b are solutions to the MCP.

Remark 8.4.3. The generic properties that characterize structural controllability [Dion et al.,

2003] imply that almost all parameters of both dynamics and input matrices satisfying a

given structural pattern are controllable. Although, in Theorem 8.4.2 the dynamics’ simple

matrix A is fixed, i.e., a numerical instance with specified structure, density arguments are

provided to the numerical realizations of the input vector with certain structure that ensure

controllability of the system.

Although Theorem 8.4.2 ensures that almost all parameterizations provide a feasible solu-

tion to the MCP, we need to determine one parameterization that guarantees controllability,

which can be determined by solving the following optimization problem.

B∗ = arg minB∈Rn×m

0

Bl,k = 0 if Bl,k = 0, l, k = 1, . . . , n.(8.4)

Remark 8.4.4. In fact, suppose the objective function in the optimization problem (8.4) is

given by f(B). Then, this can be chosen to satisfy additional design constraints. For instance,

86

f(B) = cᵀB1, where c could capture an actuation cost, i.e., entry ci captures how desirable is

to actuate xi, and 1 is a vector of ones with appropriate dimensions. Subsequently, one may

need additional constraints such that the total actuation budget r available is bounded, for

instance, |f(B)| ≤ r and Bi,j ≥ 0 to avoid negative entries that will restrain the objective goal.

Alternatively, f(B) can also be considered to be nonlinear, while capturing control-theoretic

properties; in particular, it can be a function of the controllability Grammian [Pasqualetti

et al., 2014], with some appropriate constraints to ensure the problem to be well defined.

Next, we show that the (sparsest) pattern given by Lemma 8.4.1 with the optimization

problem (8.4) leads to a numerical realization that is a solution to the MCP.

Lemma 8.4.5. Given vii∈J with vi ∈ Cn, the procedure of finding b∗ ∈ Rn that is a solution

tob∗ = arg min

b∈Rn‖b‖0

s.t. vi · b 6= 0, for all i ∈ J ,(8.5)

is polynomially (in |J | and n) reducible (by Algorithm 4) to a minimum set covering

problem, with numerical entries determined using the optimization problem (8.4).

Proof. By Lemma 8.4.1, given vii∈J , problem (8.5) is polynomially (in |J | and n) reducible

to a minimum set covering problem. Now, given a solution b to (8.3), the optimization

problem (8.4) can be used to obtain a numerical instantiation b with the same structure as

b such that vi · b 6= 0 for all i ∈ J , which incurs polynomial complexity (in |J | and n).

Furthermore, it is readily seen that any feasible solution b′ to (8.5) satisfies ‖b′‖0 ≥ ‖b‖0 =

‖b‖0. Hence, b obtained by the above recipe is a solution to (8.5) and the desired assertion

follows by observing that all steps in the construction have polynomial complexity (in |J | and

n).

Now, we state one of the main results of the chapter.

Theorem 8.4.6. The solution to the MCP can be determined by first identifying the sparsity

of the input vector as in Lemma 8.4.1, followed by determining the numerical realization of

the non-zero entries as in Lemma 8.4.5.

Proof. The proof follows by invoking the PBH eigenvector test. The left-eigenbasis is available

by Assumption 1, the problem in (8.5) is a restatement of the MCP.

Next, based on the previous solution to the MCP, we extend the result to find a dedicated

solution to the MCP.

Theorem 8.4.7. Let b ∈ Rn be a solution to the MCP as described in Theorem 8.4.6, b its

sparsity and N ⊂ 1, . . . , n the indices where b is non-zero, i.e., N = i : bi = ?, and i =

87

1, . . . , n. If B ∈ 0, ?n×n has exactly one non-zero entry in the i-th row, where i ∈ N ,

then the output B ∈ Rn×n of (8.4), when B and the left-eigenbasis of A are considered, is a

solution to the MCP.

Proof. The feasibility of the solution is ensured by proceeding similarly to Theorem 8.4.2,

when the left-eigenbasis of the dynamics’ matrix is considered to invoke the PHB eigenvector

criterion. The optimality follows similar steps to those presented in Lemma 8.4.5.

Before characterizing the solutions to the rMCP, we notice that this problem is computa-

tionally challenging. Specifically, we obtain the following result which follows from noticing

that a particular instance of the rMCP is the MCP (an NP-hard problem).

Theorem 8.4.8. The rMCP is NP-hard.

Therefore, without incurring in additional computational complexity and similar to the

reduction proposed from MCP to the set covering problem, we can characterize the dedicated

solutions to the rMCP as follows.

Theorem 8.4.9. Let v1, . . . , vn be a left-eigenbasis of A, and s the number of possible

input failures. Further, consider the set multi-covering problem (S1, . . . ,S(s+1)n, U ≡1, . . . , n; d), where the demand is d(i) = s+1 for i ∈ U , and Sk = j : [vj ]l 6= 0, and l−1 = k

mod n for k ∈ K ≡ 1, . . . , (s+ 1)n. Then, the following statements are equivalent:

(i) M∗ is a solution to the set multi-covering problem (S1, . . . ,S(s+1)n,U ≡ 1, . . . , n; d);

(ii) Bn(M∗) is a dedicated solution to rMCP, where [Bn(M∗)]i,l = 1 for l = i mod n and

i ∈M∗ ⊂ K, and zero otherwise.

Proof. First, we observe that, by construction of the sets S1, . . . ,S(s+1)n and the demand

function d(i), for i ∈ 1, . . . , n, there exists always s + 1 entries matching every non-zero

entry of the vectors in a left-eigenbasis. This implies that if at most s sensors fail, at least

one entry of a column c of B is such that for each left-eigenvector v.c 6= 0, implying viᵀB 6= 0

for i ∈ 1, . . . , n. Hence, the system is controllable by the PBH eigenvector test, and we

have a feasible solution. Now we need to show that the solution is optimal, i.e., there is not

another solution with less dedicated inputs to the rMCP. We will proceed by contradiction,

so assume that there is a solution to a demand function d(i) = w for i ∈ 1, . . . , n and some

w < s + 1. Then, for some entry of a left-eigenvector v it is only ensured the existence of w

columns in B whose inner product is not zero. Therefore, if w dedicated inputs fails, i.e., the

corresponding columns of B are now zero, then B is such that vᵀB = 0, for some eigenvector

v. Thus, contradicting the assumption that there is a sparser solution to the rMCP.

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Remark 8.4.10. A matrix Bn(M′) described by the concatenation of (s+ 1) solutions to the

MCP achieves feasibility to the rMCP, but it is not necessarily an optimal solution – see

Section 8.5 for a counterexample.

Next, we characterize the solutions of the rMCP, i.e., not only the ones that are dedicated.

Towards this goal, we introduce the following merging procedure. Let two distinct effective

inputs i and j, associated with two non-zero columns of the input matrix, bi and bj , be such

that they do not share non-zero entries k, i.e., [bi]k 6= [bj ]k for k ∈ 1, . . . , n. These two

inputs are said to be merged into one input bi′, where [bi

′]k = [bi]k when [bi]k 6= 0, and

[bi′]k = [bj ]k when [bj ]k 6= 0, for k ∈ 1, . . . , n. Further, we implicitly assume that bi

′takes

the place of bi, and bj is set to zero. In other words, the effective input i is associated with

bi′

and the effective input j is discarded.

Theorem 8.4.11. Let Bn(M∗) ∈ Rn×(s+1)n be a dedicated solution to the rMCP as described

in Theorem 8.4.9. In addition, let B ∈ 0, ?n×(s+1)n be the sparsity of the matrix resulting

of the merging procedure between any of the effective inputs in Bn(M∗). Then, the matrix

B ∈ Rn×n obtained using the optimization problem (8.4), with B and the left-eigenbasis of A,

is a solution to the rMCP.

Proof. The proof follows similar steps to those presented in Theorem 8.4.7. In particular,

recall the merging procedure, and the guarantees obtained in Theorem 8.4.9.

Although we reduced the rMCP to a set multi-covering problem, it is interesting to notice

that these are ‘equivalent’ in the sense that the decision version of the rMCP is NP-complete.

Theorem 8.4.12. The MCP and rMCP are NP-complete.

Proof. From [Olshevsky, 2014], we have that the MCP is NP-hard, and, in particular, the

minimum set covering problem can be polynomially reduced to it. Therefore, we just need to

show that the MCP assuming that A comprises only simple eigenvalues and the left-eigenbasis

is known, i.e., under our assumptions, can be reduced polynomially to the minimum set

covering problem.

To this end, note that, given the set vii∈J of left-eigenvectors of A, the MCP is equivalent

to problem (8.5), the latter being polynomially (in |J | and n) reducible to the minimum set

covering problem (see Lemma 8.4.5). Since |J | = n, the overall reduction to the minimum

set covering problem is polynomial in n.

Similar arguments hold for the rMCP. It was shown to be NP-hard, in Theorem 8.4.8, and

a reduction to the minimum set multi-covering problem can be obtained by Theorem 8.4.9.

Therefore, from Theorem 8.4.12, we have the following observation.

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Remark 8.4.13. A solution of the MCP almost always coincides with a numerical realization

of a solution to the associated minimal structural controllability. Combining this with the

fact that the MCP is NP-complete when the eigenvalues of A are simple (see Theorem 8.4.12),

it follows that the set of simple dynamics’ matrices that lead to NP-complete problems has

zero Lebesgue measure.

Also, we notice that if a problem is NP-hard, then it does not mean that all instances are

not polynomially solvable; notwithstanding, these can be solved exactly [Hua et al., 2009,Hua

et al., 2010].

Remark 8.4.14. The NP-completeness, stated in Theorem 8.4.12, allows us to consider the

subclasses of the set multi-covering problem that are known to be polynomially solvable, to

identify polynomially solvable subclasses of the rMCP. This enables a new characterization of

solutions to the question posed in [Aguilar and Gharesifard, 2014], regarding the existence of

polynomial algorithms to determine controllable graph structures.

Additionally, by the construction proposed in Theorem 8.4.9 and the result in Theo-

rem 8.4.12, if the set multi-covering problem obtained possess additional structure, then this

can be leveraged to use polynomial algorithms to approximate the solutions with close-to-

optimal solutions (see Algorithm 5).

Furthermore, Algorithm 5 leverages the submodularity properties [Bach, 2011] of the set

multi-covering properties to obtain a dedicated solution to the rMCP. Submodularity proper-

ties ensure that the associated polynomial greedy algorithms have sub-optimality guarantees

while performing well in practice [Bach, 2011]. Subsequently, we can obtain the following

result.

Theorem 8.4.15. The matrix Bn(M′) obtained using Algorithm 5, with B and the left-

eigenbasis of A, is a feasible solution to the rMCP. Further, the computational complexity of

Algorithm 5 is O(sn), and it ensures an approximation optimality bound of O(log n).

Proof. Algorithm 5 terminates when each element of the universe set U is covered s+ 1 times

(steps 4-5) by the sets of the set multi-covering problem indexed by J . In other words,

it terminates when we obtain a solution to the set multi-covering problem. By designing

Bn(M′), withM′ = J , we build a matrix that corresponds to dedicated inputs. Thus, using

Theorem 8.4.9, since J is a solution to the set multi-covering problem, then Bn(M′) is a

dedicated solution to the rMCP.

First, notice that the output of Algorithm 5, i.e., Bn(M′), is a feasible solution since the

algorithm stops when each of the elements in the universe of the set multi-cover is s+ 1 times

covered.

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Algorithm 5 Approximate Solution to the rMCP

Input: Left-eigenbasis v1, . . . , vn associated with A ∈ Rn×n and the number s of possible

input failures.

Output: Dedicated solution Bn(M′) ∈ Rn×(s+1)n.

Step 1. Let S1, . . . ,S(s+1)n, where Sk = j : [vj ]l 6= 0, and l − 1 = k mod n for k ∈ K ≡1, . . . , n(s+ 1).Step 2. set U i = ∅, with i = 1, . . . , s . denote the indices in U that are covered i times and

the indices of the sets covering them, respectively.

Step 3. set J = ∅Step 4. for i = 1, . . . , s+ 1

set U i = k : |k ∈ U : k ∈ Sj , j ∈ J | ≥ i . the indices that are already covered by at

least i sets

Step 5.while U i 6= Uselect Sj with largest number of indices in U \ U i

set J ← J ∪ jset U i ← U i ∪ Sjend while; end for

set M′ ← JStep 6. set Bn(M′), where [Bn(M′)]i,l = 1 for

l = i mod n and i ∈M′ ⊂ K, and zero otherwise.

The computational complexity of Algorithm 5 is obtained by the overall complexity of

steps 1, 4 and 5. In step 1, we need to compute (s + 1)n sets, in step 5 we need to consider

at most n sets, and, in step 4, (s + 1) iterations are performed, each with the number of

steps of step 5, yielding (s + 1)n computational steps. Summing up the complexity of each

step, Algorithm 5 has, in the worst case, complexity of order O(sn). In addition, notice that

the performance attained in a multi-set covering problem is the same as in the rMCP, as a

consequence of Theorem 8.4.12. Furthermore, the solution obtained incurs in an optimality

gap of at most O(log n) since the algorithm implements the greedy algorithm associated with

submodular functions, as it is the case of the multi-set covering problem, and the result

follows.

Finally, by invoking Theorem 8.4.11 and Theorem 8.4.15, we obtain the following result.

Theorem 8.4.16. Let Bn(M′) ∈ Rn×(s+1)n be a dedicated solution to the rMCP as described

in Theorem 8.4.15. In addition, let B ∈ 0, ?n×(s+1)n be the sparsity of the matrix resulting

of the merging procedure between any of the effective inputs in Bn(M′). Then, the matrix

B ∈ Rn×n obtained using the optimization problem (8.4), with B and the left-eigenbasis of A,

91

achieves feasibility to the rMCP and is computed in polynomial time.

8.4.1 Numerical and Computational Remarks

Now, for the sake of completeness, we discuss the implications of waiving Assumption 2 and

the impact on the input vector in the MCP. The results readily extend to the general solution

to the rMCP. Towards this goal, we need the following result.

Theorem 8.4.17 ([Pan and Chen, 1999]). Let A ∈ Cn×n be a matrix with simple eigenvalues.

The deterministic arithmetic complexity of finding the eigenvalues and the eigenvectors of A

is bounded by O(n3)

+ t (n,m) operations, where t(n,m) = O((n log2 n

) (logm+ log2 n

)),

for a required upper bound of 2−m‖A‖ on the absolute output error of the approximation of

the eigenvalues and eigenvectors of A and for any fixed matrix norm ‖ · ‖.

More precisely, Theorem 8.4.17 states that in practice, only a numerical approximation

of the left-eigenbasis is possible in polynomial time. In this case, let ε = 2−m‖A‖ be as in

Theorem 8.4.17, then the results stated in Lemma 8.4.1 and Lemma 8.4.5 (see also Algorithm 4

and the optimization problem (8.4)) can only be used in an ε-approximation of the left-

eigenbasis of the dynamics’ matrix. Therefore, the ε-approximation of the left-eigenbasis may

lead to the following issues:

(i) an entry in the left-eigenvector is considered as zero, where in fact it can be some non-

zero value that (in norm) is smaller then ε. Consequently, the sets generated using Algorithm 4

(see also Lemma 8.4.1) do not contain the indices associated with those non-zero entries. Thus,

additional sets need to be considered to the minimum set covering, which implies that the

structure of the input vector may contain more non-zero entries than the sparsest input vector

that is a solution to the MCP. In other words, we obtain an over-approximation of the sparsest

input vector that is a solution to the MCP.

(ii) an entry of the ε-approximation in a left-eigenvector of the left-eigenbasis is non-zero.

Then, it does not represent an issue when computing the structure of the input vector as

described in Lemma 8.4.1 (see also Algorithm 4), but it can represent a problem when deter-

mining the numerical realization by resorting to the optimization problem (8.4). Nonetheless,

by Theorem 8.4.2 it follows that such issue is unlikely to occur.

To undertake a deeper understanding of which entries fall in the first issue presented above,

several methods to compute eigenvectors can be used and solutions posteriorly compared,

see [Demmel et al., 2000] for a survey on different methods and computational issues associated

with those.

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8.5 Illustrative Examples

To illustrate the first main result of this chapter, to find a solution to the MCP, consider the

dynamics’ matrix A

A =

6 −3 3 2 −1

0 8 0 0 0

4 3 7 2 1

0 0 0 6 0

−4 −3 −3 −2 3

, (8.6)

where σ(A) = 2, 4, 6, 8, 10 consists of distinct eigenvalues, so the matrix A is simple and

our results are applicable. Consequently, to obtain the solution to the MCP, we first compute

the left-eigenvectors of A that are as follows:

v1 = [ 1 1 0 0 1 ]ᵀ, v2 = [ 0 0 1 0 1 ]ᵀ, v3 = [ 0 0 0 1 0 ]ᵀ,

v4 = [ 0 1 0 0 0 ]ᵀ and v5 = [ 1 0 1 1 0 ]ᵀ.

Using Algorithm 4, since vi for i = 1, . . . , 5, we obtain Sjj=1,...,5, where the j-th set corre-

sponds to the set of indices of the left-eigenvector which have a non-zero entry on the j-th

position. In particular, we obtain

S1 = 1, 5 ,S2 = 1, 4 ,S3 = 2, 5 ,S4 = 3, 5 ,S5 = 1, 2 ,

and the universe set is given by U = 1, 2, 3, 4, 5 . Now, it is easy to see that a solution to

this minimum set covering problem is the set of indices I∗ = 2, 3, 4, since U = S2 ∪ S3 ∪ S4

and there is no pair of sets, i.e., I ′ = i, i′ with i, i′ ∈ 1, . . . , 5 such that U = Si ∪ Si′ .Therefore, a possible structure of the vector b that is a solution to the MCP is

b = [ 0 ? ? ? 0 ]ᵀ. (8.7)

Additionally, to find the numerical parametrization of b, under the sparsity pattern of b,

we have to solve the following system with three unknowns: b2, b3, b4 6= 0 and b3 + b4 6= 0. By

inspection, a possible choice is b = [ 0 1 1 1 0 ]ᵀ, but the numerical parametrization can

be obtained by invoking the optimization problem (8.4), with the set of left-eigenvectors of A

given by vjj∈1,...,5 and the structure of b given by b in (8.7). For the sake of completeness,

we, the controllability matrix is given by

C = [ b Ab A2b A3b A4b ] =

0 2 44 608 7184

1 8 64 512 4096

1 12 120 1176 11520

1 6 36 216 1296

0 −8 −104 −1112 −11264

,

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and the rank(C) = 5, implying that (A, b) is controllable.

Observe that the single-input solution obtained with b = [ 0 1 1 1 0 ]ᵀ, can be

immediately translated into a solution with two effective inputs, by Theorem 8.4.7. In

particular, two possible solutions are B = [ b1 b2 ] with b1 = [ 0 1 1 0 0 ]ᵀ and

b2 = [ 0 0 0 1 0 ]ᵀ, and B = [ b1 b2 b3 ], with b1 = [ 0 1 0 0 0 ]ᵀ, with b2 =

[ 0 0 1 0 0 ]ᵀ and b3 = [ 0 0 0 1 0 ]ᵀ, where the latter is a dedicated solution. Al-

ternatively, if we consider, for instance, B = [ b1 b2 ], with b1 = [ 0 1 0 0 0 ]ᵀ and

b2 = [ 0 0 −1 1 0 ]ᵀ, then vᵀB = 0 for the left-eigenvector v = [ 1 0 1 1 0 ]ᵀ, and

the pair (A,B) is uncontrollable. Thus, as prescribed in Theorem 8.4.7, by the optimization

problem (8.4), one can obtain a new realization of B that ensures controllability of (A,B);

e.g., the same b1, and b2 = [ 0 0 1210 1 0 ]ᵀ.

Notice that a systematic polynomial approximation to the MCP can be obtained by con-

sidering the rMCP with the number of input failures s = 0. By doing so, we obtain the same

sparsity to b, i.e., b, as in the aforementioned example, and the subsequent analysis follows.

We also observe that the approximate solution is a solution to the MCP.

Now, we illustrate how to find a solution to P. Let us apply the developments of Sec-

tion 8.4, when we consider the dynamics’ matrix in (8.6). First, if we consider that at most

one input fails, we use Algorithm 4, where a set multi-covering problem is considered with

the sets as in Section 8.4, universe U = 1, . . . , 5 and with a demand function d(i) = 2

for i = 1, . . . , 5, i.e., each element must be covered twice. Subsequently, by inspection, we

conclude that the sets S2 and S4 need to be considered twice, since the elements 5 and 4

only appear in these sets, respectively. After this, we need to cover the element 2 and to

this end we can choose S3 or S5 or twice one of them, so a possible solution to the multi-set

covering problem is M∗ = 2, 3, 4, 2, 3, 4. Therefore, Bn(M∗) is a solution to the rMCP,

and, in particular, the solution is the same as concatenating twice a dedicated solution to the

MCP, see Remark 8.4.10. Further, Algorithm 5 produces an optimal solution as often occurs

in practice.

In fact, if we apply our results when s inputs are allowed to fail, i.e., d(i) = s + 1 for

i = 1, . . . , 5, we notice that the sets S2 and S4 need to be considered s + 1 times since

the elements 5 and 4 only appear in these sets, respectively. Besides, we need to cover the

element 2, so we can choose either S3 or S5 s + 1 times, which implies that B(M∗), with

M∗ = 2, 3, 4, . . . , 2, 3, 4 where the elements 2, 3 and 4 appear s + 1 times, is a solution.

Similarly, the solution consists of concatenating s+ 1 times a dedicated solution to the MCP,

and the same remarks are applicable, i.e., Remark 8.4.10.

However, the concatenation of s+ 1 solutions to the MCP is not always a solution to the

rMCP when at most s inputs are allowed to fail. Let us consider the dynamics’ matrix and

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associated left-eigenvectors as follows:

A =

4 −2 2

−1 3 1

1 −1 5

and V =

| | |v1 v2 v3

| | |

=

1 0 1

1 1 0

0 1 1

. (8.8)

First, we note that σ(A) = 2, 4, 6, so A is simple, and we can apply our results. Secondly,

the structure of the left-eigenvectors of A is given by v1 = [ ? ? 0 ]ᵀ, v2 = [ 0 ? ? ]ᵀ and

v3 = [ ? 0 ? ]ᵀ. Further, we consider that at most one input failure is likely to occur, i.e.,

s = 1. Then, we can invoke Algorithm 4 to build the sets for the set multi-covering problem,

which are as follows: S = S1,S2,S3, with S1 = 1, 2, S2 = 2, 3 and S3 = 1, 3, and

U =⋃3i=1 Si = 1, 2, 3. By inspection, we obtain thatM′ = 1, 2, 3 is the optimal solution,

where the indices cover each element of U twice. Further, observe that a solution to the

dedicated input MCP always has size equal to two, and in this case, the concatenation of

two solutions lead to a solution that has one more input than the optimal solution obtained.

Observe that this is a small dimensional example that incurs into a solution that is already

33% worst than the optimal. Alternatively, if we apply Algorithm 5 to approximate the

solution to the rMCP, we obtain one that is optimal, i.e., B(M′) where M′ = 1, 2, 3.


In this chapter, we addressed two minimal controllability problems, with the goal of charac-

terizing the input configurations that actuate the minimal subset of variables yielding con-

trollability, under a specified number of failures. The problems explored were shown to be

NP-complete, and a polynomial reduction of these to a set multi-covering problem was pro-

vided. In particular, the strategies followed by us separate the discrete and continues nature

of the minimal controllability problems. Subsequently, we discussed greedy solutions to the

minimal controllability problems that yields feasible (but sub-optimal) solutions to rMCP.

Directions for future research in this line of work include the use of the obtained inputs’

structure and consider methods such as coordinate gradient descent to minimize an energy

cost, and to consider the case where the model is not exactly known. Additionally, it would be

interesting to assess the computational complexity of the rMCP without the assumption on

the spectrum of the dynamics’ matrix, as well as to provide polynomial algorithms to obtain

approximated solutions with suboptimal guarantees.

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Chapter 9

The robust minimal controllability

problem for switched linear

continuous-time systems

Now, we extend the results of Chapter 8 to switched LTI systems. These systems have a set

of discrete modes among which they may switch. Similarly to the previous chapter, the goal

is to encounter the minimal number of actuators and their placement, such that the system

is controllable in the scenario where a specified number of actuators may fail. We solve two

versions of the problem. In the first version, we may have a different actuators’ placement for

each mode, and, in the second version, we aim to find the same actuators’ placement across all

modes that ensure controllability of the system. We present algorithms to solve both versions

of the problem. Moreover, we provide algorithms to approximate the solution of the problem.

However, due to the combinatoric nature of the second version of the problem, we may only

approximate the first version in polynomial time.

Again, we achieve the purpose of ensuring information security in the area of control

systems. We published this work in [Ramos et al., 2018b].

9.1 Introduction

Switched systems are paramount in an extensive number of applications, such as control

of mechanical systems, process control, automotive industry, power systems, aircraft/traffic

control, see for instance [Lin and Antsaklis, 2009, Sun, 2006]. The systems belonging to the

subclass of switched systems whose subsystems are described by linear differential equations

are called switched linear systems. These systems alone consist of a line of research with

growing attention [Lin and Antsaklis, 2009], and several works aim to study the properties

97

of this class such as controllability, observability and reachability [Cheng, 2005, Ji et al.,

2007,Sun, 2006,Sun et al., 2002].

Recent works studied controllability under the scope of uncertain switched linear systems,

where the state matrices’ entries of each mode are only known to be zero or non-zero [Liu et al.,

2013]. A switched linear system is said to be structurally controllable whenever there exists

a numerical realization of the non-zero entries of the state matrices leading to a controllable

switched linear system. In [Ramos et al., 2013], the authors introduced a framework to

model check structural properties of switched linear systems, and propose its use to check

the structural controllability of each subsystem. In [Pequito and Pappas, 2017], the authors

addressed the structural minimal controllability problem for switched linear continuous-time

systems, finding the minimum number of inputs that need to be considered to attain structural

controllability of the system. However, the state matrices’ entries may be linearly dependent,

and the system is structurally controllable but not controllable by the same set of actuators.

In contrast, in this chapter, we propose to address the scenario where we have knowledge

of the state matrices entries and that these matrices are simple. We aim to ensure the

controllability of the system, the ability to drive the system from an initial state to the

desired state, extending the results of Chapter 8, published in [Pequito et al., 2017].

We assume that either we have access to ‘common’ transitions and knowledge of the

existing modes of the switching system, or that the controller is equipped with supervisory

capabilities enabling the system to switch between modes, as considered in same engineering

applications as in [Pequito and Pappas, 2017,Petreczky et al., 2015]. More specifically, given a

switched linear system with continuous time, we address the problem of finding the minimum

number of inputs/actuators and state variables we need to actuate, ensuring the system’s

controllability under two scenarios (when a specified number of inputs may fail):

(i) design an input matrix, for each system’s mode, that controls the system actuating a small

number of state variables;

(ii) design a common input matrix that controls the system actuating a small number of state

variables.

Main contributions of this chapter consist in addressing (i) and (ii) while providing in-

sights on how the obtained solutions can be exploited to improve the computational complexity

of the proposed algorithms. We reduce both problems to the well studied set multi-covering

problem [Chekuri et al., 2012]. Also, we show that (i) is NP-complete when we use a suffi-

cient condition for controllability. These results allow us to use known polynomial complexity

algorithms that approximate the set multi-covering problem, to get approximations for (i)

and (ii). However, due to the combinatorial nature of (ii), only (i) may be approximated in

polynomial time with those approximation algorithms.

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Chapter structure. Section 9.2 states the problems we aim to solve and Section 9.4

illustrates the main results with examples. Finally, Section 9.5 concludes the Chapter.

9.2 Problems Statement

Consider a large-scale dynamical system with dynamics modeled by a switched linear continuous-

time system (SLCS). Conceptually, we can see a switched linear continuous-time system

(SLCS) as a set of linear continuous-time systems (LCS), where each element of the set is

called a mode, together with a set of discrete events that cause the system to switch between

modes. Subsequently, an SLCS for which some actuators may fail, due to either a mali-

cious entity tempering with the actuators or natural phenomena reasons, may be described

as follows:

x(t) = Aσ(t)x(t) +BM\Aσ(t)σ(t) u(t), (9.1)

where σ : R+ → M = 1, . . . ,m is a piecewise switching signal, that only switches once in a

given dwell-time, x(t) ∈ Rn the state of the system, and u(σ(t)) ∈ Rp is a piecewise continuous

input signal. Moreover, BM\Aσ(t)σ(t) consists of the subset of columns with indices inM\Aσ(t),

the set M = 1, . . . , p is the set of inputs’ labeling indices and Aσ(t) the set of indices of

affected (i.e., malfunctioning) inputs, for each mode σ(t). Additionally, as discussed in the

introduction, we focus on the scenario where we have the knowledge of the switching signal,

as well as dwell-time, as in [Pequito and Pappas, 2017,Petreczky et al., 2015] and references

therein.

To ease the notation, we refer to the system in (9.1) by the pair (Aσ(t), BM\Aσ(t)σ(t) ). Each

mode of the system corresponds to the time interval where the switching signal is constant,

σ(t) = i and i ∈ 1, . . . ,m. In other words, it corresponds to an LCS system, which we

denote by the pair (Ai, BM\Aii ). It is worth noticing that in each mode the dynamics’ matrix

could have a different dimension. For instance, we may want to model a power system such

that some of its components (e.g., the generators) may be working, depending on the mode

the system is in. Hence, for a mode where some generators are not working, the dynamics’

matrix may be designed with a small number of state variables. Moreover, we can include

this behavior taking n as the maximum of the dimensions of each mode’s dynamics’ matrix

Ai, i ∈ M, and assuming fixed the system variables’ order. Hence, when a variable does not

play a role in a mode, its dynamics’ matrix has zeros in the respective row and column.

Furthermore, from a systems’ engineering perspective, we often want to ensure that the

systems possess properties such as controllability. The SLCS (9.1) is controllable if, for any

initial state x(0) = x0 and any desired state xd there exists a time instance tf > 0, a switching

signal σ : [0, tf [→ M and an input u : [0, tf [→ Rp, s.t. x(tf ) = xd. In other words, we

99

can always design a control law that drives the system from an initial state to any desired

state in a finite amount of time. Thus, for each mode, an extra set of actuators must be

in place to ensure that the system is still controllable if some inputs fail. Besides, due to

economic restrictions, i.e., since more actuation capabilities incur in higher cost, it is of utmost

importance to deploy the minimum number of actuators that can still control (9.1), whenever

some specified maximum number of actuators may fail for each mode as in (9.1). We refer

to this problem as robust minimal controllability problems for SLCS (rMCPS). Subsequently,

given the system (9.1), the rMCPS can be posed as follows:

Problem Statement 1. Determine matrices B1 ∈ Rn×(s1+1)n, . . . , Bm ∈ Rn×(sm+1)n that

are a solution to the minimization problem

minB1,...,Bm

m∑i=1

‖Bi‖0 (9.2)

s.t. (Ai, BM\Aii ) is controllable for all i ∈M, and

|Ai| ≤ si and Ai ⊂M for all i ∈M,

where the dimension of Bi is n × (si + 1)n to guarantee that there exists a solution to the

problem.

Similarly, we can model the case where we want to design a common input matrix, i.e.,

Bi = Bj = B for all i, j ∈ M, that controls the SLCS and for which a certain number of

inputs may fail, as finding an input matrix BM\A such that

x(t) = Aσ(t)x(t) +BM\Au(t) (9.3)

is controllable. Notice that A can be seen as Aσ(t), i.e., the inputs that may fail are the same

across dynamic’s switching. Therefore, another problem we are interested in is as follows:

given a common actuator placement, i.e., (9.3), the common rMCPS (crMCPS) consists in

the following problem.

Problem Statement 2. Determine matrix B from the minimization problem

minB∈Rn×(s+1)n

r ‖B‖0 (9.4)

s.t. (Ai, BM\A) is controllable for all i ∈M, and

|A| ≤ s and A ⊂M for all i ∈M,

where the dimension of B is n× (s+ 1)n to assure that there exists a solution to the problem.

Note that, concatenating s times the identity matrix results in an input matrix that is

a feasible solution to both problems, where the only relevant columns of B are the non-zero

100

ones. Although problems (9.1) and (9.3) seem to be very similar, the proposed solutions are

quite diverse, and in fact, they exhibit different computational complexity issues.

Additionally, to solve Problem 1 and Problem 2, we require two technical assumptions

that we now detail.

Assumption 1. The dynamics’ matrix of each mode, Ai with i ∈ M, is simple, i.e., Ai has

distinct eigenvalues.

Note that, many applications have dynamics’ matrices satisfying Assumption 1, e.g., dy-

namical systems modeled as random networks of the Erdos-Renyi type [Tao and Vu, 2016], or

benchmark dynamical systems in control system engineering [Ogata and Yang, 1970, Siljak,

2011].

Assumption 2. A left-eigenbasis of Ai is known for each mode i ∈ M (the set of the left-

eigenvectors of Ai).

Assumption 2 is a technical restriction. In general, the left-eigenbasis is acquired by

numerical methods and, hence, we obtain approximated eigenvectors up to a floating-point

error.

9.3 The Robust Minimal Controllability Problem

In this section, we present the main results of the chapter. We address problem rMCPS (9.2)

and crMCPS (9.4), by ‘decoupling’ the problems into their discrete and continuous optimiza-

tion properties. We start by identifying the structure of the solutions and, after, a numerical

realization of them that ensures controllability under the possible input failures. We introduce

the minimum set multi-covering problem, that we use in Algorithm 6 to build a solution to the

rMCPS and the crMCPS. Then, we find a dedicated solution to the rMCPS, in Theorem 9.3.6,

that is used together with Algorithm 8 to find a general solution to the problem, in Theo-

rem 9.3.7. A solution to the crMCPS is constructed, in Corollary 9.3.8, using Algorithm 7.

Finally, we show that the rMCPS is NP-complete, in Theorem 9.3.9.

We start by noticing that the Popov-Belevitch-Hautus (PBH) eigenvalues controllability

test gives us a sufficient controllability condition for SLCS (9.1) or (9.3).

Proposition 9.3.1. Given an SLCS, if for each mode i ∈M the (Ai, Bi) is controllable, then

the SLCS is controllable.

Hence, Proposition 9.3.1 bestow a polynomial method (in m and n) to check a sufficient

condition for the controllability of an SLCS. In other words, for each mode i ∈ M, and for

each eigenvalue λ ∈ σ(Ai), we only need to compute the rank of [Ai − λIn Bi].

101

However, this criterion does not inform about which entries of each Bi should be non-zero,

as well as with which particular values, to ensure the rank condition. That is, we can verify

in polynomial time that each Bi is a solution. Notwithstanding, we notice that the rMCPS is

computationally challenging to solve since a particular instance of the rMCPS, i.e., when (9.1)

has one mode, we get the minimal controllability problem (MCP) that is known to be NP-

hard, we can polynomially reduce an NP-hard problem to this problem, recall Theorem 8.4.8.

Thus, the rMCPS is as difficult as the latter, which leads to the following result.

Corollary 9.3.2. The rMCPS (9.2) and the crMCPS (9.4) are NP-hard.

Instead of a naıve usage of the PBH eigenvalue test that leads to a strictly combinato-

rial procedure for solving the SLCS, we may consider the PBH test for controllability using

eigenvectors, which allow us to design a sufficient condition for the controllability of an SLCS.

Proposition 9.3.3. Given (9.1), if for each mode i ∈ M and for each left-eigenvector v of

Ai we have that v†Bi 6= 0, then the system is controllable.

Proposition 9.3.3 plays a central role in this chapter’s main results.

As previously mentioned, we first address the discrete part, which requires us to introduce

the following constructs.

Definition 9.3.4. (Minimum Set Multi-covering Problem [Chekuri et al., 2012]) Given a

universe with m elements, U = 1, 2, . . . ,m, a collection of n sets S = S1, . . . ,Sn, where

Sj ⊂ U , for j ∈ 1, . . . , n, s.t.⋃nj=1 Sj = U , and a demand function d : U → N that imposes

the number of times the element i needs to be covered. The minimum set multi-covering

problem consists in finding a smallest set of indices I∗ ⊆ 1, . . . , n s.t. ∪j∈ISj = U and

every element i ∈ U is covered, at least, d(i) times, i.e.,

J ∗ = arg minJ⊆1,...,n

|J |

s.t. |j ∈ J : i ∈ Sj| ≥ d(i) .

A particular case of the problem in Definition 9.3.4 is when each element needs to be cov-

ered once, d(i) = 1 for all i ∈ U , called the minimum set covering problem, see [Chekuri et al.,

2012]. These two problems are ubiquitous in the fields of combinatorics, computer science,

and complexity theory. They are NP-complete problems for which efficient approximation

algorithms are known and well studied [Vazirani, 2013].

These problems are particularly useful to leverage the controllability criterion using the

PHB criterion to ensure the feasibility of the sparsest input. In particular, for the LTI case,

we have the following approach.

102

Lemma 9.3.5 ([Pequito et al., 2017]). Given a non-empty collection of non-zero vectors

vjj∈J , with vj ∈ Rn, the procedure of finding b∗ ∈ Cn that is a solution to

b∗ = arg minb∈Rn

‖b‖0

s.t. vk · b 6= 0, for all j ∈ J(9.5)

is polynomially reducible (in |J | and n) to a minimum set covering problem.

Next, we can build upon this problem and the notion of controllability for the SLCS, and

under the assumptions posed in Section 9.2, to find the sparsest set of vectors that ensure

controllability. Specifically, since we have different modes, the goal is to consider the sparsest

sets of inputs across the different eigenvectors of the left-eigenbasis.

Subsequently, we start by presenting an algorithm (Algorithm 6) that receives a collection

of structural vectors and outputs a setup for a set-covering problem. The sets we build, Sik,have two indices, i and k, the first matches the mode the structural vectors belong to, and

the second ranges from 1 up to the number of such vectors in that mode. A pair (i, j) belongs

to Sik whenever the jth structural vector of mode i is non-zero at index k.

Algorithm 6 Polynomial reduction of the structural optimization problem (9.5), to a set-

covering problem

Input: Consider the eigenvectors of the left eigenbasis of the different modes, vji i∈Mj∈1,...,|Ji|

, and

J =⋃

i∈M Ji.Output: The setup for a set multi-covering problem, S = Sij i∈M

j∈1,...,|Ji|and U , a set with n sets,

and the universe of these sets, respectively.

1: set Sji = ∅, for i ∈M and j ∈ 1, . . . , |Ji|2: for i ∈M

for j = 1, . . . , |J |for k = 1, . . . , n

if [vij ]k 6= 0 then

Sik = Sik ∪ (i, j)3: set S =

Sij

i∈Mj∈1,...,|Ji|

and U =⋃V∈S V.

To make our approach easier to follow, we first characterize the dedicated solutions to the

rMCPS, i.e., the solution where each input actuates a unique state variable.

Theorem 9.3.6. Given a left-eigenbasis, vi1, . . . , vin, for each Ai and for each mode i ∈M, and

given the number of possible input failures for each mode, s1, . . . , sm, consider the multi-set

covering problem (S,U ; d), where

• S =⋃i∈MSi1, . . . ,Si(si+1)n;

103

• U =⋃i∈M Ui, with Ui = (i, 1), . . . , (i, n);

• d(i, j) = si + 1, for (i, j) ∈ U ;

where Sik = (i, j) : [vij ]l 6= 0 and l− 1 = k mod n, for i ∈M and k ∈ K ≡ 1, . . . (si + 1)n.Then, the following conditions are equivalent:

(i) I∗ is a solution to the set multi-covering problem (S,U ; d) and I∗i = (i, a) : (i, a) ∈ I∗;(ii) the set of matrices B1 (I∗1 ) , . . . , Bm (I∗m), where Bi ∈ 0, 1n×(si+1)n, is a dedicated

solution to the rMCPS (9.2), with [Bi(I∗i )]j,k = 1 for k = j mod n and (i, j) ∈ I∗ ⊂ K, and

[Bi(I∗i )]j,k = 0 otherwise.

Proof. By Theorem 8.4.9, Bi(I∗i ) is an optimal feasible dedicated solution to controls mode i,

equivalent to a solution I∗i that covers Ui, for demand d(i, j) = si + 1. By Proposition 9.3.3,

the set of matrices B1 (I∗1 ) , . . . , Bm (I∗m) is a feasible dedicated solution to rMCPS (9.2),

equivalent to a solution of the multi-set covering problem (S,U , d).

Note that the first entry of the pairs constituting the universe of the set multi-covering

problem is what identifies the mode of the input matrix when recovering a solution to the set

multi-covering problem to a solution of the rMCPS (9.2).

The crMCPS (9.4) requires a different approach because we want to design an input

matrix that is equal to every mode. Intuitively, this is computationally more demanding,

since we want to minimize the size of the union of the solutions for each mode, i.e., we want

to actuate a small number of state variables that, across the modes, ensure the system to be

controllable. The first step to solve the problem consists in applying Algorithm 6, to build

the sets of S. The second step is to find a solution for the crMCPS (9.4), using S, U and the

demand function d(i) = s+1. We need to, carefully, choose one solution of indices of the state

variables that we need to actuate, which ensures each mode to be, not only, controllable, but

also robust to s input failures, maximizing the state variables in common across the modes.

To achieve this, we build an algorithm that needs to find all the solutions to several set multi-

covering problems, translating to all possible solution to the crMCPS (9.4) and, afterward,

we need to select one that actuates the smallest number of state variables across all modes.

We summarize this procedure in Algorithm 7 that selects a minimal number of state variables

across all modes that we need to actuate such that Proposition 9.3.3 holds, yielding a solution

to the crMCPS (9.4). Besides, note that Algorithm 7 has worst-case complexity exponential

(in m and n), since each set multi-covering problem may have an exponential number of

solutions.

Now, we go further and characterize the general solutions to the rMCPS (9.2) and the

crMCPS (9.4), not only dedicated ones. We derive the global solutions based on the dedicated

104

Algorithm 7 Find a minimal set of state variables of a problem (9.4) that need to be actuated

Input: An instance of the problem (9.4), a collection of sets S and a universe U , the output of

Algorithm 6 applied to the set of eigenvector for each dynamics matrix Ai, and demand function d,

withvji

i∈M

j∈1,...,nand d(i) = s+ 1, respectively.

Output: A set of state variables’ indices to actuate s.t. the problem instance is controllable,

I ⊆ 1, . . . , n.

1: for i ∈Mset Si = St

k : t = i and k ∈ 1, . . . , nset Ui = k : (i, k) ∈ Ufor j = 1, . . . , n

set Bi as the set of all possible covers for Uiwith the collection of sets Si and demand d

2: set X ∗1 , . . . ,X ∗m = arg minX1∈B1,...,Xm∈Bm|⊔

i∈M Xi|3: set I =

⊔i∈M X ∗i and B = Bii∈M

solutions by combining them. Towards this goal, we propose a merging procedure in Chapter 8

that we summarize in Algorithm 8. The procedure tries to combine in the smallest possible

number of inputs, the entries of the dedicated inputs while ensuring that the PHB eigenvectors

criterion holds. Specifically, Algorithm 8 picks two compatible inputs, i.e., with different

structure and structural inner-product 0. In this procedure, when we combine two compatible

inputs, we set the first one to actuate the variables that both actuate, and we discard the

second one (the respective column is set to zero).

Algorithm 8 Merging procedure

Input: An input matrix B ∈ 0, ?n×m.

Output: The matrix B = [ b1 ... bm ] with inputs merged.

1: while ∃i,j : bi 6= bj 6= 0 and bi · bj = 0

set bi = bi + bj and bj = 0

Up to this point, Algorithm 8 is about the structure of the input matrix B, and we now

build a numerical realization of it. Subsequently, we need to solve the following problem

to perform the second step required to obtain a solution to the rMCPS (9.2) and the crM-

CPS (9.4), i.e., a parametrization B∗ of the structural matrix B (a feasible solution to be

orthogonal to a given set of m = |J | vectors, vjj∈J ):

B∗ = arg minB∈Rn×m

0

s.t. B · vj 6= 0, for all j ∈ J and B has the structure of B.(9.6)

105

As a consequence, we can obtain a solution for the rMCPS (9.2) as described in the

following result.

Theorem 9.3.7. Let Bi(I∗i )mi=1, where Bi ∈ 0, 1n×(si+1)n, be a dedicated solution to the

rMCPS (9.2), obtained with Theorem 9.3.6. Further, let Bimi=1, where Bi ∈ 0, ?n×(si+1)n

be the sparsities of the matrices resulting from the merging procedure, Algorithm 8, between any

of the effective inputs, for each Bi(I∗i ). Then, the set of matrices B∗i mi=1, where each matrix

B∗i ∈ Rn×n is obtained using the optimization task (9.6) for inputs Bi and left-eigenbasis of

Ai is a solution to the rMCPS (9.2).

Proof. By Theorem 9.3.6, then we have a feasible solution to the rMCPS (9.2), and, by

construction, Algorithm 8, we preserve the feasibility by merging only compatible inputs.

Also, by construction, the optimization problem (9.6) ensures that the obtained numerical

realization of the solution’s sparsity verifies the PHB eigenvector criterion. Hence, we obtain

a solution to the rMCPS (9.2).

In fact, the above also applies to the crMCPS (9.4), considering a matrix B, instead of

the set of matrices Bimi=1. Hence, the next result readily follows.

Corollary 9.3.8. Let B(I∗i ), with B ∈ 0, 1n×(s+1)n, be a dedicated solution to the crM-

CPS (9.4), obtained with Theorem 9.3.6. Further, let B ∈ 0, ?n×(s+1)n be the sparsity of

the matrix resulting from the merging procedure, Algorithm 8, between any of the effective

inputs. Then, the matrix B∗ obtained using the optimization task (9.6) for inputs B and

left-eigenbasis of A1, . . . , Am is a solution to the rMCPS.

It is worth noticing that the decision version of the rMCPS (9.2), when using the sufficient

condition of Proposition 9.3.3, is equivalent to the set multi-covering problem and the rM-

CPS (9.2) is NP-complete for that controllability sufficient condition. For the crMCPS (9.4),

since we lose the information about the modes when we build the input matrix B, the deci-

sion version of the problem is not equivalent to the decision version of the set multi-covering

problem when using the sufficient condition of Proposition 9.3.3.

Theorem 9.3.9. By using the sufficient condition for controllability in Proposition 9.3.3, the

rMCPS (9.2) is NP-complete.

Proof. Using Proposition 9.3.3 as the controllability condition, the rMCPS (9.2) is equivalent

to the set covering problem, by Theorem 9.3.6 and Theorem 9.3.7, the result follows.

106


In this section, we illustrate the main results from this chapter. In order to do so, we fix an

SLCS with two modes and dynamics matrices A1 and A2 given by

A1 =

2 0 0 −3 0

0 3 0 0 0

2 1 4 3 0

0 0 0 5 0

−2 −1 −3 −7 1

; A2 =

2 −1 0 −3 0

0 3 0 0 0

2 1 4 3 0

0 0 0 5 0

−2 −1 −3 −7 1

.

The dynamics matrices’ left-eigenvalues are, respectively,

V1 =

| |v1

1 . . . v51

| |

=

0 1 0 1 0

0 0 1 1 0

1 0 0 1 0

1 1 0 0 1

1 0 0 0 0

; V2 =

| |v1

2 . . . v52

| |

=

0 1 0 1 0

0 0 1 0 0

1 0 0 1 0

1 1 0 0 1

1 0 1 0 0

.

The eigenvalues of each mode are σ(A1) = σ(A2) = 1, 2, 3, 4, 5. By design, the spectrum is

equal, but do not need to be, as the only assumption is that the state matrices need to be

simple.

9.4.1 Example I

Next, we explore this example when we consider that no input can fail (s = 0) as an instance

of both the rMCPS (9.2) and the crMCPS (9.4), and when, in both cases, some inputs may

fail.

The rMCPS (9.2) scenario

First, we consider we want to design input matrices for each mode that actuate the minimal

number of inputs that, by Proposition 9.3.3, ensures the system to be controllable. Applying

Algorithm 6, we obtain the following set: S11 = (1, 3), (1, 4), (1, 5), S1

2 = (1, 1), (1, 4),S1

3 = (1, 2), S14 = (1, 1), (1, 2), (1, 3) and S1

5 = (1, 4), that correspond to the first mode

of the system. For the second mode of the system, we get sets S21 = (2, 3), (2, 4), (2, 5),

S22 = (2, 1), (2, 4), S2

3 = (2, 2), (2, 5), S24 = (2, 1), (2, 3) and S2

5 = (2, 4). This yields

the universe U = ∪2i=1(i, 1), (i, 2), (i, 3), (i, 4), (i, 5). Now, solving the associated set multi-

covering (set covering) problem (S,U ; d = 1), we obtain as a solution that U = S11 ∪S1

4 ∪S21 ∪

S22 ∪ S2

3 .

107

This leads to a structure of a solution to the rMCPS (9.2) of B1 = [ ? 0 0 ? 0 ]ᵀ and

B2 = [ ? ? 0 0 0 ]ᵀ. Subsequently, we can check that by setting every non-zero entry of

B1 and B2 as 1, by Proposition 9.3.3, we obtain a solution to the problem.

The crMCPS (9.4) scenario

Now, our aim is to design a common input matrix that controls the system. First, we

need to compute all possible solutions, for each mode of the system. For both modes,

the universe of the associated set covering problem is U = 1, 2, 2, 3, 4, 5. Now, we ap-

ply Algorithm 7 and we get, for the first mode, S1 = 1, 4, 5, 1, 4, 2, 1, 2, 3, 4,and, thus, each of the following sets of indices constitute a solution that covers the universe

B1 = 1, 4, 1, 2, 4, 1, 3, 4, 1, 4, 5, 1, 2, 3, 4, 1, 2, 4, 5,1, 3, 4, 5, 1, 2, 3, 4, 5, that is, for each set X ∈ B1, we have that U =

⊔i∈X [S1]i. Note that

5 only belongs to the first set of S1, and 3 is only in the forth set of S1. Hence, these two

sets need to belong to the solution. Analogously, for the second mode, we have that S2 =

3, 4, 5, 1, 2, 4, 2, 5, 1, 3, 4, and all solutions of the associated set covering prob-

lem are B2 = 1, 2, 3, 1, 3, 4, 3, 4, 5, 1, 2, 3, 4, 1, 2, 3, 5 , 1, 3, 4, 5, 1, 2, 3, 4, 5 . By

combining the two sets of possible solutions, we obtain the result that consists in selecting, for

instance, 1, 4 ∈ B1 together with 1, 3, 4 ∈ B2. Thus, 1, 3, 4 are the set of state variables

that we need to actuate, in both modes, to attain controllability. Then, an input matrix with

dedicated inputs may have the following sparsity

B =

? 0 0 0 0

0 0 ? 0 0

0 0 0 ? 0

ᵀ

.

Now, resorting to Algorithm 8, we get the input matrix pattern with merged columns B =

[ ? 0 ? ? 0 ]ᵀ, and we can check that B = [ 1 0 1 1 0 ]ᵀ is a solution to the problem.

Note that the solutions to the problem (9.2) and (9.4) instances are distinct. In fact, if we

set B1 and B2 in Section 9.4.1 as B, we get a solution to the problem (9.2) instance, that is

not minimal in each mode.

9.4.2 Example II

Now, we explore the scenario where a set of inputs may malfunction in the switching system.

We want to account for them when designing the inputs, and the respective variables that

they control, to still be able to control the system.

108

The rMCPS (9.2) scenario

Suppose now that in the first mode, there are not inputs that may fail, but, in the second

mode, one input may fail. In other words, s1 = 0 and s2 = 1, which translates to have, in the

corresponding set multi-covering problem, d(1, j) = 1 and d(2, j) = 2 for j ∈ U , with U and

collection os sets S as in 9.4.1.

Now, a solution to the problem is U = S11 ∪ S1

4 ∪ S21 ∪ S2

2 ∪ S23 ∪ S2

4 , translating to the

pattern for the input matrices, when considering dedicated inputs,

B1 =[? 0 0 ? 0

]ᵀ, and B2 =

? 0 0 0 0

0 ? 0 0 0

0 0 ? 0 0

0 0 0 ? 0

ᵀ

.

We apply Algorithm 8 and obtain that a single input controls the second mode, and each non-

zero value of the inputs matrices being 1 verifies Proposition 9.3.3, i.e., the input matricesB1 =[1 0 0 1 0

]ᵀand B2 =

[1 1 1 1 0

]ᵀare a solution to the problem instance.

The crMCPS (9.4) scenario

Finally, suppose the objective is to design a single input matrix that not only ensures that

the system is controllable, but also that the system remains controllable whenever, at most,

one input fails. In other words, s = 1, which we is reflect in the demand function for the set

multi-covering problems of Algorithm 7, d = 2.

By recalling the sets B1 and B2 from Section 9.4.1, we know that a solution for the first

mode must have twice the indices 1 and 4 so that we cover elements 3 and 5 twice. Hence,

a possible and minimal solution is the set of indices I = 1, 1, 2, 3, 4, 4. In fact, we can

check that this also produces a solution for the second mode. Hence, the pattern of the

solution, when considering dedicated inputs, and a solution, after applying Algorithm 8, are,

respectively,

B =

? ? 0 0 0 0

0 0 ? 0 0 0

0 0 0 ? 0 0

0 0 0 0 ? 0

0 0 0 0 ? 0

, and B =

[1 1 1 1 0

1 0 0 1 0

]ᵀ.

We can find others solutions by changing the merging order. If we set B1 and B2 in Sec-

tion 9.4.2 as B, we get a solution to the problem (9.2) instance, the same we obtained.

However, we want to ensure the system to be robust to one input failure for each mode, while

109

in Section 9.4.2, we want to ensure that the system is robust to one input failure only in the

second mode.


In this chapter, we addressed two robust minimal controllability problems for switched linear

continuous-time systems, extending the results of Chapter 8. The first is to design an input

matrix for each mode that ensures the switched system to be controllable. The second is

to design a common input matrix guaranteeing the switched system to be controllable. We

showed that the first problem is NP-complete. We reduce both problems to set multi-covering

problems in a two-step, solving first the discrete nature of the problem and afterward the con-

tinuous one. These reductions allow deploying approximated and efficient algorithms to solve

set multi-covering problem instances, to get solutions to rMCPS (9.2) or the crMCPS (9.4)

instances that are feasible and have sub-optimality guarantees.

Future work involves exploring the structure of the problems to attain optimal solutions,

leveraging the eigenbasis’ structure. Further, we want to consider, besides the number of

inputs for controllability, obtaining a certain controllability index, minimizing the number of

inputs and the number of times that we need to actuate the system. Last, we want to relate

macroscopic interconnections between dynamical systems, leading to a modular approach to

the actuation placement that ensures controllability.

110

Chapter 10

On the index of convergence of

Boolean matrices with commutative

SD-decomposition

Finally, we explore a digraph decomposition used in structural control systems, to propose a

new bound for the index of convergence of Boolean matrices that have a digraph decomposition

with certain properties. This work is submitted for publication, see [Ramos and Caleiro, 2018].

10.1 Introduction

In this chapter, we present a new bound for the index of convergence (transient) for a large

class of Boolean matrices that emerge in several application domains. The index of conver-

gence and the period of a matrix are paramount in applications such as transportation systems,

production plants cyclic scheduling, network synchronizers, and distributed algorithms in the

scope of routing or resource allocation, see [Akian et al., 2006]. For instance, the termination

time of the Full Reversal algorithm for message routing in computer networks is equivalent

to the index of convergence of the dynamical system’s matrix [Charron-Bost et al., 2011].

10.2 Preliminaries & Terminology

We next introduce the notation and preliminaries specific to this chapter.

10.2.1 Boolean Matrices

We start by recalling Boolean matrices and the algebra we use to operate them. Next, in

tandem with Boolean matrices, we also overview directed graphs, and we present known

111

results for Boolean matrices based on directed graphs.

Let B = 0, 1. We use the max-min algebra [Gavalec, 1997] to operate elements of B. If

a, b ∈ B, then a ⊕ b = maxa, b, and a ⊗ b = mina, b. If k, n,m ∈ N1 (N1 ≡ N \ 0) and

k, n,m > 0, then a vector v ∈ Bk and a matrix A ∈ Bn×m are a Boolean vector and a Boolean

matrix, respectively. We denote the inner product of u, v ∈ Bn by u v, using the max-min

algebra, i.e., u v =⊕n

i=1 ui ⊗ vi. We denote by In the n× n identity matrix, dropping the

n when it is obvious from the context. Given two matrices A ∈ Bn×m and B ∈ Bm×k, the

product of A and B is a matrix C = A ⊗ B ∈ Bn×k, obtained by the usual matrix product,

but using the inner product . Let s ∈ N, we denote the s-th power of the matrix A ∈ Bn×n

as A⊗s. It is inductively defined as A⊗s = In, if s = 0, and A⊗s = A ⊗ A⊗(s−1), otherwise.

We say that two square matrices A,B ∈ Bn×n commute whenever A⊗B = B ⊗A.

Boolean matrices emerge in several applications. For instance, they emerge in [Liu and

Wang, 2007,Satta, 1994,Prosser, 1959,Cheng, 2011], designated structured matrices, and they

are used to encode the structure of real/complex matrices, to model physical systems where

we do not know the exact values of system variables, but we know the location of zeros. We

used structured matrices in Chapter 8 and Chapter 9, and introduced this notion in Chapter 7,

where we used ? no denote a non-zero entry. In this chapter, because we only use Boolean

matrices, we no longer need to use ? so that we distinguish a non-zero entry from a real entry

equal to 1. Therefore, we use B = 0, 1 to improve readability, and to follow the standard

notation from this area.

The set Bn×n, with the ⊗ times operation, is a semigroup of order 2n2. Also, given

A ∈ Bn×n, the sequence of powers (A⊗i)i∈N forms a finite subsemigroup 〈A〉 of Bn×n. Thus,

there is a least positive integer k = k(A) such that A⊗k = A⊗(k+t), for some integer t > 0.

Further, there is a least positive integer p = p(A) such that A⊗k = A⊗(k+p). Such k and

p are called the index (of convergence) or transient, and the period (of convergence) of A,

respectively.

A Boolean matrix A ∈ Bn×n is said to be reducible if there exists a permutation P ∈ Bn×n

such that P ⊗ A⊗ P ᵀ =

[B 0

D C

], where B,C are square matrices. A is irreducible if it is

not reducible.

A Boolean matrix A ∈ Bn×n is primitive if and only if A⊗k = 1, where 1 is the n × nmatrix with every entry equal to 1. If A is primitive and irreducible then p(A) = 1. If

A⊗k = 1, then k is called the primitive exponent of A, denoted by γ(A). If A is primitive

then k(A) = γ(A).

112

10.2.2 Digraphs

Boolean matrices may be represented as directed graphs (digraphs). A digraph G is a pair

(V,E), where V is a finite set of points in N1, called vertices and E ⊆ V × V is a relation

between vertices such that if u, v ∈ V and (u, v) ∈ E, then we say that the ordered pair (u, v)

is an edge that starts in u and ends in v. Given a Boolean matrix A ∈ Bn×n, its digraph

representation G(A) = (V,E) has V = 1, . . . , n and E = (i, j) : Aij = 1 and 1 ≤ i, j ≤ n.In turn, A is called the adjacency matrix of G(A). Similarly, given a digraph G = (V,E), we

can obtain its equivalent Boolean matrix.

Next, we introduce some graph theoretic notions [Bollobas, 2012] to make the manuscript

self-contained. Let G be a digraph, a walk of size k ∈ N from vertex u to vertex v is a sequence

of vertices 〈v1, v2, . . . , vk, vk+1〉 such that (vi, vi+1) ∈ E for 1 ≤ i ≤ k. A path is a walk that

does not repeat vertices. A cycle in G, 〈v1, v2, . . . , vk, v1〉, is a path 〈v1, v2, . . . , vk〉 such that

(vk, v1) ∈ E. The girth of G is the shortest size of a cycle contained in the G. If G does not

have cycles, then it is a directed acyclic graph (DAG). A self-loop DAG is a digraph with a

self-loop in each node such that, if we remove the self-loops, we obtain a DAG. A directed

tree is a DAG in which a node is assigned to be the root, and there is exactly one path from

the root to each node. A directed r-tree is a directed tree such that each node has, at most, r

edges to other nodes. In a directed tree, a leaf is a node that does not have outgoing edges.

The height of a directed tree is the maximum of the lengths of paths from the root to any leaf.

A balanced directed r-tree is a directed r-tree such that the lengths of any two paths between

the root to a leaf differ from 1. A self-loop directed r-tree is a digraph with a self-loop in each

node such that, if we remove the self-loops, we obtain a directed r-tree.

The shortest path problem is the problem of finding a path between a source vertex s

and a target vertex t in G, such that we minimized the number of its edges. The problem of

finding the path with a maximum number of edges from vertex s to vertex t in G is the longest

path problem, lp(G). We may solve the shortest path problem in polynomial time [Cormen

et al., 2001]. The longest path problem is NP-hard, though for DAGs it has a linear time

solution [Sedgewick and Wayne, 2011].

A trail in a digraph G is a walk that does not repeat edges. A circuit is a trail that starts

and ends in the same vertex. The diameter of G is the size of the longest shortest path between

any pair of vertices. A sub-graph of a digraph G = (V,E) is a digraph H = (V ′, E′) such that

E′ ⊆ E, V ′ ⊆ V and E′ ⊆ V ′×V ′. A digraph G = (V,E) is strongly connected (SCD) if there

exists a path between every pair of vertices (u, v) ∈ V × V . A strongly connected component

(SCC) of a digraph is a maximal sub-graph that is strongly connected. The cyclicity of a

strongly connected digraph G is the greatest common divisor of the lengths of all circuits of

the graph. The cyclicity of a digraph G is the least common multiple of the lengths of its

113

SCCs.

A Boolean matrix A is irreducible if and only if G(A) is strongly connected.

10.2.3 Known Results

Now, we present a set of results that explore properties of Boolean matrices by means of

digraphs properties. A more detailed survey may be found in [Li and Shao, 1993]. The first

result about the index of a Boolean matrix A is the following.

Proposition 10 ( [Wielandt, 1950]). If the matrix A ∈ Bn×n is primitive and G(A) is strongly

connected, then k(A) ≤ (n− 1)2 + 1.

The next result provides a tighter bound for the index of a Boolean matrix, using the

girth of its associated digraph.

Proposition 11 ( [Denardo, 1977,Dulmage et al., 1964]). If A ∈ Bn×n is a primitive matrix

such that G(A) is strongly connected with girth g, then the value of k(A) is O(g · n).

Another result sharpening the bounds of the index of a Boolean matrix, using the cyclicity

of the digraph representation, is the following.

Proposition 12 ( [Schwarz, 1970]). Let G(A) be a non-primitive strongly connected digraph,

then the index of A verifies k(A) ≤ (n − 1)2 + 1. Further, if the cyclicity of G(A) is δ, then

k(A) is O(n2/δ).

The previous result suggests that the higher the cyclicity of G(A), the lower the index

of A. Notwithstanding, the girth of a strongly connected digraph is always greater or equal

to the cyclicity, and Propositions 11 and 12 suggest a necessary trade-off between these two

quantities to attain a small index of convergence. A more recent upper bound was provided

in [Kim, 1979], generalizing Propositions 11 and 12, as follows.

Proposition 13 ( [Kim, 1979]). Let G(A) be a strongly connected digraph, with n vertices,

girth g and cyclicity δ, then the index of A is at most n+ g ·(⌊

nδ

⌋− 2).

Even more recently, in [Merlet et al., 2017] the authors identify the matrices that actually

achieve two particular bounds on the indices of Boolean matrices, which are generalizations

of the bounds of Wielandt and Dulmage-Mendelsohn.

10.3 Index of Boolean matrices with commutative SD-decom-

position

Here, we present a new bound for a class of Boolean matrices that we next detail. First, we

introduce a decomposition of a digraph, inspired by results from structured control systems,

114

see [Ramos et al., 2015].

Definition 10.3.1. Given a digraph G with adjacency matrix A, its SCC and DAG decom-

position, SD-decomposition, consists of S, the adjacency matrix of the subgraph of its SCCs,

and D, the adjacency matrix of the subgraph of the DAG that results from removing the SCCs

of G, such that A = S ⊕ D. A has a commutative SD-decomposition (CSDD), whenever

S ×D = D × S.

Observe that for G(A) = (V,E), A ∈ Bn×n, we can compute an SD-decomposition in

Θ(|V |+ |E|) time, using Tarjan’s algorithm to compute the SCCs of G(A), see [Tarjan, 1972].

Further, we can check if an SD-decomposition commutes in O(n2.3728639), see [Le Gall, 2014].

Hence, given A ∈ Bn×n, we can check if its SD-decomposition commutes in O(n2.3728639).

Next, we characterize the digraphs that have a CSDD.

Proposition 14. Let A ∈ Bn×n, let G(A) = (V,E) be its digraph representation and let

A = S⊕D be its SD-decomposition, with digraph representations G(S) = (V,ES) and G(D) =

(V,ED), respectively. Then, A has a CSDD if and only if for all i, j ∈ V there is a path of

size two starting in i and ending in j that passes in some vertex k ∈ V with (i, k) ∈ ES and

(k, j) ∈ ED if and only if there is a path of size two starting in i and ending in j that passes

in some vertex k′ ∈ V with (i, k′) ∈ ED and (k′, j) ∈ ES.

Proof. Let (S ⊗ D)ij = 1. This means that there exists k s.t. Sik = 1 and Dkj = 1 and,

thus, there is a path starting in i and ending in j that passes in some vertex k ∈ V with

(i, k) ∈ ES and (k, j) ∈ ED. (S ⊗D)ij = 0, otherwise. Hence, A has a CSDD if and only if

(S ⊗D)ij = (D ⊗ S)ij for all i, j ∈ V .

To illustrate Proposition 14, we consider the two digraphs depicted in Figure 10.1 and

their respective SD-decompositions. In each digraph, the red edges correspond to S and the

black edges correspond to D. In digraph (a) there is a path between vertices 1 and 4 with a

red edge before a black edge, but there does not exist a path with a black edge before a red

edge. For this reason, (a) does not have CSDD. In digraph (b) there is a path with a black

edge before a red edge between two distinct vertices, if, and only if, there is a path with a red

edge before a black edge. Therefore, (b) has CSDD.

In Table 10.1, we present the number of Boolean matrices with CSDD in Bn×n. Although

we do not have a closed form for the proportion of size n CSDD Boolean matrices, we notice

that all strongly connected digraphs correspond to matrices with a CSDD. Furthermore, up

to n = 5, around half of the matrices have a CSDD.

115

(a)

(b)

Figure 10.1: Two digraphs and their respective SD-decompositions.

n 1 2 3 4 5 6

SCD 1 4 144 25 696 18 082 560 47 025 585 664

CSDD 1 12 260 30 444 18 819 092 47 543 429 052

Bn×n 1 16 512 65 536 33 554 432 68 719 476 736

Table 10.1: Number of size n CSDD Boolean matrices vs. size n Boolean matrices.

In fact, the class of Boolean matrices that commute has interesting properties, as studied

in [Katz et al., 2012]. In this work, the authors investigate the Frobenius normal forms of

commuting matrices, and they show how the intersection of eigencones of commuting matrices

can be described, considering connections with Boolean algebra, to prove that commuting

irreducible matrices in the max-min algebra have a common eigennode.

Notwithstanding, several applications use digraphs that may have a CSDD. In [Gao et al.,

2010], the authors make use of DAGs for which they add self-loops in some vertices in the

context of automata theory. In [Lin, 2012], the authors use the same kind of digraphs to

model and verify multithreaded programs. Also, in [Fletcher et al., 2012] the authors defined

an extended DAG (EDAG), a digraph that becomes a DAG when the self-loops are deleted

and where for each path there is, at most, one self-loop. They used EDAGS to study the

expressive power of navigational query languages on graphs that represent binary relations.

Theorem 10.3.2. Let A ∈ Bn×n, with G ≡ G(A) its digraph representation. Let A =

S ⊕D be its SD-decomposition, with SCCjlj=1 the strongly connected components of G(S),

and G(D) be the DAG that corresponds to G when we remove its SCCs. Further, let m =

lcm(k(SCC1), . . . ,K(SCCl)) and d = |lp(G(D))|+ 1. If [S,D] = 0 then

k(A) ≤

min2(m+1)·(d+1) − 1, 2n

2 if S,D 6= 0

m if D = 0

d if S = 0

.

116

Proof. Let A ∈ Bn×n and G ≡ G(A) be its associated digraph representation. Further, let

A = S ⊕ D be the respective SD-decomposition, where the matrix S corresponds to the

adjacency matrix of SCCjlj=1, the strongly connected components of G, and D corresponds

to the remaining digraph, which is a DAG. In fact, without loss of generality, we can see S

as a block diagonal matrix, where each block corresponds to an SCC. In other words, we can

rename the vertices of the digraph in such a way that the matrix is block diagonal, and this

corresponds to applying a permutation to the rows of A. The matrix D is the DAG built

from G when we remove the SCCs. Let us compute At for t ∈ N1. Let S,D 6= 0, since A has

a CSDD, we can use the binomial expansion A⊗t = (S ⊕D)⊗t =

t⊕i=0

S⊗(t−i) ⊗D⊗i. Now, we

explore the behavior of the powers of D. Since D is the adjacency matrix of a DAG, we know

there exists d ≤ n such that d− 1 is the size of the longest path between any pair of vertices.

Therefore, we have that D⊗d = D⊗(d+1) = 0, in other words, there are not walks in a DAG

with size larger than d− 1 and k(D) = d.

Finally, we study the behavior of the powers of S. Let kj ≡ k(SCCj) denote the index of

its corresponding block in S. Then k(S) = lcm(k1, . . . , kl). Let m denote the value of k(S).

Putting all pieces together, to compute A⊗t, we only need to consider products of ordered

pairs in X = I, S, . . . , S⊗m × I,D, . . . ,D⊗d \ (I, I). More precisely, we can eventually

have different powers of A for each sum of product of pairs of subsets of X. There are

M = (m+ 1) · (d+ 1) ordered pairs to consider, andM∑i=1

(M

i

)= 2M − 1 such subsets. Thus,

we have that k(A) ≤ 2M − 1 = 2(m+1)·(d+1)− 1, and since k(A) ≤ 2n2, the number of different

n × n Boolean matrices, it follows that k(A) ≤ min2(m+1)·(d+1) − 1, 2n2. If S = 0, then

k(A) ≤ d, and if D = 0, then k(A) ≤ m.

The result of Theorem 10.3.2 allows us to apply one of the known bounds to each SCC,

Propositions 10-13, and get a bound for the index of convergence of a Boolean matrix that,

contrarily to Theorem 10.3.2, does not depend on the index of convergence of other matrices.

Corollary 10.3.3. Let A ∈ Bn×n, with G ≡ G(A) its digraph representation. Let A = S ⊕Dbe its SD-decomposition, with SCCjlj=1 the strongly connected components of G(S) and

G(D) be the DAG that corresponds to G when we remove its SCCs. Further, let k(SCCi) ≤ ki(a bound for the index of SCCi), for 1 ≤ i ≤ l, let m =

∏li=1 ki, and let d = |lp(G(D))|+ 1.

If [S,D] = 0 then

k(A) ≤

min2(m+1)·(d+1) − 1, 2n

2 if S,D 6= 0

m if D = 0

d if S = 0

.

117

We can identify some classes of digraphs for which the bound from Theorem 10.3.2, ap-

proximated by Corollary 10.3.3, is clearly better than the size of all possible powers, i.e., when

2(m+1)·(d+1) − 1 2n2.

• balanced directed r-trees (r ≥ 2): m = m = 0, d = logr n+ 1 and k(A) ≤ 2logrn+2− 1 ≤4n− 1 = O(n);

• self-loop balanced directed r-trees with n vertices: m = m = 1, d = logr n + 1 and

k(A) ≤ 22(logrn+2) − 1 ≤ 16n2 − 1 = O(n2);

• DAGs with n vertices: m = m = 0, d ≤ n+ 1 and k(A) ≤ 2n+2 − 1 = O(2n);

• self-loop DAGs with n vertices: m = m = 1, d ≤ n+ 1 and k(A) ≤ 22(n+2)− 1 = O(2n);

• Digraphs with a giant SCC and an edge from a vertex in the giant SCC to each

vertex (outside the giant SCC) that only has a self-loop, see Figure 10.2(c): k(A) ∈O(

2(n+g·(bnδ c−2)))

.


To illustrate the bounds of Theorem 10.3.2 and Corollary 10.3.3, we consider some examples

of Boolean matrices having a CSDD.

Consider the family of matrices Ai1∞i=1, where Ai1 ∈ B(2i+1)×(2i+1), i ∈ N, with CSDS of

Si1 ⊕Di1, as the adjacency matrices of the digraphs depicted in Figure 10.2 (a).

Ai1 = Si1 ⊕Di1,

Si1 =

B

. . .

B

1

, Di1 =

0 1

. . ....

0 1

0 · · · 0 0

, where B =

[0 1

1 0

].

The digraph G(Ai1) has n = 2i+ 1 vertices and it has s+ 1 SCCs, one self-loop and s cycles

of size 2. Hence, using Theorem 10.3.2, we have that m = 1 + 1, using Proposition 10, and

d = 1 + 1, which implies that k(Ai1) ≤ 2(2+1)(2+1) − 1 = 29 − 1 = 511, instead of the know

bound of 2(2i+1)2 .

118

...

(a)

(b)

(c)

Figure 10.2: A family of digraphs G(Ai1)∞i=1 in (a); digraph G(A2) in (b); and digraph G(A3)

in (c). The SCCs are represented by the red edges in (a), each cycle is an SCC, and they are

represented by different (not black) colors, one per SCC, in (b) and (c). The DAG of each

digraph is represented by the black edges.

Now, consider the matrix A2 with SD-decomposition of S2 ⊕D2

A2 =

1 0 0 0 0 1 0

1 1 0 0 1 0 0

1 1 1 0 1 1 1

0 1 0 0 1 0 0

1 1 0 1 0 1 0

0 1 0 1 1 1 0

0 1 1 1 1 1 1

,

119

S2 =

1 0 0 0 0 1 0

1 1 0 0 1 0 0

0 0 1 0 0 0 1

0 1 0 0 1 0 0

1 1 0 1 0 1 0

0 1 0 1 1 1 0

0 0 1 0 0 0 1

and D2 =

0 0 0 0 0 0 0

0 0 0 0 0 0 0

1 1 0 0 1 1 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 1 0 1 1 1 0

.

In this example, the G(A2) has two SCC. They are depicted in the red edges, SCC1, and in

the blue edges, SCC2, of Figure 10.2 (b), corresponding to G(S2). The black edges edges of

Figure 10.2 (b) represent the DAG subgraph of G(A2), G(D2). By Proposition 11, we have

that the girth of both SCCs is 1 because both have a self-loop, a cycle of size 1. Hence,

k(SCC1) ≤ 2×1 = 2 and k(SCC2) = 5×1 = 5. Moreover, the longest path of G(D2) has size

d = 1. By Corollary 10.3.3, we have that k(A2) ≤ 2(2×5)×(1+1) − 1 = 4.194.303, much smaller

(10.000 times smaller) than the obvious bound of 262 = 68.719.476.736. Last, we consider

the 5-bus electrical power system [Ramos et al., 2013], depicted in Figure 10.3, with digraph

representation of the dynamics state matrix of Figure 10.2 (c).

l2 l3

G1 G2

L1

G3

L2

l1 l4

l5

l6

Figure 10.3: Graph representation of a 5-bus power system.

The SD-decomposition, A3 = S3 ⊕ D3, is commutative and G(A3) has l = 3 SCCs, see

Figure 10.2 (c). Using Proposition 13 with Corollary 10.3.3, we have that the girth and

cyclicity of G(S3) are g = 2 and δ = 1, respectively. Thus, m = 16 + 2× (16− 2) = 44, d = 1

and k(A3) ≤ 2(44+1)×(1+1) − 1 = 290 − 1, instead of the previous 2(182) = 2324 known bound.

120


In this chapter, we explored the index of convergence of Boolean matrices, and we presented

a new bound for the class of Boolean matrices that have a CSDD. Our result makes use

of previously know bounds to present a new bound for the class of Boolean matrices with

commutative SD-decomposition, enabling us to use previous known bounds to each one of the

constituents of the decomposition. Future work includes exploring the exact proportion of size

n square Boolean matrices that has CSDD, and to find more important classes of digraphs

for which the new presented bound is helpful.

121

Chapter 11

Conclusions and Future Work

The main goal of this dissertation was to study information security aspects in ranking,

recommender, and control systems, and to develop systems in these classes that are more

robust to attacks. We used the area of information theory, in particular of Kolmogorov

complexity theory, graph theory, clustering and collaborative filtering, and the area of control

systems theory as a valuable tools to approach the fields of ranking and recommender systems,

improving the state-of-the-art in both fields by designing systems more robust to attacks and

noise, and also to design systems that are more efficient in terms of computational complexity.

Moreover, we used ideas from control systems theory and applied them in the context of

ranking systems. We studied the bribing effect in ranking systems and the optimal and

profitable strategies to attack them.

The Part I of this dissertation used information theory and ideas from control systems,

Part II, in the context of e-commerce. In Chapter 3, we created a multipartite reputation-

based ranking system that clusters users using similarity measures, based on users’ tastes.

The system is more user personalized. Further, we proved the convergence and efficiency

of the system, and we tested it against the state-of-the-art ranking systems, obtaining more

robustness to noise and attacks. In Chapter 4, we studied the effect of bribing in ranking

systems, we calculated the optimal bribing strategies, and we tested the resistance to bribery of

bipartite ranking systems versus multipartite ranking systems, being the latter more resistant

than the former.

As future work, we would like to evaluate the personalization perspective of our ranking

system, i.e., to measure how better the proposed ranking system produces items’ rankings

for each user when compared to bipartite ranking systems. Hence, we would like to measure,

through different metrics, if our ranking generates preferences that are close to the individual

preferences. Also, we would like to consider a game theory model with the sellers as players, to

study the interactions between bigger and smaller players (big and small companies), and the

123

scenario where sellers can bribe users to not only to increase, but also to decrease a competitor

item’s ranking. Moreover, it would be interesting to study the optimal bribing strategies in

the cases where the reputations are dynamically computed (change when the ratings change),

as in the ranking system we propose in Chapter 3.

By exploring the similarity measures that we proposed, we introduced, in Chapter 5, a

recommender system that is highly parallelizable. Besides, the KS similarity measure, that

we introduced to compute how related each pair of users, and pair of items, is has optimal

time complexity. By evaluating its performance with synthetic and real datasets, we obtained

results similar to, and sometimes better than, the state-or-the-art recommender systems. We

presented a novel group recommender system, by exploring the KS similarity measure that

we proposed in the grouping phase, in Section 6. When evaluated with real data, our method

revealed results that are statistically the same as the ones obtained with the standardly used

similarity, the Pearson correlation. However, a big advantage of our similarity measure is

that it achieves measurable gains in terms of time complexity. In the future, we would like to

provide a description of the groups that we generate, so that the companies/sellers know what

characterizes users in terms of their preferences. Also, it would be interesting to study the

optimal bribing strategies such that the bribing company can push the recommendation order

of their products to the top-N recommendation list presented to users or groups of users.

In Part II of this manuscript, we studied the security aspects of control of both LTI (linear-

time invariant) systems and switched LTI systems. Chapter 8 addresses the robust minimal

controllability (rMCP) problem in the context of LTI systems. The problem consists in finding

one of the smallest set of state variables that we need to actuate such that the underlying

dynamical system is controllable, even in the event of failure of a subset of controllers, for

instance, due to an external agent attacking the system. We characterized the exact solutions

to this problem. We showed that the problem is NP-complete and provided approximation

algorithms with polynomial complexity to solve it. In Chapter 9, we extended the results of

Chapter 8 to switched LTI systems. Under this scenario, we identified two relevant minimal

controllability problems, depending on whether we want to place a fixed set of actuators or one

set of actuators for each mode of the system. Again, we provided approximation algorithms

to solve both versions of the problem. Notwithstanding, only for the second version of the

problem the approximation algorithms may have polynomial complexity. Future research

includes exploring the structure of the inputs and to consider methods, such as coordinate

gradient descent, to minimize an energy cost.

124

InformationTheory and Security

E-commerce ControlSystems

RecommenderSystems

RankingSystems

LTIsystems

SwitchedLTI systems

Ch. 5,6 Ch. 3,4Ch. 4 Ch. 8 Ch. 9

Part I Part II

Figure 11.1: Future work directions.

Moreover, it would be interesting to find solutions for the rMCP of both Chapter 8 and

Chapter 9 without the assumption about the eigenvalues of the state matrices. Another inter-

esting avenue for future research is to model the dynamics of ranking systems, or recommender

systems, by means of a dynamical system. By doing this, we may study the smallest subset

of users that a company/seller needs to bribe in order to control the ranking of their products

or of competitor products, or the order of the appearance of an item in a recommendation

list. These future research lines correspond to the dashed edges of Figure 11.1.

Last, because the area of structural control theory is strongly connected with Boolean

matrices and graph theory, we used ideas from the first area and explore a digraph decom-

position to provide a novel and more general bound for the index of convergence of Boolean

matrices. In the future, we would like to explore more digraph decompositions to extended

our new bound.

125

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