A Game Theoretic Approach towards Social Network Multi-Objective Analysis

David Beirão da Cruz e Silva

Project in Industrial Engineering and Management

Master Degree (MSc) in Industrial Engineering and Management

June 2014

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Acknowledgements

I would like to start by expressing my deepest gratitude and appreciation to Professor José Rui

Figueira for his valuable and constructive suggestions and continuous support from the inception to

the conclusion of this research project. His willingness to give his time is extremely appreciated and

recognized.

To my father, Professor Edgar Cruz e Silva, I cannot find the words to thank and homage your

example and inspiration. It is from you that I retain my unswerving motivation, desire to excel and

need to outperform myself continuously.

To my caring and loving girlfriend, Isabel, I would like to thank your support and encouragement. It

was extremely motivating to see your dedication to help me redacting this work. Without your caring

and attentive support, this research project would have been a much more extenuating task.

To my mother, Professor Odete Cruz e Silva, and brother, Cristovão Cruz e Silva, thank you for your

support, proof-reading and revising of all the contents. All your advices and critics were welcomed and

critical to the quality of this research project.

Finally, I would like to extend my thanks to the staff of Instituto Superior Técnico’s library for their

courtesy and politeness. The requisition of certain important and relevant literature was only possible

due to their attention and special care.

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Resumo

A análise de redes sociais estuda a forma como entidades interagem num sistema complexo

multiagente com relações socioeconómicas entre elas. A análise de redes sociais, tipicamente, é

incapaz de ter em conta características da rede e da não-rede. A teoria de jogos é sugerida como

abordagem capaz de considerar a interacção de agentes autónomos, inteligentes e racionais. A

optimização multiobjectivo permite considerar os objectivos, por vezes discordantes, de vários

agentes. Como objectivos deste projecto estão a compreensão da forma como a estrutura da rede

explica propriedades de larga-escala tal como comportamentos e a importância de cada agente. O

ponto de entrada e o enquadramento para o investigador operacional na análise de redes sociais é,

também, apresentado. Especificidades de formulação e a adaptação de conceitos clássicos de

análise de redes sociais são exploradas. A área científica da análise de redes sociais é dividida em:

análise centrada nos nós/arcos e análise centrada na rede. A revisão de literatura considerou estas

duas perspectivas. Um breve caso de estudo é utilizada para exemplificar como a análise de redes

sociais pode beneficiar com teoria de jogos e optimização multiobjectivo. Conclui-se que mais

desenvolvimento na formulação de programação linear multiobjectivo é essencial. Futuras áreas de

investigação dentro da análise de redes sociais são propostas.

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Abstract

Social network analysis studies how entities with socio-economic relations behave in a complex multi-

agent system. This project stems from the fact that the existing methods and techniques for social

network analysis are unable to account for both network and non-network features. Game theory is

suggested as an approach capable of dealing with the interaction of several autonomous, intelligent

and rational agents. Multi-objective optimization is proposed as a methodology accounting for each

agent’s, possibly conflicting, objectives. The aims of this project are to understand the extent to which

the network structure explains large-scale properties as well as behaviours and the importance of

agents. Moreover, the entry point and the future framework for operations researchers in social

network analysis are identified. Formulation issues and the representation of classical social network

analysis concept are also explored. Social network analysis is divided into two perspectives:

node/edge centric analysis and network centric analysis. A literature review considering these two

perspectives is presented. A simple case study is provided in order to exemplify how social network

analysis could benefit from game theory and multi-objective analysis. This project, generally,

concludes that one should better design game theoretic multi-objective linear programming problems

so that social network analysis concepts are effectively modelled. Future avenues for research allying

game theory and operations research to social network analysis are suggested.

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Contents

ACKNOWLEDGEMENTS II RESUMO III ABSTRACT IV

CONTENTS V

LIST OF TABLES VI

LIST OF FIGURES VII

ACRONYMS VIII

1 INTRODUCTION 1

1.1 PROBLEM CONTEXTUALIZATION 1 1.2 MOTIVATION 4 1.3 RESEARCH TOPIC DEFINITION 6 1.4 MAIN OBJECTIVES 6 1.5 PROJECT STRUCTURE 6

2 PROBLEM DEFINITION 8

2.1 CONCEPTS AND DEFINITIONS 8 2.1.1 SOCIAL NETWORK ANALYSIS 8 2.1.2 MULTI-OBJECTIVE OPTIMIZATION 11 2.1.3 GAME THEORY 12 2.2 SPECIFIC OBJECTIVES 17

3 LITERATURE REVIEW AND STATE OF THE ART 18

3.1 HISTORY 18 3.2 STATE OF THE ART 18 3.2.1 NODE/EDGE CENTRIC ANALYSIS 21 3.2.2 NETWORK CENTRIC ANALYSIS 28

4 CASE STUDY: STRATEGIC CORPORATE NETWORK FORMATION 32

5 CONCLUSION 34

REFERENCES   36  

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List of Tables

Table 2.1 Matching Pennies Game in Strategic Form _____________________________________ 13 Table 2.2 Rock-Paper-Scissors Game in Strategic Form __________________________________ 13 Table 2.3 Coordination Game in the Strategic Form ______________________________________ 14 Table 2.4 Battle of the Sexes Game in the Strategic Form _________________________________ 14 Table 2.5 The Prisoner’s Dilemma Game in Strategic Form ________________________________ 15

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List of Figures

Figure 2.1 Example of a graph H where H = 7 and H = 5 _______________________________ 8 Figure 2.2 Example of a digraph H* where H* = 7 and H* = 5 ____________________________ 9 Figure 2.3 Induced subgraph of H* ___________________________________________________ 10 Figure 2.4 Spanning subgraph of H* __________________________________________________ 10 Figure 2.5 Graph I with N = 1, 2, 3, 4, 5, 6, 7 and A = 1,3 , 3,7 , 5,2 , 5,4 , 5,6 , 5,7 , 6,3 , (7,1) 11 Figure 2.6 Tree T with N = 2, 3, 4, 5, 6 and A = (5,2), (5,4), (5,6), (6,3) _____________________ 11 Figure 2.7 Perfect Information Extensive Form Game ____________________________________ 16

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Acronyms

BOINF – Bi-Objective Integer Network Flow

BON – Bi-Objective Network Flow

CSR – Corporate Social Responsibility

DSNA – Dynamic Social Network Analysis

GT – Game Theory

LP – Linear Programming

MOLP – Multi-Objective Linear Programming

MOO – Multi-Objective Optimization

NE – Nash Equilibrium

OR – Operations Research

PII – Political Independence Index

R&D – Research and Development

SCA – Sustainable Competitive Advantage

SNA – Social Network Analysis

SV – Shapley Value

TU – Transferable Utility

WWW – World Wide Web

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1 Introduction

The work here presented assesses the state of the art of Social Network Analysis (SNA) from a Game

Theory (GT) and Multi-objective Optimization (MOO) point of view and tries to identify future avenues

for research and study. This research project takes advantage of a set of concepts already developed

on bi-objective network flows by Professor José Rui Figueira and colleagues (Eusébio & Figueira

2009; Eusébio et al. 2014; Gómez et al. 2013) as its starting point. The present document discusses

the context of the problem to be investigated, the problem definition, the approach used, as well as the

techniques and methods found in the main Operations Research, Game Theory and Social Networks

journals. A critical analysis is also presented, and a final discussion, considering in particular the main

aspects to be explored in future work.

1.1 Problem contextualization

SNA, a branch of Network Science, addresses how entities relate within a system. Entities can be

countries, organizations, companies, groups or even individuals. The relations between entities can

represent any type of socio-economic meaningful tie (Leinhardt 1977; Watts 2001) or, in some cases,

flows of some type (del Pozo et al. 2011). A network structure is an intuitive representation of several

real world multidisciplinary domains such as politics, epidemiology, online social networking, labour

markets, purchasing decisions, organizational behaviour, communication, scientific collaboration,

computer communications in the World Wide Web (www), strategic interactions, technology and

product adoption, transaction engagement, protein interactions, brand choice, drug prescription,

farmer’s crop choice, military systems, sociology, social psychology, economics, anthropology, human

geography, marketing, among others. In fact this technique has even been applied to terrorist

organizations as a tool for counterterrorism practice (Lindelauf et al. 2013).

SNA permits analysing complex multi-agent systems (Szczepański et al. 2012). It is difficult to

clearly specify and define what makes social networks complex systems. A brief and simple

explanation would be the fact that it comprises several components, several relations and a myriad of

potential actions. As documented by Alderson (2008), these causes are called “complexity of size”,

“complexity of interconnection” and “complexity of interaction”. This scientific area has expanded

dramatically over the last years and has been and continues to be applied to several diverse scientific

problems and research topics. SNA attempts to study the dynamic nature of the network, that is, not

only to explain, but also to predict its behaviour and social evolution, thus capturing a lot of attention

from policymakers, engineers and managers (Alderson 2008). Recent technology has provided

scientists and analysts with an increasing ability to gather tremendous amounts of information. SNA

surges as a quantitative modelling technique that is able to elegantly deal with large volumes of data,

which are often incomplete and not completely representative of real world problems (Lindelauf et al.

2013). The ever increasing amount of scientific publications in SNA, attest to its growing importance

(Alderson 2008). Social behaviour is explained by the structure of the network itself (Leinhardt 1977;

Freeman 1978; Smith et al. 2014; Dawande et al. 2012). Given that SNA is a tool evaluating the

overall network structure, it represents a functional advantage when considering interrelationships

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within groups of agents (Ressler 2006). It is very important to clearly distinguish “dynamics of the

network” from “dynamics on the network”, as will be shown in detail in Section 3 (Alderson 2008).

Among some of the most commonly addressed problems in SNA, are the determination of the

importance of individual entities in an existing social network (Lindelauf et al. 2013; Szczepański et al.

2012; Freeman 1978; Smith et al. 2014; Kolaczyk et al. 2009; Goyal & Kearns 2012; Everett & Borgatti

2010); the consequences of a decision making process and behaviour adoption of entities under

social influence (Hernández et al. 2013; Kleinberg & Ligett 2013); the determination of the importance

of a group of entities in an existing social network (Barrat et al. 2008; Grossmann & Dominguez 2009;

Merida-Campos & Willmott 2007; Bogomolnaia & Jackson 2002; Everett et al. 2005; Everett & Borgatti

1999a); the identification of the opportunities or restrictions for entities within a defined network

structure (del Pozo et al. 2011; Smith et al. 2014; Grossmann & Dominguez 2009); the understanding

of social network formation dynamics (Watts 2001; Galeotti et al. 2006) and the construction of

algorithms to help address or deal with these problems (Dawande et al. 2012; Brautbar & Kearns

2010). Note that the importance of an entity or group of entities within social networks is closely

related to the concept of power (or, similarly, influence) that, due to the SNA nature, is mainly

relational (del Pozo et al. 2011). Also, as was briefly referred previously, it is interesting to understand

that the social network structure may pose itself, simultaneously, as an opportunity and a restriction. In

del Pozo et al. (2011), the social network is seen as a restriction to the cooperation of entities. Taking

Smith et al. (2014) and Dominguez (2008) as examples, the position of an entity in the network

defines opportunities, either as a means to having access or control over other entities or as the

definition of party formation opportunities for all entities involved1. Depending on the context and

problem formulation the network can have these two fronts (opportunity or restriction).

The present project will explore the evolving technique of SNA. Moreover, it will explore MOO, from

an Operations Research (OR) point of view, and Game Theory (GT).

OR addresses how operations are conducted within an organization and is applicable to

transportation, communications, healthcare, military, public services and telecommunication problems,

among several others. In fact, OR follows the typical scientific method: (1) Observe, (2) Define

Problem, (3) Gather Data, (4) Formulate Mathematical Model, (5) Develop Procedure, (6) Test Model,

(7) Refine Model, (8) Prepare for Application and (9) Implement (Hillier & Lieberman 2010). OR is

commonly referred to as an operational tool for decision making and must provide the means to attain

useful conclusions for the decision maker (INFORMS 2014). That is, OR can provide a holistic view,

allowing for the consideration of the trade-off among the components of an organization, considering

what is best for the organization as a whole. OR allows engineers and managers to put the resources

available to best use, assisting in the proper management of limited resources. It is very important to

understand that applying OR to SNA results in some structural differences which are better explained

by Alderson (2008).

1 Here, however, if negative ties are allowed, within the same framework, the network may, also, define restrictions (threats). Smith et al (2014) state that if actor A is linked to actor B, who is under threat from actor C, actor A has an increased nodal power since actor B is more dependent upon him. It is quite clear how the network may define either restrictions or opportunities.

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Real world decision making must take into account several, frequently non-commensurable and

conflicting, objectives. In this sense, OR may be improved by defining more than one objective. MOO

is, therefore, concerned with mathematical programming problems, where one must strive towards

more than one objective (to be maximized or minimized). Since more than one objective is being

pursued, dominance appears as a central concept in order to filter alternative solutions (Ehrgott 2005).

The purpose can be to find all the non-dominated/efficient solutions. The concept of dominance allows

researchers to filter the solutions into two sets: the set of non-dominated/efficient solutions and the set

of dominant/inefficient solutions. One of the concerns of this project may in fact be finding all the non-

dominated/efficient solutions.

GT permits scientists to study the decision making process and tries to explain the behaviour of

players through mathematical models. This approach is commonly used in economics, psychology,

sociology and political science. GT provides a framework for the understanding and analysis of

strategic scenarios properly formulated (Turocy & Stengel 2003). GT models mainly concern

cooperation or conflict decisions between more than one entity (typically called players) and takes into

account each player’s selfishness (Goluch 2012). Four questions will be answered in order to frame a

game theoretic analysis. The questions as defined by Jackson (2011) are: (1) who are the entities?,

(2) what actions are possible for each entity?, (3) when do the interactions occur? and (4) what does

each entity get from these interactions?. These can be explained as follows: (1) Interestingly, similarly

to SNA, the entities may be countries, organizations, companies, groups or individuals; (2) actions that

will, in any way, affect players’ payoff should be available for choice; (3) the actions of the players may

occur simultaneously or sequentially - this may have a direct impact in the information available to

each player when deciding which action to take; (4) the payoff for a player can be viewed as the

balance between costs and benefits resulting from each set of possible actions. The definition of these

payoffs may, in many cases, be the most difficult task. Also, strategic thinking and/or rational decision

making (Varoufakis 2001), often the main assumptions of GT, may not necessarily be true for all

players. Additionally, a strategy is the reference for each player’s behaviour. In other words, a strategy

is the “reference point” of a player. Thus, taken together, a game is defined by its players, their

preferences, the actions and information available and, finally, how these actions impact each player’s

payoffs. The concept of strategy is defined in Section 2.

A game is, generally, a mathematical model representing an interacting or, similarly, decision

making situation (Turocy & Stengel 2003). One can distinguish between cooperative game theory and

non-cooperative game theory. Cooperative game theory studies the relative power each player of a

coalition has and, alternatively, how it should divide its payoff. A coalition is a “high level” concept

referring to a group of players that cooperate in order to maximize their payoff (Turocy & Stengel

2003). Nash’s famous model is within the cooperative game theory framework, which focuses on the

bargaining process (Nash 1950). Non-cooperative game theory is mainly concerned with strategic

choices and their analysis (Turocy & Stengel 2003). The main functional difference between

cooperative and non-cooperative game theory is that the latter takes into account the timing and

ordering of the actions, not considered by Nash’s traditional model (Nash 1950). The models can

assume different levels of information, either complete or incomplete information. One can also

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identify differences between pure strategies and mixed strategies (a probability distribution over Pure

strategies). Pure Strategies make sense when one can identify dominated strategies and/or saddle

points. There is also a third perspective, when there aren’t any dominated strategies or saddle points,

but rather focuses on maximizing (or minimizing) the minimum expected gain (maximum expected

loss). Note that a detailed description of all concepts presented is given. These will be provided and

discussed in Section 2.

1.2 Motivation

Predicting outcomes as a function of behaviour provides an important advantage and the driving force

to understand SNA. Furthermore, addressing SNA from a GT approach with an OR framework can

provide significant advantages. Moreover, an OR framework permits the study and analysis of

emerging areas of interest still not properly explored. The novelty of the present work is the

understanding and quantification of the added value which arises from crossing SNA with GT and

MOO, in an OR framework. The lack of an extensive literature on this topic attests the innovativeness

of the area.

As previously discussed, real world problems are multi-dimensional. Therefore, MOO is more

appropriate to deal with real world problems. MOO provides better insights into the dynamics of the

problems being solved (Eusébio & Figueira 2009; Eusébio et al. 2014; Gómez et al. 2013).

Engineers must understand how to model and control the function of systems. The use of OR in

SNA allows engineers to use inverse optimization in order to explain observed functions of systems

(Alderson 2008). Thus, it is quite evident that OR helps in validating and controlling models typically

built in SNA (Alderson 2008). Note that these are two crucial steps of the scientific method.

Furthermore, OR may help answer the question of why social networks form. Much attention has been

given to how networks form and to what are, formally, relevant network structures (Alderson 2008). In

fact, it is of extreme importance, for the operations researcher, to understand why the network is

forming or, in other words, what is the problem being solved by the network2. It must be pointed out

that, in an OR framework, the researcher focuses on the performance of the systems, resource and

material restrictions and trade-offs in the design of the system (Alderson 2008).

SNA uses the network structure to explain social behaviour. A GT approach allows non-network

features to be considered as well, such as preferences and individual based parameters. The

incorporation of this additional information allows for more realistic models (Lindelauf et al. 2013).

Basically, one can take into account, simultaneously, the “network structure”, “additional information”

on individuals and the “level of differentiation” when ranking these individuals (Lindelauf et al. 2013).

GT provides SNA with the ability to consider the interests that make entities interact (Watts 2001; del

Pozo et al. 2011; Hernández et al. 2013; Bogomolnaia & Jackson 2002). This may, to some extent,

help in answering the question raised by OR in SNA, mainly in identifying the problem being solved by

the social network (Alderson 2008). Additionally, GT allows for the analysis of networks where entities

are trying to undermine each other (Smith et al. 2014). Agents, in a social network, struggle to improve

their wellbeing (Hernández et al. 2013). GT allows the researcher to account for the influence of

2 The answer to this question may depend a lot on the scientific area of study.

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others (Hernández et al. 2013) and each agent’s identity (Galeotti et al. 2006) when defining entities’

payoffs. It even allows for reasoning with respect to the potential contribution each entity would have

as a member of all possible groups functioning in the network (Szczepański et al. 2012). Of peripheral

interest is the fact that SNA itself may also provide gains in GT research, namely, on the definition and

comprehension of the coalition formation process (Turocy & Stengel 2003; Amer et al. 2007).

By applying GT and MOO to SNA one can simultaneously consider, in an individual identity,

preferences and global individual based parameters (such as individual’s financial means or skills) as

well as the existence of simultaneous, even potentially conflicting, objectives in social behaviour. The

motivation for this work is, in fact, to understand the extent to which this is already done and to identify

the most promising future avenues and approaches for research.

In order to better understand the integration of SNA with MOO and GT a brief conceptual example

is provided. Consider a social network formation model. Each node of the network represents a firm

and each tie a partnership between two firms. Ties are formed by choice of the firms. The model must

be able to capture social and/or economic incentives for firms to form partnerships. It is quite obvious

that firms, in order to achieve a sustainable competitive advantage (SCA), must act strategically. In

this context, GT allows for the consideration of each firm’s identity. Each firm’s identity can be defined

as its business area. Suppose that firm A prefers firm B if it is from a different business area (giving it

a distinct identity). Also, firm A may prefer firm B to others depending on its interest in partnering with

firms from specific business areas. Preference is a result of identity differentiation. Service providers

from different business areas increase the competitiveness and the ability to achieve a SCA

(Thompson et al. 2014). In fact, firms are selfish and try to maximize the realized utility3. Moreover, the

model allows for the inclusion of individual parameters such as the financial means, client portfolio,

R&D abilities, corporate social responsibility (CSR) reputation of each entity, among other possible

parameters. MOO is able to reflect conflicting objectives among firms. Each firm has its personal

objective and tries to create partnerships such as to help it achieve its strategic goals. In some cases,

competing firms may have incompatible objectives that prevent partnerships from being celebrated.

Thus, MOO is a valuable tool given that it even allows for different notions of objective and/or utility.

Firm A may want to maximize profits (its utilities must be defined by this metric) whilst firm B may

want, for example, to maximize its CSR reputation in order to publicly approve its brand’s License to

Operate (Edelman 2014). This is a two-sided link formation model; meaning mutual consent is needed

to form a partnership. Note that this assumption also relates to GT since it represents cooperative

decisions. Each firm would have its objective function and the number of decision variables would be

the number of potential partnerships between firms; if no restrictions exist this would be equal to the 2-

combination of the number of firms. By adding restrictions to the creation of certain partnerships (such

as the existence of firms with substitute products) the number of variables would decrease. The two

firms associated to each partnership would simultaneously control that variable. Given this, a set of

non-dominated/efficient solution would be obtained.

3 No effort has been put into the definition of the utility since it goes beyond the scope of this example.

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1.3 Research topic definition

The topic of research will be addressed in a multidisciplinary context. The major focus point is SNA

from a GT approach. A MOO methodology within an OR framework will be explored. The initial

research is aimed towards better understanding the link between engineering and management in

applying SNA with MOO and GT. The research topic may be stated as: “ A Game Theoretic Approach

towards Social Network Multi-objective Analysis”. As stated, the focus is on social behaviour analysis,

control and prediction. The ex-ante practical applications defined were mainly related with user

attributes and behaviour analysis, customer interaction, business intelligence and analysis, military

intelligence and counter-intelligence, and public engagement.

This section will also list the research keywords employed. The research keywords were used to

expedite the first phase of the research methodology. Game Theory, Social Network Analysis, Multi-

objective Optimization, Multi-criteria Optimization, Algorithmic Game Theory, Centrality Measures,

Cooperative Games, Non-cooperative Games, Network Optimization and Operations Research were

the most important research keywords considered. Some similar concepts and extensions were also

included.

In order to gather the needed information to fruitfully redact this project, advantage was taken from

data previously produced and released into the scientific community. Particularly, the study resorted to

the most recent publications in scientific journals. In fact, given the novelty of the area the material of

choice was recent scientific publications rather than textbooks.

1.4 Main objectives

In SNA, generally, one wants to understand how the relations between entities are arranged, how the

behaviour of individuals depends on their location in this arrangement, and how the qualities of the

individuals influence the arrangement (Leinhardt 1977). Therefore, the main objectives of this work are

threefold (Alderson 2008), and can be expressed in the following questions:

1. Can large-scale properties of social networks be explained by the network structure?

2. Are there universal laws ruling the structure and behaviour of social networks?

3. Can key pivotal agents be identified?

In order to fulfil these objectives the research must be guided in a way that one clearly understands

how social networks impact behaviour and which structures are expected to naturally emerge in

society (Jackson 2008).

1.5 Project structure

The following research project is structured as follows. The present section provides a brief

introduction and overview, problem contextualization, the project’s motivation, the research topic

definition and the main objectives. Section 2 defines the problem and introduces the scientific areas of

SNA, GT and MOO by discussing main concepts, notations and definitions. Moreover, it defines the

specific objectives of the research project. Section 3 presents the state of the art of SNA technique

implementation and GT and MOO general concepts and frameworks. This analysis is structured into

two perspectives: node/edge centric point of view and network centric point of view. Moreover, this

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section provides a very brief historic contextualization of the evolution of SNA. In Section 4 a case

study is analysed. Finally, in Section 5 a conclusion and some general concluding remarks are

developed as well as the identification of possible future work.

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2 Problem definition

SNA could largely benefit from the ability of accounting for more than one objective simultaneously

and, also, from the consideration of each entity’s preferences and identity. GT and MOO, intuitively,

seem to allow for the inclusion of these and other relevant issues in SNA, as presented in Section 1.

The purpose of this work is to understand the implications of applying a GT approach and MOO

methodologies to SNA and, also, to categorize the area of research. Furthermore, the present

document will also probe important future research directions in SNA allied to GT and MOO.

2.1 Concepts and definitions

Fundamental Concepts and definitions of SNA, GT and MOO are provided in this section. The

purpose is to contextualize all the work presented in Section 3. Note that only a brief introductory

clarification is given. In Section 3 several of these concepts, and other new ones, will be further

explored.

2.1.1 Social Network Analysis

A graph is a pair 𝐺 = (𝑉,𝐸) of sets where 𝐸 ⊆ 𝑉 !. Note that 𝐴 ! represents all the subsets of 𝐴 with

𝑘 elements. The elements of 𝑉 and 𝐸 represent the set of vertices (or nodes) and edges of 𝐺 ,

respectively. (𝑖, 𝑗) ∈ 𝐸 is an edge connecting vertex 𝑖 and vertex 𝑗 and is normally written 𝑖𝑗. Note that

edges aren’t directed, therefore, 𝐺 = (𝑉,𝐸) is a graph, 𝑖𝑗 = 𝑗𝑖,∀𝑖, 𝑗 ∈ 𝑉. The order of 𝐺 is the number of

vertices in 𝐺 and is denoted by 𝐺 . The number of edges in 𝐺 is its size and is denoted by 𝐺 . The

set of vertices of 𝐺 is called 𝑉(𝐺) and the set of edges of 𝐺 is called 𝐸(𝐺). Figure 2.1 shows an

example of a graph 𝐺 obtained using Visone Software (Brandes & Wagner 2003); all graphs were

obtained through this software. Graph 𝐻 in Figure 2.1 has a set of vertices 𝑉 = 1, 2, 3, 4, 5, 6, 7 and a

set of edges 𝐸 = 1, 7 , 2, 5 , 3, 6 , 4, 5 , (5, 6) .

Figure 2.1 Example of a graph 𝑯 where 𝑯 = 𝟕 and 𝑯 = 𝟓

A digraph (also known as directed graph) is a graph 𝐺 = 𝑉,𝐸 assigning to every arc an initial and

terminal node; ordered pairs of distinct nodes. In opposition to edges, arcs are directed. Here, 𝑖𝑗 ≠ 𝑗𝑖.

If 𝑖 = 𝑗 then 𝑖𝑗 is called a loop. Oriented graphs are digraphs neither with loops nor with multiple edges

connecting the same two nodes. Figure 2.2 shows a digraph 𝐻∗ = (𝑉,𝐸) obtained through the

refinement of the graph in Figure 2.1 with 𝐸 = 5, 2 , 5, 4 , 5, 6 , 6, 3 , (7, 1)

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Figure 2.2 Example of a digraph 𝑯∗ where 𝑯∗ = 𝟕 and 𝑯∗ = 𝟓

A network may be seen as a graph with some extra information (Ahuja et al. 1993). A directed network

𝑁 = 𝐺, 𝑐, 𝑙, 𝑢, 𝑏 = (𝑉,𝐸, 𝑐, 𝑙, 𝑢, 𝑏) consists of a digraph 𝐺 by adding the values 𝑐!", 𝑙!" and 𝑢!" to each

𝑖𝑗 ∈ 𝐸, and the value 𝑏! to each 𝑖 ∈ 𝑉. Here the set 𝑉 will be called the set of nodes of the network. It is

more common to denote a network by 𝑁 = (𝑁,𝐴, 𝑐, 𝑙, 𝑢, 𝑏) where 𝑁 and 𝐴 are the sets of nodes and

arcs of 𝑁, respectively. The sets 𝑐, 𝑙, 𝑢 and 𝑏 are the numerical values associated to the graph and

may represent costs, lower capacities, upper capacities and supplies/demands. The definition of an

undirected network should be quite clear; therefore, it will not be discussed. The digraph presented in

Figure 2.2 corresponds to the graphical representation of the network; the sets 𝑐 , 𝑙 , 𝑢 and 𝑏 ,

associated to the arcs and nodes, do not need to be represented graphically.

Arc 𝑖𝑗 has node 𝑖 and node 𝑗 as endpoints. Node 𝑖 is called its tail and node 𝑗 its head. Arc 𝑖𝑗 is an

outgoing arc of (emanates from) node 𝑖 and an incoming arc of (terminating at) node 𝑗. A multiarc is

one element of a set of two or more arcs that have the same endpoints (the same tail and head).

Multiarcs can also be called parallel arcs.

The indegree of a node 𝑖 is the number of arcs terminating at 𝑖. The outdegree of node 𝑖 is the

number of arcs emanating from 𝑖. Therefore, the degree of a node is the sum of its indegree and

outdegree.

𝐴 𝑖 is the arc adjacency list of node 𝑖 and represents the set of arcs emanating from 𝑖, that is,

𝐴 𝑖 = 𝑖𝑗 ∈ 𝐴: 𝑗 ∈ 𝑁 . Nodes 𝑖 and 𝑗 are adjacent if 𝑖𝑗 ∈ 𝐴. The node adjacency list of node 𝑖 is the set

of nodes adjacent to node 𝑖, denoted by 𝑁𝑎 𝑖 = 𝑗 ∈ 𝑁:  𝑖𝑗 ∈ 𝐴 . Ahuja et al. (1993) shows that 𝐴(𝑖)

equals the outdegree of node 𝑖.

A subraph of 𝐺 is a graph4 such that 𝐺! = (𝑁!,𝐴!) where 𝑁! ⊆ 𝑁 and 𝐴! ⊆ 𝐴. The subgraph of 𝐺

induced by 𝑁! is characterized by having all arcs of 𝐴 in 𝐴! given that 𝑖, 𝑗   ∈ 𝑁!. A spanning subgraph

of 𝐺 is one such that 𝑁! = 𝑁 and 𝐴! ⊆ 𝐴 . Figure 2.3 shows a subgraph of 𝐻∗ induced by 𝑁! =

2, 5, 6, 7 . Figure 2.4 shows a spanning subgraph of 𝐻∗ with 𝐴! = 5, 2 , 5, 4 , 6, 3 , (7, 1) .

4 Since a graph is a subset of a network most definitions and concepts are shown for graphs. Hereon digraphs will be called graphs, for simplification.

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Figure 2.3 Induced subgraph of 𝑯∗

Figure 2.4 Spanning subgraph of 𝑯∗

A walk is a subgraph of 𝐺 and a sequence of nodes and arcs 𝑖 −  𝑎! −   𝑖 + 1 −  𝑎!!! −⋯−  (𝑗 − 1)  −

 𝑎!!! − 𝑗 where 𝑎! = 𝑘, 𝑘 + 1  ∨ 𝑎! = 𝑘 + 1, 𝑘 ∈ 𝐴,∀𝑘 ∈ 𝑁. Notice, in Figure 2.2, the walk 4 – 5 – 2 –

5 – 6; for simplicity the arcs may be omitted. A directed walk is a walk were 𝑎! = 𝑘, 𝑘 + 1 ∈ 𝐴. In

Figure 2.2 the walk 5 – 6 – 3 is a directed walk.

A path is a walk with no repeated nodes and a directed path is a directed walk with no repetition of

nodes. A cycle is a path added with arc 𝑗𝑖 or 𝑖𝑗. A directed cycle is a directed path added with arc 𝑗𝑖. A

graph is acyclic if it contains no directed cycles. In Figure 2.2 the walk 4 – 5 – 2 is a path and the walk

5 – 6 – 3 is a directed path. In Figure 2.5 the path 5 – 7 – 3 – 6 – 5 is a cycle while the path 1 – 3 – 7 –

1 is a directed cycle. The graph from Figure 2.4 is acyclic.

Nodes 𝑖 and 𝑗 are connected if there exists at least one path between them. If all combinations of

two nodes from a graph are connected the graph itself is connected. Furthermore, if a graph has at

least one directed path between all pairs of nodes, it is strongly connected. Graph 𝐼 from Figure 2.5 is

connected. In Figure 2.5 the component defined on the node set 1, 3, 7 is strongly connected.

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Figure 2.5 Graph 𝑰 with 𝑵 = 𝟏,𝟐,𝟑,𝟒,𝟓,𝟔,𝟕 and 𝑨 = 𝟏,𝟑 , 𝟑,𝟕 , 𝟓,𝟐 , 𝟓,𝟒 , 𝟓,𝟔 , 𝟓,𝟕 , 𝟔,𝟑 , (𝟕,𝟏)

A cut partitions a node set 𝑁 into 𝑆 and 𝑆 = 𝑁 − 𝑆. The arcs having one endpoint in each partition

define the cut. The graph in Figure 2.4 can be obtained through a cut in the graph in Figure 2.2. In this

case the set of arcs in the cut is (5, 6) .

A tree is an acyclic connected graph while a forest is a graph with no cycles. In other words a

forest is the junction of several trees. Figure 2.2 is a forest and Figure 2.6 is a tree.

Figure 2.6 Tree 𝑻 with 𝑵 = 𝟐,𝟑,𝟒,𝟓,𝟔 and 𝑨 = 𝟓,𝟐 , 𝟓,𝟒 , 𝟓,𝟔 , (𝟔,𝟑)

Networks are far more complex than graphs. Notice that focus was given to the graphs that help

define a network. Little or nothing was said about the sets of numerical values. The reason for this is

twofold. Firstly, those can have different interpretations depending on the application of the network

and, secondly, GT and MOO concepts are also discussed, which help define these sets of numerical

values. It is also important to know why one should consider studying networks instead of graphs.

Alderson (2008) addressed this issue by showing that networks have much more interest and

applicability for engineers and operation researchers.

2.1.2 Multi-objective Optimization

MOO is essential to deal with problems when accounting for several criteria simultaneously. Here,

multi-objective linear programming (MOLP) is needed instead of typical linear programming (LP). The

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compact notation to represent MOLP is max 𝐶!𝑥:  𝐴𝑥 ≤ 𝑏, 𝑥 ≥ 0 . Where 𝐶 ∈ ℝ(!×!) is the objective

matrix with 𝑓 = (𝑓!,… , 𝑓!) the vector of objective functions and 𝑦 = (𝑦!,… , 𝑦!) the vector of objective

functions’ values, 𝑥 = 𝑥!,… , 𝑥! is the variable vector, 𝐴 ∈ ℝ !×! is the constraint matrix of an LP

and 𝑏 ∈ ℝ! is the right hand side vector of a (MO)LP. Notice that one could, instead of maximizing,

minimize the objective functions. The concept of optimum from typical OR must be replaced by the

concept of efficiency. Interestingly it is possible to adapt some typical methods to find efficient

solutions (like the simplex method) to the MOLP problem. In MOO, ℝ! is the decision space where 𝑛

is the number of decision variables. Also, 𝑋 ⊂ ℝ! is the feasible region in the decision space. In a

problem with 𝑘 objectives, ℝ! is the criterion space and the feasible region in the criterion space is

denoted by 𝑌 ≔ 𝑓  (𝑋)  ⊂ ℝ!.

Let 𝑦 and 𝑦! represent two criterion space vectors, 𝑦 dominates 𝑦! if 𝑦 ≥ 𝑦! and 𝑦 ≠ 𝑦! , that is,

𝑦! ≥ 𝑦!!  ∀𝑖 ∈ 1, 𝑝 and 𝑦! > 𝑦!!  ∃𝑖 ∈   [1, 𝑝]. Also, if 𝑦 ≥ 𝑦! or, equivalently, 𝑦! > 𝑦!!  ∀𝑖 ∈ 1, 𝑝 , 𝑦 strongly

dominates 𝑦!. The non-dominated set 𝑌! is the set of non-dominated criterion space vectors. A non-

dominated criterion vector 𝑦 is a vector such that ∄𝑦 ∈ 𝑌:  𝑦 ≥ 𝑦 and 𝑦 ≠ 𝑦. A non-dominated criterion

vector 𝑦 ∈ 𝑌 is supported if, and only if, 𝑦 ∈ 𝑌! and 𝑦 ∈ 𝑏𝑑  𝑌! . Where 𝑌! = 𝑐𝑜𝑛𝑣  (𝑌!! + ℝ!!) with

ℝ!! = 𝑦 ∈ ℝ!:  𝑦 ≥ 0 , 𝑌!! + ℝ!

! = 𝑦 ∈ ℝ!: 𝑦 = 𝑦! + 𝑦!!, 𝑦! ∈ 𝑌!! , 𝑦!! ∈ ℝ!! and 𝑌!! the set of integer

non-dominated criterion vectors. The boundary of 𝑌! is denoted by 𝑏𝑑  𝑌!. Notice that if the purpose

was to maximize the objective function one would consider 𝑌! instead. Moreover, 𝑐𝑜𝑛𝑣   𝑌 =

𝜆!𝑦!!!!! : 𝜆! = 1!

!!! , 𝜆! ≥ 0, 𝑖 = 1,… , 𝑝, 𝑦! ∈ 𝑌, 𝑌 = 𝑝 . A supported-extreme non-dominated criterion

vector is an extreme point of 𝑌!.

A solution 𝑥 ∈ 𝑋 is efficient if, and only if, ∄𝑥 ∈ 𝑋:  𝐶!𝑥 ≥ 𝐶!𝑥 and 𝐶!𝑥 ≠ 𝐶!𝑥. Let 𝑋! represent the

set of efficient solutions.

The detection of efficiency and domination can be done, for example, through domination sets and

dominance displaced cones. The popular simplex method can also be adjusted to deal with MOLP to

find the efficient and non-dominated regions, and is called the bi-objective simplex method. One could

also use, when dealing with networks, the bi-objective network simplex method. Other scalarization

techniques are also available such as weighted-sums, Chebychev functions and 𝜀-constraint, among

others. A detailed description of these methods and techniques will not be discussed since it is beyond

the scope of this document.

2.1.3 Game Theory

Common issues raised in a GT framework are the actions that each player should take; if different

scenarios influence behaviour; what patterns of behaviour should the designer of the system expect;

how can preference alteration affect the outcome; if communication would have any effect on the

behaviour observed; if decisions can be repeated; and does the assumption that players are rational

have any limitation (Fudenberg & Tirole 1984).

Some conceptual clarification is needed before tackling mathematical definitions. Self-interested

agents aren’t necessarily interested only in themselves or in harming others. The fact is that they have

their own description of the states of the world and act accordingly. Each agent has its utility function.

This function describes and quantifies the preferences of the agent across several alternative

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decisions. The utility function may even be designed to represent the uncertainty inherent to the

decision. Agents act in such way as to maximize their expected utility. This may, in fact, be a crossing

point with OR or, even, MOO.

In GT, the decision makers are the players and these may be countries, organizations, companies,

groups or individuals. Examples of actions for players to decide between are, for instance, to enter or

not a bid in an auction or to decide on which political party to vote. Finally, the motivation of each

player may be profit or any other factor; that is why GT utility functions are used since they allow for

the representation of more than one factor and/or trade-offs between factors for different players.

The most simple game one can define is a finite n-person normal form game 𝑁,𝐴, 𝑢 where

𝑁 = 1, 2,… , 𝑛 is a finite set of players, 𝐴! is the set of possible actions for player 𝑖 where 𝑎 =

𝑎!, 𝑎!,… , 𝑎! ∈ 𝐴 = 𝐴!×𝐴!×…  ×𝐴! is an action profile and 𝑢!:𝐴 → ℝ is the utility function for player 𝑖

with 𝑢 = 𝑢!, 𝑢!,… , 𝑢! the profile of utility functions. Typically these games are represented in the

strategic form, also called the matrix form. Table 2.1and Table 2.2 are two pure competition games in

the strategic form. Using Table 2.1 as an example, the rows represent one player and the columns the

other 𝑁 = (1, 2) and 𝐴! = 𝐴! = 𝐻𝑒𝑎𝑑𝑠,𝑇𝑎𝑖𝑙𝑠 . Both Table 2.1 and Table 2.2 are pure competition

games because they have exactly two players with opposite interests. Typically these games are

characterized by 𝑎 ∈ 𝐴, 𝑢! 𝑎 + 𝑢! 𝑎 = 𝑐 where 𝑐 is a constant. This allows for simplifications since

one can only worry about one of the player’s utility function. Note that the matching pennies games in

Table 2.1 and the rock-paper-scissors game Table 2.2 are special cases where 𝑐 = 0, typically called

zero-sum games. In Table 2.1 player 1 (rows) wants to match while player 2 (columns) wants to

mismatch therefore making it a pure competition game. In Table 2.2 the well-known rock-paper-

scissors game is being played which is a pure competition game. Curiously, both are zero-sum

games.

It is also possible for players to have exactly the same objective, designated as cooperation

games. In this case there can exist more than two players and ∀𝑎 ∈ 𝐴,∀𝑖, 𝑗 ∈ 𝑁, 𝑢! 𝑎 = 𝑢!(𝑎). Table

2.3 shows an example of a coordination game. Imagine the game is the decision on which side of the

road to drive. It is quite evident that both players would want to have the same decision since it is the

only way either one receives positive utility.

Table 2.1 Matching Pennies Game in Strategic Form

Heads Tails Heads 1,−1 −1, 1 Tails −1, 1 1,−1

Table 2.2 Rock-Paper-Scissors Game in Strategic Form

Rock Paper Scissors Rock 0 −1, 1 1,−1 Paper 1,−1 0 −1, 1

Scissors −1, 1 1,−1 0

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Table 2.3 Coordination Game in the Strategic Form

Left Right Left 1,1 0,0

Right 0,0 1,1

The interest arises when combinations of pure competition and coordination games appear. Table 2.4

shows a general game involving concepts of both competition and coordination. In this case Table 2.4

represents the battle of the sexes game. Imagine a couple wanting to go to the cinema. They can

either watch a movie called “Bombs and Explosions” denoted by B or one called “Flowers and Love”

denoted by F. The male member of the couple obviously prefers to watch B while the female member

prefers F. However, if each decides to see a different movie they will not benefit from a night out

together (meaning zero utility).

Table 2.4 Battle of the Sexes Game in the Strategic Form

B F B 2,1 0,0 F 0,0 1,2

The games presented so far are very simple and involve only two players, but these concepts can be

applied to complex situations. A good example is the popularized beauty contest game presented by

Keynes (1936). These complex games are extremely difficult to analyse. In this context one must

understand strategic reasoning.

Strategic reasoning concerns the actions of competitors and the decision one should take in

consequence of those. In GT players are considered to be rational, therefore, each player best

responds to others. Let 𝑎!! = 𝑎!,… , 𝑎!!!, 𝑎!!!,… , 𝑎! and 𝑎 = 𝑎!! , 𝑎! . The set of best responses of

player 𝑖 is denoted by 𝐵𝑅(𝑎!!). The definition of a best response is as follows 𝑎!∗ ∈ 𝐵𝑅 𝑎!!  if and only

if ∀𝑎! ∈ 𝐴,  𝑢!(𝑎!∗, 𝑎!!)  ≥  𝑢!(𝑎! , 𝑎!!).

If all players best respond to each other, equilibrium is achieved. Nash (1950) called this

equilibrium the Nash Equilibrium (NE). NE is an action profile maximizing each player’s utility given the

action of others. It is also a stable profile and no players have incentives to deviate from that action

profile. A question could be raised about the feasibility of the NE, namely, if it will, in fact, be played.

The main issue concerning the NE is the fact that players don’t know for certain what others will do.

The NE is defined as follows 𝑎 = 𝑎!,,… , 𝑎! is a (pure strategy5) NE if and only if ∀𝑖 ∈ 𝑁, 𝑎! ∈ 𝐵𝑅(𝑎!!).

The game in Table 2.3 has two NE (or pure strategies) corresponding to the action profiles

𝑎 = 𝐿𝑒𝑓𝑡, 𝐿𝑒𝑓𝑡 and 𝑎! = 𝑅𝑖𝑔ℎ𝑡,𝑅𝑖𝑔ℎ𝑡 . The game in Table 2.4 behaves similarly. The matching

pennies game in Table 2.1 has no pure strategy NE. Table 2.5 presents another well-known game

called the prisoner’s dilemma. In Table 2.5 the action profile 𝑎 = 𝐷𝑒𝑓𝑒𝑐𝑡,𝐷𝑒𝑓𝑒𝑐𝑡 is the only NE.

5 The definition of pure strategy will be discussed later in the document

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Table 2.5 The Prisoner’s Dilemma Game in Strategic Form

C D C −1,−1 −4, 0 D 0,−4 −3,−3

For now a strategy will consist of the choice of an action. If 𝑠! and 𝑠!! are two possible strategies for

player 𝑖 and 𝑆!! the set of all possible strategies for the other players; 𝑠! strictly dominates 𝑠!! if

∀𝑠!! ∈ 𝑆!! , 𝑢!(𝑠! , 𝑠!!) > 𝑢!(𝑠!!, 𝑠!!); 𝑠! very weakly dominates 𝑠!! if ∀𝑠!! ∈ 𝑆!! , 𝑢!(𝑠! , 𝑠!!)  ≥ 𝑢!(𝑠!!, 𝑠!!). If a

strategy dominates all others it is dominant and, also, a profile of strategies made of dominant

strategies is, necessarily, a NE. A NE of strictly dominant strategies is unique. In Table 2.5, for player

1 (rows) the strategy 𝑠! = 𝐷 is strictly dominant and, for player, the strategy 𝑠! = 𝐷 is strictly

dominant; this is in accordance with the previous comment about the existence of a unique NE of the

prisoner’s dilemma game in Table 2.5.

NE may not imply that the outcome from an outsider perspective is the best possible. An outcome

𝑜 may be as good as, or better than, an outcome 𝑜! and a player may strictly prefer outcome 𝑜 over 𝑜!.

In this case 𝑜 Pareto-dominates 𝑜!. Moreover, an outcome 𝑜∗ is Pareto-optimal if there is no outcome

Pareto-dominating it. This is an important concept, which, in its essence, is the reason behind the

name of the prisoner’s dilemma game. Analysing the game in Table 2.5 allows one to notice that the

NE is the only non Pareto-optimal outcome since it is strictly dominated by the outcome resultant from

the action profile 𝑎 = 𝐶𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒,𝐶𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒 .

In a game such as the matching pennies game from Table 2.1 it wouldn’t be very beneficial to play

any deterministic strategy (pure strategy). Therefore the following definition arises, a pure strategy is

one of playing deterministically one single action while a mixed strategy is that of playing more than

one action with positive probability. The randomness behind this behaviour may be beneficial since it

can confuse opponents or help players deal with uncertainty. The actions with positive probability in a

mixed strategy are its support. 𝑆! the set of all possible strategies for player 𝑖 and 𝑆 = 𝑆!×…  ×𝑆! the

set of all strategy profiles, one must apply the concept of expected utility where

𝑢! 𝑠 = 𝑢! 𝑎 ×!∈! 𝑃(𝑎|𝑠) and the probability that an action profile 𝑎 is played given that strategy

profile 𝑠 is the multiplication of the probability that each player is playing its part of that given action

profile 𝑃 𝑎 𝑠 = 𝑠!(𝑎!)!∈! . Given this, the definitions of best response and NE for action profiles are

generalized for strategy profiles. Nash (1950) shows that every finite game has a NE. Mixed strategies

effectively represent cases of repeated play.

The simplest way for a player to decide its strategy is to explore the concept of dominance,

however, a player may be pessimistic and want to maximize his/her worst-case payoff or even feel

that everyone wants to harm him. In these cases a maxmin strategy6 is applicable. The maxmin

(safety) value is the minimum payoff guaranteed by a maxmin strategy. The maxmin strategy for

player 𝑖 is arg𝑚𝑎𝑥!!𝑚𝑖𝑛!!!𝑢!(𝑠! , 𝑠!!) and the maxmin value is 𝑚𝑎𝑥!!𝑚𝑖𝑛!!!𝑢!(𝑠! , 𝑠!!). In two player zero-

sum games one commonly refers the minmax strategy. Since it is a zero-sum game by minimizing an

opponent’s utility one maximizes his/her own utility. The formal definitions are very similar and easy to

6 When considering pure strategies some authors call this the maxmin criterion

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extrapolate. The maxmin/minmax value and/or strategy can be obtained through graphical procedures

where one identifies a saddle point or through linear programming, which can be an entry point for

OR.

Besides the normal form game representation one can use the extensive form as an alternative

that is able to make the temporal structure explicit. There are two variants: perfect information and

imperfect information extensive form games. A (finite) perfect information extensive form game is

denoted by 𝑁,𝐴,𝐻,𝑍,𝜒, 𝜌,𝜎, 𝑢 where 𝑁 is the set of players, 𝐴 is a single set of actions, 𝐻 is a set of

choice nodes where players take an action, 𝜒:  𝐻 → 2! is an action function associating to each choice

node a set of possible actions, 𝜌:  𝐻 → 𝑁 is a player function identifying who plays at each node, 𝑍 is a

set of terminal nodes that are disjoint from 𝐻, 𝑢 = 𝑢!,… , 𝑢! ;  𝑢!:ℤ → ℝ is the utility function and

𝜎:  𝐻×𝐴 → 𝐻 ∪ 𝑍 is a successor function mapping a choice node and action to a new choice or terminal

node. If ℎ!, ℎ! ∈ 𝐻 and 𝑎!, 𝑎! ∈ 𝐴, if 𝜎 ℎ!, 𝑎! = 𝜎(ℎ!, 𝑎!) then ℎ! = ℎ! and 𝑎! = 𝑎!. The pure strategies

for player 𝑖 are 𝜒  (ℎ)!!!",! ! !! . This definition allows for the definitions of mixed strategy, best

response and NE previously used. In fact, all extensive form games can be converted to normal form

games.

Figure 2.7 Perfect Information Extensive Form Game

The perfect information extensive form game in Figure 2.7 has two players as depicted next to the

choice nodes. Several other concepts and definitions for perfect information extensive form games

could be given but it is important to keep in mind the purpose of this section. The objective is only to

provide the fundamental concepts.

One may want to consider cases where players have no knowledge on others actions (or even

their own) through an imperfect information extensive form game   𝑁,𝐴,𝐻,𝑍,𝜒, 𝜌,𝜎, 𝑢, 𝐼 , which is a

perfect information extensive form game plus an information set. This information set comprises nodes

player 𝑖 cannot distinguish. Formally, 𝐼 = 𝐼!,… , 𝐼! where 𝐼! = 𝐼!,!,… , 𝐼!,!! is a partition of ℎ ∈

𝐻: 𝜌 ℎ = 𝑖 that follows the property 𝜒 ℎ = 𝜒(ℎ!) and 𝜌 ℎ = 𝜌(ℎ!) whenever ∃𝑗: ℎ ∈ 𝐼!,!and ℎ! ∈ 𝐼!,! .

Pure strategies are the cross product 𝜒(𝐼!,!)!!,!∈!! . Mixed strategies, best response and NE are easily

defined. In imperfect information extensive form games, there may also be discussion about

behavioural strategies. This will not be discussed. The definitions for repeated, stochastic and

17

Bayesian games will not be provided since these go beyond the scope of this section. However

coalitional games are highly relevant.

In coalitional games focus is given towards how groups of agents act rather than individuals. It

does not concern how individuals choose within a coalition nor how they coordinate. A coalitional

game with transferable utility (TU) assumes that utility may be distributed among members of the

coalition, individuals are happy when this utility or payoff is distributed in a universal currency and

each coalition can be assigned a single value of utility. A TU coalitional game is a pair 𝑁, 𝑣 where 𝑁

is the set of players and 𝑣:  2! → ℝ associates to each coalition 𝑆 ⊆ 𝑁 a real valued payoff 𝑣  (𝑆). A TU

coalitional game may help understand which coalition(s) form and how its members divide payoff. A

game 𝐺 = 𝑁, 𝑣 is superadditive if ∀𝑆,𝑇 ⊂ 𝑁 , if 𝑆 ∩ 𝑇 = ∅ , then 𝑣 𝑆 ∪ 𝑇 ≥ 𝑣 𝑆 + 𝑣(𝑇) . In

superadditive games the grand coalition will form. In terms of what each member of the coalition

should receive, Shapley (1953) suggested some axioms to help answer the question of what is a fair

division of payoff. Denoting 𝜓!(𝑁, 𝑣) as the utility given to player 𝑖 in the coalition formed in game

𝐺 = (𝑁, 𝑣). Interchangeable players relative to 𝑣 are those that for all 𝑆 that contains neither 𝑖 nor 𝑗,

𝑣   𝑆 ∪ 𝑖 = 𝑣  (𝑆 ∪ 𝑗) . Shapley’s (1953) axiom of symmetry states that for any 𝑣 if 𝑖 and 𝑗 are

interchangeable then 𝜓! 𝑁, 𝑣 = 𝜓!(𝑁, 𝑣), meaning they should receive the same payoff. A dummy

player is one that for all 𝑆:  𝑣   𝑆 ∪ 𝑖 = 𝑣  (𝑠). Shapley’s (1953) axiom states that if player 𝑖 is a dummy

player then 𝜓! = 0. Shapley’s (1953) additivity axiom states that for the sum of any two coalitional

games 𝑣 = 𝑣! + 𝑣!, 𝜓! 𝑁, 𝑣! + 𝑣! = 𝜓! 𝑁, 𝑣! + 𝜓!(𝑁, 𝑣!) given that 𝑣! + 𝑣! 𝑆 = 𝑣! 𝑠 + 𝑣!(𝑠). Given

these axioms the Shapley Value (SV) is denoted by 𝜙! =!!!

𝑆 ! 𝑁 − 𝑆 − 1 ! 𝑣 𝑆 ∪ 𝑖 −!⊆!\ !

𝑣(𝑆) , for each player 𝑖. The SV is the unique payoff division that satisfies the axioms of symmetry,

dummy player and additivity. The SV captures the marginal contribution of each player 𝑖 and weights it

according to all the possible ways of obtaining that contribution. Another famous allocating rule is the

core. This concept will not be explored here.

Notice that, following the presented nomenclature; 𝐴 can either represent the set of available

actions for players or, in an SNA framework, the set of arcs in the graph as well as 𝐺 may represent a

game or a graph. Whenever two of these concepts appear together they will be properly distinguished.

2.2 Specific objectives

As discussed by Alderson (2008) there may be an opportunity for the operations researcher in SNA.

This project intends to explore upon that statement and to identify what is the entry point for the

operations researcher in SNA as well as to, not only, identify, but also, understand the framework for

OR and engineering (generally speaking) within SNA. Further, it is necessary to define the state of the

art in SNA considering GT and MOO and, if necessary, to define a framework for future research.

Additionally, it is of primary need to understand how SNA problems with a GT approach can be

formulated through MOO and how classical SNA concepts can be modelled; such as centrality

measures, namely, from a GT perspective, the SV.

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3 Literature Review and State of the art

This section provides a brief historic review of the evolution of SNA before discussing the state of the

art in SNA. Considering the research topic definition from Section 1; focus will be given to SNA when

allied to GT and MOO. It has not been possible to identify literature corresponding exactly to a GT

approach and MOO methodology in SNA; only literature partially corresponding to the research topic

definition was found. In fact, this would be expected due to the innovativeness of the research topic.

Some effort will have to be employed in order to ally MOO and SNA literature with GT and SNA

literature.

3.1 History

Moreno and Jennings (1932;1934) are the formal founders of SNA but, at that time, named it

Sociometry. At the same time Warner (1941; 1941) designed a study and involved students and

colleagues to study social networks. There was a third group, headed by Lewin, in the thirties, also

developing work on social networks and social psychology, that made relevant contributions (Lewin et

al. 1938). Lewin’s group made several notable contributions to SNA to what it is presently (Festinger

et al. 1950; Newcomb 1961). The work of Travers and Milgram (1969) was also very significant for the

evolution of social networks study. Likewise, Granovetter (1973) also made relevant contributions to

the advance of SNA, namely in network structure understanding. Watts (1999) took advantage of

Milgram’s and Granovetter’s work and stated that most natural and man-made networks are highly

clustered yet far reaching.

Prior to 1970 several centres started working and researching on social networks, but only in the

early 1970s did White (1971; 1976) help to popularize SNA, with his deep knowledge on the topic and

ease of communicating it with his colleagues. In the late 1990s several physicists also began

publishing on social networks.

Currently SNA is a research area catching the attention of many scientists and researchers from

the most diverse areas. More recently it has even developed interest in the OR community where the

present document inserts itself. A detailed insight on the history and origins of SNA is not provided;

instead, only a brief introduction and contextualization. For a more detailed review of SNA’s origins

and history consult (Borgatti & Foster 2003; Barabási 2003; Watts 2004; Freeman 2004).

3.2 State of the Art

This project takes advantage of some already existing MOO literature (Eusébio & Figueira 2009;

Gómez et al. 2013; Eusébio et al. 2014) as its starting point. Eusébio and Figueira (2009) developed

an algorithm for finding all the non-dominated, and consequent efficient solutions, in MOO problems.

In fact they only show it for a bi-objective optimization problem. The issue explored is an integer

network flow problem, which implies that the model is a connected digraph. Hence, the problem being

solved is a bi-objective integer network flow (BOINF) problem.

Eusébio and Figueira (2009) explored not only MOO, but also integer programming. The use of MOO

is intended to allow for a better understanding of the real world decision making process. The method

19

proposed by Eusébio and Figueira (2009) solves a sequence of 𝜀-constraint problems through a

branch-and-bound algorithm. The used approach is implicit enumeration. The non-integer solutions

are obtained through parametric programming and a typical network simplex algorithm. The bi-

objective network flow problem (BONF) can be stated as min 𝐶!𝑥:  𝐴𝑥 = 𝑏, 𝑙 ≤  𝑥 ≤ 𝑢 with 𝑥 ∈ 𝑋 .

Some assumptions are taken: the costs (elements of the objective matrix), lower and upper bounds of

the arcs and demand/supply of the vertices are finite and integral. The integer nature of the problem

raises a difficulty, there exist two types of non-dominated solutions (criterion vectors): supported and

unsupported. According to the problem presented, the 𝜀 -constraint method is as follows

min 𝑓! 𝑥 : 𝑓!   𝑥 ≤ 𝜀, 𝑥 ∈ 𝑋! where 𝜀 is a scalar value. Eusébio and Figueira (2009) propose a

network simplex algorithm and a parametric programming based algorithm so as to achieve 𝑌! and

𝑋! . Subsequently, a branch-and-bound method (allied to 𝜀-constraint) obtains 𝑌!! and 𝑋!! . These

algorithms are proved to be able to get all the non-dominated criterion vectors. Eusébio and Figueira

(2009) test their algorithms and methods with an illustrative example. The networks were generated

with NETGEN network generator (Klingman et al. 1974) and a multiple linear regression model was

applied through SPSS software version 17.0 (SPSS 2009). Eusébio and Figueira (2009) conclude that

the proposed algorithm keeps the network structure of the problem while scanning the feasible space

𝑋. The algorithm cannot be applied to large size instances but requires small CPU time for small and

medium size instances. Also the proposed method could be adjusted to consider only desirable

regions of 𝑋 according to the decision maker’s preferences. In terms of future research Eusébio and

Figueira (2009) also suggest that an improved version of the network simplex algorithm is developed

in order to reduce CPU time and consider large size instances and, also, to extend the proposed

method and algorithms to more than two objectives.

Eusébio et al. (2014) further developed their research (Eusébio & Figueira 2009) and propose a new

algorithm that is able to obtain a representation of the non-dominated criterion vectors of a BOINF

problem. The purpose for this work is to facilitate the decision maker’s task of choosing one solution

over all others. The less options the analyst finds the easier it is to take a decision. It is important,

however, to ensure the quality of this subset of solutions. The model and approach are similar to that

of their previous work (Eusébio & Figueira 2009) added the fact that criteria for the subsets are

included. These are coverage (𝛾), uniformity (𝛿) and cardinality ( 𝑌! ). Let 𝛾 ∈ ℝ! and  𝑌! ⊆ 𝑌! then 𝑌!

is a 𝛾-representation of 𝑌! if for all 𝑦 ∈ 𝑌! there exists 𝑦! ∈ 𝑌! such that 𝑑(𝑦, 𝑦!) ≤ 𝛾, where 𝑑:ℝ! → ℝ

is a distance measure. Letting 𝛿 ∈ ℝ!, 𝑌! is a 𝛿-uniform representation of 𝑌! if for all 𝑦! , 𝑦! ∈ 𝑌! , 𝑦! ≠

𝑦! it holds that 𝑑(𝑦! , 𝑦!) ≥ 𝛿. Also, the cardinality of the subset 𝑌! is denoted by 𝑌! . Eusébio et al.

apply a subroutine of the 𝜀-constraint network flow algorithm. The logical implementation sequence

proposed is to apply a representation algorithm and, then, the 𝜀-constraint network flow algorithm. A

FindEpsilonp procedure is also run to define the value of 𝜀. Eusébio et al. (2014) were able to develop

an algorithm with the ability to obtain 𝛾 -representations and 𝛿 -representation of a set of non-

dominated criterion vectors. The computation time is a function of the complexity of the network

(namely the number of nodes and arcs). As future avenues for research, they identify that the study of

20

the distribution of non-dominated criterion vectors could help to gain further insight on how to improve

the proposed algorithm.

Gómez et al. (2013) use a different perspective towards MOO in networks; several centrality

measures are extended to consider two criteria instead of only one, by modelling them as bi-criteria

network flow problems (equivalent to BONF problems). In fact Gómez et al. (2013) refer to Freeman et

al. (1991) as one of the first authors to relate centrality measures with network flow optimization

problems in the context of social networks. Network analysis focuses a lot on the concept of centrality.

It measures the importance of each node in a defined network. Understanding the conceptual

complexity of centrality may be very challenging. In a social networks framework, centrality may

represent the ability for a node to communicate directly with others or, even, its positioning as an

intermediary in communication, thus having control over information flow or being able to eavesdrop

communications in the network. Degree centrality is the outdegree of a node 𝑖 or, equivalently, 𝐴(𝑖) .

Closeness centrality is defined as the ability of node 𝑖 to communicate with other nodes in the network

using the minimum possible number of intermediaries. Notice that degree centrality is a special case

of closeness centrality where no intermediaries are allowed. Betweenness centrality is defined as the

ability of node 𝑖 to be an intermediary in communications between two other nodes in the network.

Notice that betweenness centrality is a measure of control over communication. Several variants of

these foundational centrality measures have been proposed. The most common centrality measures

found in SNA software are shortest path closeness, shortest path betweenness, information centrality,

eigenvector centrality, Katz centrality, eigenvector centrality for non symmetric relations, centrality

based on GT and betweenness centrality based on random walks. Game theoretic based centralities

account for the interaction of agents in a cooperative game. These measures of centrality (commonly

the SV) present high computational complexity in their calculation; some approximation to compute

the SV have been already proposed (Castro et al. 2009). Gómez et al. (2013) claim that, to their

knowledge, centrality measures based on GT have only been developed for unweighted networks.

Gómez et al. (2013) propose a multi-criteria approach towards centrality. This allows for the explicit

consideration of different dimensions of a problem when applying centrality analysis. Besides these

typical centrality measures, flow centrality measures are available for valued digraphs (networks).

Flow betweenness (FB) represents the amount of communication in the network that has node 𝑖 as an

intermediary and its normalized value is calculated as follows 𝐹𝐵!! =!!"

!!"∈!

!!"!"∈! with 𝑖, 𝑗, 𝑘 ∈ 𝑁, 𝑖 ≠ 𝑗, 𝑖 ≠

𝑘, 𝑗 ≠ 𝑘 and where 𝑓!" and 𝑓!"! represent the maximum flow from node 𝑖 to node 𝑗 and the maximum

flow from node 𝑖 to node 𝑗 requiring node 𝑘 as an intermediary, respectively. Notice that 𝑓!"! can be

obtained through 𝑓!"! = 𝑓!" − 𝑓!"

! where 𝑓!"! is the maximum flow from node 𝑖 to node 𝑗 after node 𝑘 is

removed from the network. Flow closeness (FC) represents the flow between nodes as a power

measure and is calculated as follows 𝐹𝐶! = 𝑓!"!∈! with 𝑗 ≠ 𝑘. The influence of node 𝑖 over node 𝑗 is

simply defined as maximum flow from node 𝑖 to node 𝑗. This flow centrality measure can be extended

to sets of nodes. The influence of set 𝑇 over set 𝑅 given that 𝑇,𝑅 ⊆ 𝑁 and 𝑇 ∩ 𝑅 = ∅, is denoted by

𝑓!" = 𝑓!"!∈!,!∈! . Gómez et al. (2013) work from these flow centrality measures and model the

21

problem as a BONF problem and propose a family of flow-cost centrality measures. These incorporate

the cost of communications besides only the information flow (communication) among nodes. Gómez

et al. (2013) also present some concepts. Assuming a BONF problem, if 𝑦, 𝑦! ∈ ℝ! are two bi-

dimensional vectors 𝑦 = (𝑦!, 𝑦!) and 𝑦! = (𝑦!!, 𝑦!!) then the lexicographic order denoted by ≥!"# is

stated as follows 𝑦 ≥!"# 𝑦! if, and only if, 𝑦! > 𝑦!! or 𝑦! = 𝑦!! and 𝑦! > 𝑦!!. If 𝑦! = 𝑦!! and 𝑦! = 𝑦!!

then 𝑦 and 𝑦! are lexicographically indifferent denoted by ~!"#. Consider 𝑌! and 𝑌!! in which elements

are lexicographically ordered, 𝑌!  ~!"#  𝑌!! if, and only if, 𝑌! = 𝑌!! , 𝑌! ≻ 𝑌!! if max   𝑌! ∩ 𝑌!! ∪

𝑌! ∩ 𝑌!! ∈ 𝑌! and 𝑌! ≽ 𝑌!! if 𝑌! ≻ 𝑌!! and 𝑌!  ~  𝑌!!. Also, Gómez et al. (2013) define that 𝑌! 𝑌!! =

ND{x ∈ ℝ!: 𝑥 = 𝑦 + 𝑦!, 𝑦 ∈ 𝑌! , 𝑦! ∈ 𝑌!!} where 𝑁𝐷 represents the set of non-dominated vectors. The

authors apply their proposed approach to centrality measures to a network of the Iranian government

in order to identify individuals that are the most influential in the Iranian government and show that the

centrality measures used should be decided according to the context of each problem. The authors

conclude that the proposed measures of centrality allow for the consideration of several

measurements with different scales and, even, in some cases, the number of intermediaries in

communication flows. They also state that this approach is more adequate for constructive

frameworks considering interaction among agents. Typical centrality measures cannot consider

simultaneously negative and positive ties. Bi-criteria centrality measures implicate much more

computation time.

Given this starting point two categories of SNA were defined: node/edge centric analysis and network

centric analysis. The previously discussed centrality measures belong to the node/edge centric

analysis category. The determination of diversity among nodes and link prediction problems are some

other cases of node/edge centric analysis. The network centric analysis includes community detection,

graph visualisation and summarization, subgraph discovery, generative models and other typical SNA

tasks. It is also exceedingly important to specify the methods considered addressing SNA problems

and tasks. This work will consider a variation of traditional optimization techniques, through OR and

MOO, and more recent game theoretic techniques.

3.2.1 Node/Edge Centric Analysis

Most of the research carried out in a node centric analysis perspective focuses on determining the

centrality of nodes in a graph (or network). The purpose is, mainly, to determine key entities with

powerful positions in the network. This analysis uses the network structure to determine a centrality

measure for each node, sometimes even sets of nodes (Amer et al. 2007; Kolaczyk et al. 2009;

Everett & Borgatti 2010; del Pozo et al. 2011; Szczepański et al. 2012; Lindelauf et al. 2013).

Amer et al. (2007) propose a concept of accessibility that allows for the determination of the

importance of players in a cooperative game. The cooperation opportunities are limited by the links of

a digraph without loops. The main objective of this concept of accessibility is to take into account each

player’s marginal contribution to the possible coalitions. The digraph represents the ability of certain

players to propose cooperation to others. Contrary to most literature, Amer et al. (2007) consider that

22

the preferable position is the head node since accessibility does not necessarily imply power over

other nodes. The authors try to rank the nodes according to each node’s marginal value in the

cooperative game and its structural position in the network. The game considered is a typical TU

coalitional game. Following the classical notation of GT, where a TU game is denoted by 𝐺 = 𝑁, 𝑣 ,

the authors denote by 𝐺! the set of all possible games on 𝑁. Furthermore, 𝐻 𝑆 denotes the set of all

the orders of elements from a set 𝑆 ⊆ 𝑁. Also, the elements 𝑇 = 𝑖!,… , 𝑖! ∈ 𝐻 𝑆 such that ∅ ≠ 𝑆 ⊆ 𝑁

are called the ordered coalitions. A consecutive subcoalition of 𝑇 is 𝑄 = 𝑖!,… , 𝑖!!! where 1 ≤ 𝑝 ≤

𝑝 + 𝑢 ≤ 𝑠 . Given the digraph 𝐷 = 𝑁,𝐴 , a connected consecutive subcoalition according to 𝐷 is

denoted by 𝑄 = (𝑖!,… , 𝑖!!!) where either 𝑢 = 0 or (𝑖! , 𝑖!!!) ∈ 𝐷, 𝑗 = 𝑝,… , 𝑝 + 𝑢 − 1. If 𝑝 = 1, (𝑖!!!, 𝑖!) ∉

𝐷 or (𝑖!!!, 𝑖!!!!!) ∉ 𝐷 it is called a maximal connected consecutive subcoalition according to 𝐷. The

game 𝑣 modified by the digraph 𝐷 is a game in the generalized characteristic function form denoted by

𝑣 𝑇 = 𝑣(𝑄!)!∈!|! ,∀𝑇 ∈ 𝐻 𝑆 ,∀𝑆 ⊆ 𝑁, 𝑆 ≠ 0 with 𝑇|𝐷 as the set of maximal connected consecutive

subcoalitions of 𝑇 according to the digraph 𝐷 and 𝑄! is the non-ordered coalition in 𝑁 formed from the

elements of 𝑄. The accessibility measures for node 𝑖 is denoted by 𝛼! 𝑣;𝐷 = !!!

𝑣! 𝑇|! , 𝑖 −!∈!(!)

𝑣!(𝑇|! where 𝑇|! denotes the consecutive subcoalition of 𝑇 with the same initial element and whose

last element is the one previous to 𝑖. Also, (𝑇|! , 𝑖) denotes 𝑇|! added 𝑖 as tail endpoint. Furthermore,

the authors prove that the concept of accessibility verifies the following properties for digraphs:

linearity; dummy player; average efficiency; if node 𝑖 is inaccessible 𝛼! 𝑣;𝐷 = 𝑣( 𝑖 ); the accessibility

of a node is not affected by adding an oriented edge leaving it; if the game is superadditive the

accessibility of a node is not affected by the addition. Note that for a complete digraph the accessibility

for a node equals its SV. Amer et al. (2007) define oriented paths as a pair (𝑁,𝑃) with

𝑃 = 𝑖!, 𝑖! ,… , (𝑖!!!, 𝑖!) ≠ 0 where 𝑖!,… , 𝑖! ∈ 𝑁 . Nodes 𝑖! and 𝑖! are the first and last nodes in the

oriented path, respectively. A convex game 𝑣 ∈ 𝐺! is one where 𝑣 𝑆! + 𝑣 𝑆! ≤ 𝑣 𝑆! ∪ 𝑆! +

𝑣 𝑆! ∩ 𝑆! ,∀𝑆!, 𝑆! ⊆ 𝑁. In convex games the addition of edges previous to the first node of an oriented

path does not decrease the accessibility of its last node. The accessibility of node 𝑖 with 𝑞 ≤ 𝑖 ≤ 𝑛, for

an oriented path 𝑃!! , is 𝛼! 𝑣;  𝑃!! = !!!

𝑛 − 1 𝑛 − 1 ! 𝑣(𝑆!!) + 𝑛 − 1 − 𝑗!!!!!!!! 𝑛 − 1 − 𝑗 !  ×

  𝑣 𝑆!!!! − 𝑣(𝑆!!!!!!) + 𝑞! 𝑣 𝑆!! − 𝑣 𝑆!!!! , where 𝑆!! = 𝑎, 𝑎 + 1,… , 𝑏 is a non-ordered coalition

formed by consecutive nodes. Amer et al. (2007) show that for an oriented path 𝑃 on 𝑁, adding a

oriented edge with opposite direction does not change the subsequent nodes’ accessibility; 𝛼! 𝑣;𝑃 ∪

𝑃! =  𝛼! 𝑣;𝑃 +  𝛼! 𝑣;𝑃! −  𝛼! 𝑣;𝑃 ∩ 𝑃! with 𝑃 and 𝑃! two oriented paths such that they have last

node 𝑖 and such that 𝑃 ∩ 𝑃! have last node 𝑖; 𝛼! 𝑣;𝐷 = 𝛼! 𝑣;𝐷! where 𝐷!  is the union of all paths in 𝐷

with last node 𝑖. The authors apply the accessibility concept to several games (unanimity games, the

pairs game, the conferences game, digraph competitions and oriented networks). Amer et al. (2007)

conclude that accessibility is an extension of the SV. This work is a clear example of GT offering a

tool, which is suitable for considering cooperation or communication in an established direction.

Kolaczyk et al. (2009) exploit the concepts of group betweenness centrality and co-betweenness

centrality as measures of the control over the flow of information for each node. The concept of group

betweenness centrality was introduced by Everett & Borgatti (1999). Kolaczyk et al. (2009) study the

23

relation between group betweenness and co-betweenness as a means to deepen the understanding

of both centrality measures. The authors also present a computationally efficient algorithm based on

Brandes’ (2001) work, in order to calculate pairwise co-betweenness. Typically, betweenness

centrality measures the number of geodesic paths passing through a node. Thus, betweenness

centrality measures the control of a given node over the communication in the network. Betweenness

centrality cannot reflect the power over communication of groups of nodes; it only provides a measure

for individual nodes. Group betweenness and co-betweenness centrality are able to represent the

influence of a group (or coalition) over the network. The former defines the betweenness of a set of

nodes through the geodesic paths that pass through at least one node of the coalition, the latter

defines it trough the number of geodesic paths passing through all the nodes of the coalition. Kolaczyc

et al. (2009) show that group betweenness of a coalition is upper and lower bounded by its individual

nodes’ betweenness centrality and by its co-betweenness centrality. A geodesic path (also known as

shortest path) between nodes 𝑖, 𝑗 ∈ 𝑁 is a path with the minimum length among all possible paths

between 𝑖 and 𝑗. The authors restrict their study to unweighted graphs; it is possible to extend these

definitions to weighted graphs. In fact, the model used by the authors is an unweighted undirected

connected graph. The betweenness of node 𝑖 is denoted as 𝐵! =!!"(!)!!"!,!∈!\ ! , where 𝜎!" represents

the number of shortest paths between nodes 𝑠 and 𝑡 and 𝜎!"(𝑖) represents the number of shortest

paths between nodes 𝑠 and 𝑡 passing through node 𝑖. The betweenness centrality may be normalized

𝐵 𝑖 = !!(!)( ! !!)( ! !!)

; recall that 𝐺 is the order of the graph. The authors use the Abilene network7 as

an illustrative example. The group betweenness centrality between two nodes 𝑖, 𝑗   ∈ 𝑁 is defined by

𝐵 𝑖, 𝑗 = 𝐶 !,! 𝑖, 𝑖 + 𝐶 !,! 𝑗, 𝑗 − 𝐶 !,! (𝑖, 𝑗) where 𝐶 !,! 𝑖!, 𝑖! = !!"(!!,!!)!!"!,!∈!\ !,! is the co-betweenness

of nodes 𝑖! and 𝑖!. The normalized group betweenness is expressed as 𝐵 𝑖, 𝑗 = !!(!,!)( ! !!)( ! !!)

. These

concepts can be extended to a set 𝐴 ∈ 𝑁. The group betweenness is 𝐵 𝐴 = !!"∗ (!)!!"!,!∉!:!!! where

𝜎!"∗ (𝐴) is the number of geodesic paths that pass through at least one element of 𝐴. The normalization

is denoted by 𝐵 𝐴 = !!(!)( ! ! ! )( ! ! ! !!)

. The numerator of the group betweenness is calculated as

follows 𝜎!"∗ 𝐴 = (−1)!!! 𝜎!"(𝑖!)!!⊆!!!!! with 𝐴 = 𝑚. Upper and lower bounds for 𝐵(𝐴) provided by

the authors will not be analysed since it is out of scope of this project. Also an algorithm for the

calculation of the pairwise co-betweenness centrality measure is provided. This algorithm extends

from that presented by Brandes (2001). It is a three-stage procedure exploiting recursions. Two simple

changes to this algorithm allow for the computation of co-betweenness for any set 𝐴 ⊆ 𝑁. Kolaczyk et

al. (2009) illustrate their algorithm on Michael’s strike network and on Zachary’s karate club network.

The authors conclude that pairwise of co-betweenness is of fundamental interest, allowing to obtain a

close approximation of group betweenness. Further, they conclude that the proposed approach gives

further insight into the composition of a coalition and, even, redundancy of certain actors (in terms of

control over the network). Kolaczyk et al. (2009) state that there is a need for the development of more

refined and efficient algorithms to obtain group betweenness and co-betweenness and that it is

7 http://www.internet2.edu

24

necessary to study in more depth the accuracy of the proposed bounds. Moreover, more work is

recommended to understand the relationship between these centrality measures and coalition

formation. In this project a question is raised concerning the methodology used by Kolaczyk et al.

(2009). To what extent are the geodesic paths representative of the used communication paths? This

question will be further developed in this project.

Everett & Borgatti (2010) explore the concepts of induced, exogenous and endogenous centrality. The

induced centrality measure, proposed by the authors, is obtained by taking any graph invariant (such

as density or maximum flow) and deriving its value, by deleting nodes and measuring the change in

the network property. According to Everett & Borgatti (2010) this concept of centrality allows for its

adaptation to any specific research problem accounting for its specifications. A graph invariant is a

measure that solely depends on the graph structure. Curiously, the presented methodology can easily

be extended to edge centrality measures. If 𝑓 is a graph invariant then the induced centrality of node 𝑖

is 𝐶! = 𝑓 𝐺 − 𝑓(𝐺 − 𝑖 ) where 𝐺 − 𝑖 is the graph without node 𝑖. The induced centrality concept

allows for the representation of well-known centrality measure as induced centralities. Graph

invariants must be defined in all graphs, not be normalized by the number of nodes in the graph, be

sensitive to node removal and be affected by the graph structure when the removal of nodes affects its

value in order to be useful. The authors name total centrality the measure that uses as graph invariant

the sum of all node’s scores of a centrality measure, and denote it as 𝐶! 𝑖 = 𝐶(𝑗)!∈! −

𝐶!(𝑗)!∈!\ ! . Total centrality represents not only the contribution of a node to the network but also its

contribution to other nodes. The former contribution is called endogenous contribution and the latter

exogenous contribution. Everett & Borgatti (2010) decompose the induced centrality into endogenous

and exogenous contributions as follows 𝑡𝑜𝑡𝑎𝑙   𝑖𝑛𝑑𝑢𝑐𝑒𝑑  𝑐𝑒𝑛𝑡𝑟𝑎𝑙𝑖𝑡𝑦 = 𝑒𝑛𝑑𝑜𝑔𝑒𝑛𝑜𝑢𝑠  𝑐𝑒𝑛𝑡𝑟𝑎𝑙𝑖𝑡𝑦 +

𝑒𝑥𝑜𝑔𝑒𝑛𝑜𝑢𝑠  𝑐𝑒𝑛𝑡𝑟𝑎𝑙𝑖𝑡𝑦. The authors consider several well-known centralities: degree, betweenness,

reverse-closeness and eigenvector centralities. Induced centrality measures have a clear

interpretation; they represent the contribution of a node to the graph invariant of the network. This

methodology allows for the construction of a centrality measure for virtually any graph invariant. Also,

induced centralities, allow for the analyst to generalize node centralities to group (or coalition)

centralities. Finally, it is possible to build a node by node matrix, showing each node’s contribution to

other nodes’ centrality.

A family of centralities applicable to directed networks is provided by del Pozo et al. (2011); a game

theoretic approach is used. Work from Gomez et al. (2003) serves as the starting point for the

development of this family of centrality measures. The proposed game theoretic centrality measure

can be divided into three subclasses similar to reception, betweenness and emission centrality. This

can be done due to the measure being defined as a vector rather than a scalar measure. The authors

analyse centrality measures for directed networks; more work on this topic can be found in White &

Borgatti (1994), Borgatti (2005), Tutzauer (2007) and Pollner et al. (2008). The work of del Pozo et al.

(2011) assumes that actors are players in a TU game and are located in a directed network. The

structure of the digraph modifies the TU game into what the authors call the “digraph restricted game”.

25

This is a generalized TU game and, as shown by Nowak & Radzik (1994), the value of a coalition

depends on its members and, also, on the order they join the coalition. The centrality of each player

will be dependent upon the interests motivating coalition formation. The technique used is very similar

to the one used by Amer et al. (2007). The directed nature of the network introduces asymmetry to the

model and, consequently, two incident nodes may have different bargaining power. The set of all

ordered coalitions of players in 𝑆 ⊆ 𝑁 is denoted by Π 𝑆 . The set of all ordered coalitions of players in

𝑁 is denoted by Ω 𝑁 = 𝑇 ∈ Π(𝑆)|𝑆 ⊆ 𝑁 . The set of players in coalition 𝑇 is denoted by 𝐻 𝑇 = 𝑆;

also, 𝑡 = 𝐻 𝑇 . In the model proposed by the authors, payoff 𝑣(𝑇) represents social or economic

possibilities of the coalition if it is formed in the order proposed by 𝑇. The set of all generalized

cooperative games with set of players 𝑁 is denoted by 𝒢!; by 𝐺! one can denote the subspace of 𝒢!

where 𝑣 𝑇 = 𝑣(𝑅) if 𝐻 𝑇 = 𝐻(𝑅). A strict linear order ≺! is established by each ordered coalition

𝑇 = (𝑖!,… , 𝑖!), and is defined as 𝑖 ≺! 𝑗 if and only if there exists 𝑘, 𝑙 ∈ 1,… , 𝑡 , 𝑘 < 𝑙: 𝑖 = 𝑖! , 𝑗 = 𝑖!. Given

𝐴,𝐵 ∈ Ω 𝑁 , 𝐴 is included in B, denoted by 𝐴 ⊂ 𝐵, if 𝐻(𝐴) ⊂ 𝐻(𝐵) and ∀𝑖, 𝑗 ∈ 𝐻(𝐴) if 𝑖 ≺! 𝑗 then 𝑖 ≺! 𝑗.

Also, 𝑖(𝑇) denotes the position of player 𝑖 in coalition 𝑇. The generalized characteristic function 𝑤! is

defined as 𝑤! 𝑅 = 1          𝑖𝑓  𝑇 ⊂ 𝑅0  𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

with 𝑅 ∈  Ω 𝑁 for any 𝑇 ∈ Ω 𝑁 \ ∅ . The family 𝑢! !⊂!\ ∅ , a basis

of 𝐺!, is defined as 𝑢! = 𝑤!(𝑆)!∈!(!) . For a given game 𝐺 = (𝑁, 𝑣)  ∈ 𝒢!, ∆!∗(𝑇) !∈!(!)\ ∅ is the set

of generalized unanimity coefficients of 𝑣 and is obtained as follows ∆!∗ 𝑇 = 𝑣 𝑇 − ∆!∗ 𝑅!⊂!,!!! =

−1 !!!𝑣(𝑅)!⊂! , where ∆!∗ 𝑇 = ∆! 𝑆 for all 𝑇 ∈ Π(𝑆). The coefficient ∆!∗ 𝑇 could be interpreted as

being the variation of payoff by forming coalition 𝑇 from its subcoalitions. A game is convex if for all

coalition 𝑆,𝑇 ⊂ 𝑁, 𝑣 𝑆 ∪ 𝑇 + 𝑣 𝑆 ∩ 𝑇 ≥ 𝑣 𝑆 + 𝑣(𝑇). A game is symmetric if 𝑣 𝑆 depends only on the

cardinality of 𝑆. The set 𝑆! is the subset of 𝐺! formed by symmetric games. A game is 0-normalized if

for all 𝑖 ∈ 𝑁, 𝑣 𝑖 = ∆! 𝑖 = 0. The set of all symmetric and 0-normalized games is denoted by

𝑆!! ⊂ 𝑆!. The work of del Pozo et al. (2011) proposes a parametric family of functions Ψ!!∈[!,!]

defined on 𝒢!, calculated as follows Ψ!! = ∆!∗(𝑇)!∈! ! ,!∈!(!)!!!!(!)

!! !!!!!!!!

. This value coincides with the

SV for games in 𝐺!. The authors denote by 𝐷! the set of all possible digraphs over node set 𝑁.

Nodes 𝑖 and 𝑗 are in the same component if there exists a sequence 𝑖!,… , 𝑖! such that, for any

𝑙 = 1,… , 𝑟 − 1, 𝑖! , 𝑖!!! or 𝑖!!!, 𝑖! ∈ 𝐸. The set of all components of the digraph is denoted by 𝑁/𝐸.

Given a component 𝐶 ∈ 𝑁/𝐸, 𝑇 ∈ Ω(𝐶) can be a non connected ordered set in the digraph. A digraph

is (weakly) connected if 𝑁/𝐸 = 1. One can denote (𝑁,𝐸\𝑖𝑗) as (𝑁,𝐸!") instead, for simplicity. A

digraph communication situation is a triplet (𝑁, 𝑣,𝐸), where 𝒟𝒞! represent the set of all digraph

communications with node/players set 𝑁. Also, 𝒟𝒞!! and 𝒟𝒞!"#! are used if the game is symmetric or if

the game is symmetric and almost positive, respectively. The proposed game theoretic centrality

measure, considering symmetric games, is 𝑘!! = Ψ!! 𝑁, 𝑣! where 𝑣! = ∆!(𝑆)𝑢!!!⊂! and 𝑢!! =

𝑤!!∈! ! ∩!!! . This measure of centrality satisfies stability, efficiency and fairness requirements. The

proposed centrality measure has a part due to each actor’s communication activity (reception and

emission) and control over others’ communication (betweenness). A centrality measure for directed

edges is, thus, proposed.

26

Szczepański et al. (2012) propose a Shapley valued based betweenness measure. The purpose is to

define a betweenness measure for, not only individual entities, but also, coalitions. Other authors have

also proposed a game theoretic network centrality measures (Gomez et al. 2003) based on GT

concepts. A problem raised by these measures is the computational complexity of calculating them.

Szczepański et al. (2012) start by refining the standard betweenness centrality measure through GT

concepts and proposing a more efficient algorithm for the computation of betweenness measures

based on the SV. The authors also show possible extensions of the proposed measure. The coalition

made of all predecessors of player  𝑖 in 𝑇 ∈ Π(𝑆) is denoted by 𝑃! 𝑖 = 𝑗 ∈ 𝑇: 𝑗(𝑇) > 𝑖(𝑇) . Similarly to

Kolaczyk et al. (2009) the authors also use the group betweenness centrality proposed by Everett &

Borgatti (1999b) as starting point. The group betweenness centrality does not allow for an individual

ranking considering all possible coalitions each node could form. Given 𝐺 = (𝑁,𝐸), the SV based

betweenness centrality of node 𝑖 is a function 𝐵!!:𝑁 → ℝ:𝐵!! 𝑖 = 𝜙!(𝑁, 𝑣) where 𝑣 is the

characteristic function such that 𝑣: 2! → ℝ: 𝑣 𝑆 = !!"(!)!!"!,!∉! with 𝑆 ⊆ 𝑁. Szczepański et al. (2012)

further develop Brandes’s (2001) and Dijkstra’s (Cormen et al. 2001) algorithms in order to develop an

algorithm and try to adapt it, so as to also deal with stress centrality. The SV based stress centrality of

node 𝑖 as a function 𝑠!!:𝑁 → ℝ: 𝑠!! 𝑖 = 𝜙!(𝑁, 𝑣)!!!!! with 𝑣: 2! → ℝ: 𝑣 𝑆 = 𝜎!"(𝑆)!,!∉! .

Szczepański et al. (2012) propose more efficient algorithms to compute SV based betweenness

measures. This centrality measure allows for the ranking of an individual node according to all

possible coalitions formed by it. The work presented could be extended to other centrality measures.

Challenging extensions would include the graph centrality, reach centrality, edge centrality, among

others.

The application of GT centrality measures to counterterrorism practice is common in the literature.

Lindelauf et al. (2013) introduce a GT approach to identify key players in a terrorist network. Tailor

made cooperative games are developed and the SV is used as measure of importance. This approach

allows for the consideration of network and non-network features, simultaneously, and ranks all nodes

in the network. Standard centrality measures only account for network features. A weighted

connectivity game is used to model the network. Recalling the standard (normalized) centrality

measures, the degree centrality is denoted by 𝐷 𝑖 = !(!)! !!

, the closeness centrality is denoted by

𝐶 𝑖 = ! !!!!"!∈!

and the betweenness centrality that has already been presented. Note that 𝑑(𝑖) denotes

the number of direct relations of node 𝑖 with other nodes and 𝑙!" the shortest distance between node 𝑖

and 𝑗. The subgraph 𝑆! denotes the graph with players of the coalition 𝑆 and their communication links

(edges of the graph). If 𝑆! is connected the coalition 𝑆 is assigned a value 1; if not the value is 0. The

connectivity game is defined as 𝑣!"##(𝑆) = 1    𝑖𝑓  𝑆!𝑖𝑠  𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑0            𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

. The connectivity game may be

adapted to a weighted connectivity game accounting for its application specification. It is then used to

compute the SV. The authors study, briefly, three different examples using different weighted

connectivity games and also apply their proposed approach to two terrorist networks. Lindelauf et al.

(2013) conclude that their GT measure is able to differentiate nodes of the network at an individual

27

level and provide more informative rankings due to the ability of it considering network and non-

network features, simultaneously. The idea of disposition matrices is proposed as further

investigation. A shared limitation among all SNA problems is the ability to gather data on individuals of

the network; this issue is of higher concern and difficulty in counter terrorist practice.

Besides the already stated node/edge centric analysis literature, other less traditional works can be

found (van den Brink & Gilles 2000; Smith et al. 2014). A digraph can be used to model dominance

relations between individuals or, even, a directed network (van den Brink & Gilles 2000). These

authors present two relational power measures: 𝛽-measure and score-measure. This dominance

relation may be called a fourth centrality measure named domination where a 𝑖𝑗 ∈ 𝐷 represents that

entity 𝑖 dominates 𝑗. The collection of all weighted digraphs on 𝑁 is denoted by 𝑊!. The dominance

weight of node 𝑗 in 𝑤 ∈ 𝑊! is denoted by 𝜆 𝑗 = 𝑐!"!∈! with 𝑐!" representing a measure of the

strength of dominance of 𝑖 over 𝑗. The generalized 𝛽-measure is obtained through 𝛽! =!!"! !!∈! with

𝜆 𝑗 > 0. The generalized score-measure 𝓈! = 𝑐!"!∈! . By defining 𝑐!" =1            𝑖𝑓  𝑖𝑗 ∈ 𝐸0    𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

one can obtain

the 𝛽 -measure and score-measure for digraphs. These two measures satisfy the dummy node

property, the symmetry axiom and the additivity axiom. The SV of the score game

𝑣! 𝑇 = 𝑐!"!∈!!∈!⊆! equals the 𝛽-measure, for unweighted directed networks.

Smith et al. (2014) also provide an interesting view on this topic. The authors apply their concept to

“politically charged networks” where allies and adversaries are present and powerful nodal positions

are defined as potential inter-actor control. The proposed measure is called Political Independence

Index (PII). The PII was developed as a complement to the power-as-access approach used in

“politically charged networks” measured, typically through degree centrality, closeness centrality or,

even, in some cases, eigenvector centrality and is, itself, a power-as-control measure. The power-as-

control approach is typically defined through betweenness centrality. Neither of these centralities

account for negative ties. The PII has been designed to consider both alliance and adversarial ties, to

account for the position of each node in the entire network and to consider that being ally to actors that

are in threat increases the focal actor’s power. The distance of node 𝑙 to edge 𝑖𝑗 is defined as

min  (𝜎!" ,𝜎!"). The PII is denoted by 𝑃𝐼𝐼! = 𝛽! 𝑃! 𝑛 ! − 𝑁!(𝑛)!!!!! where 𝑃!(𝑛) (𝑁!(𝑛)) is the number

of positive (negative) edges at distance 𝑛 of node 𝑖 , 𝛽 is an attenuation factor (assumed to be

negative) and 𝑥 ≤ !" ! !!"  ( !!"  (!)

with 𝑀 equal to the maximum number of edges incident to any node in

the network. Smith et al. (2014) compare their measure to other known power measures. Also, the

authors develop a study to understand the viability of PII as a predictive tool. As a conclusion, PII

allows for the analysis of networks where some actors are trying to undermine others; knowing that

these threats are more than just dyadic relations. The authors state that a further area of research

would be to be able to consider in the PII both network and non-network features. Also, different levels

of threat could be explored in the future. The choice of the attenuation factor (𝛽) is also a possible

future area of research.

28

3.2.2 Network Centric Analysis

The network centric analysis perspective provided in this project will focus mainly on network

formation (Watts 2001; Galeotti et al. 2006). A view on coalition formation within a given network will

also be provided (Hernández et al. 2013; Kleinberg & Ligett 2013). A brief overview on a competition

over a social network problem will be provided (Goyal & Kearns 2012). Notice the clear distinction

between “dynamics of the network”, such as network formation, from “dynamics on the network”, like

coalition formation on a given fixed network.

Watts (2001) analyses the process of dynamic network formation. The dynamic nature is due to the

self-interested individuals to form or sever links. The author shows that the network formation process

is path dependent and converges to an inefficient structure. A link can be severed unilaterally but must

be formed by agreement of both individuals. Agents are myopic thus trying to maximize their current

payoff. The payoff is defined according to the connection model (Jackson & Wolinsky 1996). The

complete connected graph over 𝑁 is denoted by 𝑔!. The set of all possible graphs is 𝑔 = 𝐺|𝐺 ⊆ 𝑔! .

The number of direct links in the shortest path between 𝑖 and 𝑗 is denoted by 𝑡(𝑖𝑗). Each agent

receives a payoff denoted by 𝑢! = 𝛿!(!")!!! − 𝑐!∈! with 𝛿 ∈  ]0; 1[ the payoff for each direct link. A

network is stable if no player wants to sever a link and if no combinations of 2 players want to sever or

form links. An efficient network 𝑔∗ is one that maximizes each agent’s payoff such that

𝐺∗ = arg𝑚𝑎𝑥! 𝑢!!!!! . The dynamic model time is a countable, infinite, set 𝑇 = 1, 2,… , 𝑡,… . Each

player receives payoff 𝑢!(𝑡) at end of time 𝑡. In each period a link is chosen to be updated with uniform

probability. If this process reaches a stable state the resulting network is stable. Watts (2001)

concludes that the efficient networks only forms if agents meet at a particular pattern. In addition, the

author concludes that an interesting future area of research would be to consider non-myopic players.

Notice that Watts (2001) considers that all actors have the same cost of maintaining links. Galeotti et

al. (2006) extend the connections model (Bala & Goyal 2000) to consider ex-ante heterogeneity in

players. The heterogeneity stems from the fact that each actor has its own costs of maintaining links

and payoffs. The purpose of this work is to understand the impact of player heterogeneity in network

structure. A strategy for player 𝑖 is a row vector 𝑠! = 𝑠!,!, 𝑠!,!,… , 𝑠!,! where 𝑠!,! = 1 if player 𝑖 has a link

with player 𝑗 or 𝑠!,! = 0 otherwise. The set of strategies for player 𝑖 is denoted by 𝑆! . Symmetric

communications are assumed. Attention is restricted to pure strategies. The space of pure strategies

of all players is denoted by 𝑆 = 𝑆!×𝑆!×…×𝑆!. A strategy profile 𝑠 = 𝑠!, 𝑠!,… , 𝑠! defines a directed

network. The set of players with whom player 𝑖 maintains a link is denoted by 𝑁! 𝑖; 𝑠 = 𝑘 ∈ 𝑁|𝑠!,! =

1 and 𝜇!! 𝑠 = 𝑁! 𝑖; 𝑠 . Also, the set of players incident to player 𝑖 is denoted by 𝑁 𝑖; 𝑠 = 𝑘 ∈

𝑁|max 𝑠!,! , 𝑠!,! = 1 and 𝜇! 𝑠 = 𝑁 𝑖; 𝑠 . The payoff to player 𝑖 is denoted by 𝑢!∗ = 𝑉!"!∈!(!;!) −

𝑐!"!∈!!(!;!) where 𝑉!" is the value player 𝑖 gets from maintaining a link with player 𝑗. Note that 𝑠!!

represents the directed network 𝑠 without player’s 𝑖 links and that 𝑠 = 𝑠! ⊗ 𝑠!!. The best response of

player 𝑖 to 𝑠!! is 𝑠! such that 𝜇! 𝑠 ≥ 𝜇! 𝑠!!⊗ 𝑠!! for al 𝑠!! ∈ 𝑆!. The set of all player’s 𝑖 best responses

to 𝑠!! is ℬℛ!(𝑠!!). A Nash network 𝑠 is a network where 𝑠! ∈ ℬℛ!(𝑠!!) for all 𝑖. The social welfare of a

29

network is denoted by 𝒲 𝑠 = 𝑢!∗!!!! and an efficient network is one where 𝒲 𝑠 ≥𝒲 𝑠! for all

𝑠! ∈ 𝑆. Galeotti et al. (2006) apply their approach to an insider-outsider model and explore the effect of

decay on the proposed payoff measure through an approach similar to the one of Watts (2001). The

authors conclude that value heterogeneity is crucial to determining the connectedness of a network

and that differences in costs of links are essential to determine the both the level of connectedness

and individual components’ architecture. Correspondingly, centrality and short distances are robust

features of equilibrium networks and these are efficient in many instances.

Hernández et al. (2013) study coordination (anti-coordination) among players in a social network.

Players have intrinsic preferences and interact through strategic complements (substitutes). In many

real world cases players seek to coordinate in order to maximize their payoff but individual

preferences are still steering the final decision. The social network is fixed and players choose from a

binary action set. The behaviour of players is assumed to be independent of each other. Every player

has an identity 𝜃! ∈ 0, 1 and chooses an action from the binary set 𝑋 = 0, 1 .The vector of actions

taken by 𝑖’s neighbours is denoted by 𝑥!! where 𝑘! = 𝑗: 𝑖𝑗 ∈ 𝐸 given graph 𝐺 = 𝑁,𝐸 . Notice that the

authors consider an undirected graph. The payoff for player 𝑖 is

𝑢! 𝜃! , 𝑥! , 𝑥!! = 𝜆!!!! 1 + 𝛿 𝐼 !!!!!!∈!! + (1 − 𝛿) 𝐼 !!!!!!∈!! , where 𝐼 !!!!! is the function indicating

the set of players choosing the same action as player 𝑖 and the parameter 𝜆!!!! = 𝛼 when the player

plays according to his preferences or 𝜆!!!! = 𝛽 otherwise, with 0 < 𝛽 < 𝛼. Also, the multiplier 𝛿 defines

the class of game being played; when 𝛿 = 1    (0) the game is one of strategic complements

(substitutes). The authors conclude that in strategic complements (substitutes) it pays off more to play

the actions that our neighbours play more (less). NE in this context (notice that this could be extended

to strategy profiles) is when no player has an incentive to deviate from an action profile (𝑥!∗,… , 𝑥!∗) and

is denoted by 𝑢! 𝜃! , 𝑥!∗,… , 𝑥!∗,… , 𝑥!∗ ≥ 𝑢! 𝜃! , 𝑥!∗,… , 𝑥!!,… , 𝑥!∗ ,∀𝑥!! ≠ 𝑥!∗ . Several definitions and

implications are analysed in perfect information settings and imperfect information settings. Hernández

et al. (2013) conclude that the main novelty in their work is that they consider intrinsic preferences and

identities for players. The authors show that players in minorities will be frustrated (choosing differently

from their preference). Further, there is a large set of Nash equilibriums for perfect information

games. Moreover, the authors believe that their proposed framework disentangles the system of

incentives that players have.

Kleinberg & Ligett (2013) propose a model with respect to the way information is shared among

friends in a social network. This model offers insight on how these networks fragment. The purpose of

this model was to understand the trade-off between the benefits of sharing information with others and

the risk of increasing gossip. In this model a fixed undirected graph represents a conflict graph where

a link 𝑖𝑗 means that players 𝑖 and 𝑗 are enemies, thus not wanting to share information. The

community is partitioned into “information-sharing groups”, which are assumed to be self-enforcing.

The problem to be solved is similar to a typical vertex colouring problem in graph theory; which

consists of assigning all nodes of the graph a colour/label such that if two nodes are connected they

30

have different colours (partition of the graph into independent sets). The smallest number of colours

needed to colour a graph 𝐺 is called its chromatic number and is denoted by 𝜒(𝐺). A 𝑐-partite is one

with chromatic number at most 𝑐. If player 𝑖 is in a group 𝑇 ∈ 𝑁 it is assumed that all members of 𝑇

know 𝑖’s personal information. Player 𝑖 receives utility equal to the number of other player in its group

provided none of these are his enemies, if any member of the group is 𝑖’s enemies he receives an

utility equal to −∞. A configuration is conflict-free it is gives non-negative utility to all individuals. A set

of nodes 𝑆 ⊆ 𝑁 breaks a conflict-free configuration if there are no conflict edges within 𝑆 and (1)

∃𝑖 ∈ 𝑆:  ∀𝑗 ∈ 𝑆\ 𝑖 could strictly improve their utilities by leaving their group and joining 𝑖 (player 𝑖 would

also improve his utility) or (2) all members of 𝑆 could strictly improve their utility by forming a new

group consisting just of 𝑆. A configuration is said to be 𝑘-stable if it is conflict-free and there is no set

of size smaller than or equal to 𝑘 that breaks it. A set of size 𝑘 that breaks a conflict-free configuration

through operations (1) and (2), previously stated, is called a 𝑘-deviation. Kleinberg & Ligett (2013)

develop an algorithm to obtain stable configurations and an improved version that is able to obtain

stable configurations in polynomial time. The authors also show some possible extension of the

proposed model, namely, instead of conflict edges one could study a model where each edge 𝑖𝑗 could

have 𝑢!" , 𝑢!" ≥ 0 associated to the utilities each player gets from being associated with the other; or a

conflict edge could have a negative finite value associated to it. As a conclusion, Kleinberg & Ligett

(2013) state that for every conflict graph a 𝑘-stable configuration exists.

Competition between firms is studied by Goyal & Kearns (2012) assuming that firms have a budget to

seed product adoption among customers in a social network. The payoff for each firm is the number of

adoptions through a competitive stochastic diffusion process. The firms, named Red and Blue, know

the structure of the social network, offer interchangeable products and have budgets 𝐾! ,𝐾! ∈ ℕ!,

respectively. Goyal & Kearns (2012) make further contributions on competitive contagion (Chasparis &

Shamma 2010; Bharathi et al. 2007). The sets of seed infections that maximize the joint expected

infection are (𝑆! , 𝑆!) and the mixed NE strategies that minimize the joint expected payoff across all NE

are (𝜎! ,𝜎!). The Price of Anarchy (efficiency of resource use) is denoted by 𝑃𝑜𝐴 = !! !!,!! !!!(!!,!!)!! !!,!! !!!(!!,!!)

where the function 𝑉! determines the payoff to firm 𝑝 = 𝑅,𝐵 . The Budget Multiplier (amplification of

ex-ante resource differences), assuming that 𝐾! ≥ 𝐾!, is 𝐵𝑀 = !! !!,!!!!(!!,!!)

× !!!!

where (𝜎! ,𝜎!) is the NE

that maximizes the Budget Multiplier. Each firm 𝑝 chooses an allocation of budget across the 𝑛 nodes

𝑎! = (𝑎!!, 𝑎!!,… , 𝑎!"), where 𝐾! = 𝑎!"!∈! , while 𝐴! denotes the set of all allocations (pure strategy

space) and 𝒜! denotes the set of probability distributions (mixed strategy space) for player 𝑝. Given

an initial allocation (𝑎! , 𝑎!) node 𝑖 is infected by 𝑅 with probability 𝑝! =!!"

(!!"!!!") and is infected by 𝐵

with probability 𝑝! =!!"

(!!"!!!"). The state of node 𝑖 in time 𝑡 is denoted by 𝑠!" = 𝑈,𝑅,𝐵 , where 𝑈

stands for uninfected. Only uninfected nodes may change state at time 𝑡. With probability 𝑓(𝛼) node 𝑖

becomes infected by 𝑅 with probability  𝑔 !!!!!!!

and becomes infected by 𝐵 with probability

𝑔 !!!!!!!

, for each time 𝑡. Given that 𝜒! denotes the random number corresponding to the number of

31

adoptions to firm’s 𝑝 product, the payoff for firm 𝑝 is 𝑉! 𝜎! ,𝜎! = Ε 𝜒!| 𝜎! ,𝜎! . NE can be applied to

the firms’ strategy profile. Goyal & Kearns (2012) propose a framework that comprises a wide variety

of competitive strategies. They identify several properties of local adoption dynamics, according to the

Price of Anarchy and/or the Budget Multiplier. As future areas of research, the authors identify that

deeper insight on the structure of equilibrium and how it is related with the network structure.

Moreover, it would be beneficial to understand the implications of endogenous budgets. Algorithmic

issues and the development of a multi-stage version of the game proposed by Goyal & Kearns (2012)

are also future areas for research.

32

4 Case Study: Strategic Corporate Network Formation

One concludes that it is necessary to specify what, in fact, is a GT approach to SNA with a MOO

methodology. In this section the SV is maximized in a MOO network formation problem. A

contextualization and some assumptions are presented. In this early stage, no attention was given to

computational, algorithmic and times constraints. The presented case study studies a strategic

corporate network formation game.

Suppose a TU game 𝑁, 𝑣 where there are 𝑁 players, each one representing an organization or a

firm, with 𝑁 = 𝑛, and each one of them gets a value or utility 𝑣(𝑖), 𝑖 = 1, 2,… , 𝑛. Organizations must

choose to cooperate or not. For simplicity, interactions occur all simultaneously.

Assume that the organizations operate in complete distinct business areas, providing non-

substitute services/products that may, or may not, be complementary. Thus, organizations seek to

establish strategic alliances and partnerships in order to keep or build on a SCA. These alliances, or

partnerships, can be acquisitions & mergers, joint ventures, R&D collaborations, technology transfers,

licensing, private labelling, joint selling and distribution, joint marketing, vendor or supplier (Thompson

et al. 2014). This case study simplifies this issue by assuming that by partnering there may or may not

exist a synergy among partnering organizations. The interest of partnering with others is a trade-off

between the cost and the benefits of each partnership or alliance.

A layout of the strategic corporate network formation game is denoted by 𝐺 = (𝑁,𝐸, 𝑣,𝐶) where 𝐸

is the set of undirected weighted edges between organizations representing alliances and

partnerships and 𝐶 is the set of costs associated with each edge. The cost of forming a partnership or

alliance between organization 𝑖 and 𝑗 is 𝑐!" = 𝑐!", ∀𝑖, 𝑗 ∈ 𝐸. The utility of player 𝑖 is the values of sales

of that organization and is denoted by 𝑣 𝑖 = 𝑆(𝑖) where 𝑆(𝑖) is the monetary value of sales of firm 𝑖.

Given a coalition 𝑆 ⊆ 𝐺 , the utility of the coalition is 𝑣 𝑆 = 1 +!"!"!"∈!

! ! × 𝑣 𝑖!∈! where

𝑆𝑦~𝑈𝑛𝑖𝑓𝑜𝑟𝑚  [0,1] represents the extent to which synergies may be explored between organizations.

Note that it is assumed that organizations do not share consumers, thus, allied to the assumption of

the inexistence of substitute products and services, justifying the positive values of 𝑆𝑦. Further, one is

assuming that the reason behind strategic alliances and partnerships is to explore synergies, which

includes diversifications and increase of market share. In reality companies may also want to explore

focus and elimination of competition. Furthermore, in the proposed MOLP problem, vertical integration

motivations of firms could be included through the variance costs of alliances and partnerships. In

order to keep the case study simple, the existences of benefits from vertical integration will not be

considered. In this case study the SV (Φ) will be maximized, corresponding to organizations’ objective

of maximizing profits.

The MOLP problem may be stated as follows max  (Φ:𝐶!𝑥 ≤ 𝐵) with 𝑥!" = 𝑥!" ∈ 0, 1 where

Φ = (Φ!,… ,Φ!) is the SV vector, 𝐵 the budget vector and 𝑥 the vector of the decision variables. A

value of 1 corresponds to the decision of the existence of the alliance or partnership. Within the vector

of decision variables, sub vectors of 𝑥 could be identified. One could divide 𝑥 in to non-partitioned sub

vectors. Each one of them is controlled by organization 𝑖 and its elements would denote the edges

with the elements of node adjacency list of node 𝑖.

33

Consider a very simple case where there are three organizations: A, B and C. The objective

function of the organizations are to maximize Φ!, Φ! and Φ!, respectively. Thus, Φ = (Φ!,Φ!,Φ!) is

the SV vector. The objective functions are subject to 𝑐!"𝑥!" + 𝑐!"𝑥!" ≤ 𝐵!, 𝑐!"𝑥!" + 𝑐!"𝑥!" ≤ 𝐵! and

𝑐!"𝑥!" + 𝑐!"𝑥!" ≤ 𝐵!, where 𝑥!" , 𝑥!", 𝑥!" ∈ 0, 1 and 𝐵! is organization’s 𝑖 budget. Remember that 𝑥!"

is only controllable by organization 𝑖 and organization 𝑗. Notice that the structure of the network may

be restricted in such way that certain 𝑥!" = 0 independently of the organizations motivation to establish

an alliance or partnership. The SV for each organization depends on the structure of the network,

which, in its place, has a different layout according to every decision of alliance or partnership. This

factor is the one responsible for the complexity of calculation the SV.

This case study allows the reader to understand, in general terms, the problem statement given

that one is using a GT approach to SNA with a MOO methodology. At a preliminary glance, one could

further develop this model so that, instead of a single stage, a multi-stage game was considered;

meaning that interactions between players would, then, occur sequentially. Additionally, instead of

having only one objective for each organization one could consider more.

A brief note must be explicated. In this simple case study, due to its already explained and justified

simplicity, nothing was developed on the calculation of the SV and, also, its computation. In future

work, considering the SV is used, it is essential to understand and explicit SV related issues.

Other case studies could be presented. Relevant economic related models could be presented,

such as coalition formation with a multi-criteria perspective according to the production possibilities

frontier or consumers’ indifference curves. These could focus on the motivations for countries to form

coalitions (such as the European Union) or for individuals (such as university students or young

couples) to decide to live together. In this project, since it is related to the Masters’ in Management

and Industrial Engineering, it was considered that a broader, business related, issue was more

appropriate.

34

5 Conclusion

From the literature review here presented and its analysis, it is obvious that OR and SNA, from a node

centric analysis point of view, cross. In fact Freeman et al. (1991) advocated so in the nineties. One

may question if there is, in fact, an entry point for the operations researcher in SNA through node

centric analysis or if it is limited to network flow optimization problems. The present project takes this

question even further and tries to pinpoint the entry point for MOO in SNA; moreover assuming a GT

approach.

MOO literature (Gómez et al. 2013) has proposed a bi-criteria perspective on centrality measures.

An interesting strategy for future research is to apply this framework to GT based centrality measures,

namely to the SV. Furthermore, this framework may, in fact, be a turning point in terms of developing

GT based centrality measures for weighted networks. One should consider that a MOO framework to

centrality measures might be more suitable in constructive frameworks: considering interaction among

agents; see the two examples given by Gómez et al. (2013).

As expected, many researchers’ work on SNA, from a node/edge centric analysis point of view,

with a GT approach gravitating around the SV. The future possibilities for research are several. The

relation between group betweenness and co-betweenness (Kolaczyk et al. 2009) may be further

explored in order to better understand the relationship between coalition robustness and redundancy

of group members. Very important work is to be done in ranking individual nodes according to all

possible coalitions formed by it (Szczepański et al. 2012; Lindelauf et al. 2013). This technique, also

based on the SV, is a future avenue for research when applying to graph, reach and edge centrality,

among others. While the SV considers non-network features, not all proposed measures are able to

do so (Smith et al. 2014). A notably interesting development would be to develop techniques that were

able to account for non-network features; thus, understanding the fungibility between network and

non-network sources of power. Finally, very few literature studies centrality measures able to account

for both positive and negative ties among players. Smith et al. (2014) propose that further work should

be dedicated to addressing centrality measures accounting for varying levels of positive and negative

valued ties.

Everett & Borgatti (2010) formalized an interesting concept. By using induced centralities, any

analyst is able to design a centrality measure tailored to the specific problem being solved. This may

be an interesting tool for the operations researcher in SNA.

From a network centric perspective, SNA literature provides interesting future areas of research.

For instance, in dynamic network formation models (Watts 2001) one could further understand how

agents meet or, even, further develop models with non-myopic players. The model proposed by

Kleinberg & Ligett (2013) could be adapted so that edges represent utility from belonging to the same

coalition or, alternatively, so that adversarial edges produce a utility different from – 𝑖𝑛𝑓. Also, similarly,

the model proposed by Goyal & Kearns (2012) could be adapted in order to formulate richer

microeconomic models, including a fully GT formulation over consumers and firms.

An interesting conclusion, despite not being explicitly a future path for research, is that the

integration of heterogeneity/identity of players disentangles the system of incentives of the players

involved.

35

An innovative future are of research is to use Dynamic Social Network Analysis (DSNA) to consider

several networks simultaneously (Lindelauf et al. 2013).

It is important to note some of the limitations found in the literature of SNA, generally speaking. In

many cases costs and benefits associated to ties are not formally defined. Thus, in many published

works it is unclear what these costs and benefits represent. When such definition is provided, it is

common for these costs and benefits (as well as the social network structure) to be very difficult to

collect. Another important critic is relative to the typical definition of betweenness centrality. The extent

to which geodesic distances are able to represent the control of an entity over the flow of information

is questionable. This issue is very relevant since it is used, in some cases, as the basis for the

definition of the GT based centrality measures.

Note that a formal crossing between MOO and GT in SNA has not been found in the literature. In

fact, the main conclusion of this project is that a GT approach to SNA with MOO methodology is,

possibly, a solution to some of the previously stated limitations, due to the OR framework (Alderson

2008). A future broad area of research is to develop multi-criteria models of SNA from a GT approach.

In that sense a brief case study is presented in order to understand and materialize what would, in

fact, be this multi-criteria model. The case study allowed concluding that, as indicated by Gómez et al.

(2013), some computational and algorithmic issues may arise.

It seems that the first step for future work in this area is to better design MOLP problems with a GT

approach as well as understand the benefits from applying OR to SNA. Later, research concerning

node/edge centric and network centric analysis could be developed for applications within OR, namely

MOO.

36

References

Ahuja, R.K., Magnanti, T.L. & Orlin, J.B., 1993. Network flows: theory, algorithms and applications 1st ed. P. Janzow & M. Peterson, eds., New Jersey: Prentice-Hall. Available at: http://www.amazon.com/Network-Flows-Theory-Algorithms-Applications/dp/013617549X.

Alderson, D.L., 2008. OR FORUM--Catching the “Network Science” Bug: Insight and Opportunity for the Operations Researcher. Operations Research, 56(5), pp.1047–1065. Available at: http://pubsonline.informs.org/doi/abs/10.1287/opre.1080.0606 [Accessed March 21, 2014].

Amer, R., Giménez, J.M. & Magaña, A., 2007. Accessibility in oriented networks. European Journal of Operational Research, 180(2), pp.700–712. Available at: http://www.sciencedirect.com/science/article/pii/S037722170600316X [Accessed April 20, 2014].

Bala, V. & Goyal, S., 2000. A Noncooperative Model of Network Formation. Econometrica, 68, pp.1181–1229.

Barabási, A.-L., 2003. Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life, Available at: http://books.google.com/books?id=rydKGwfs3UAC.

Barrat, A., Barthélemy, M. & Vespignani, A., 2008. Dynamical Processes on Complex Networks 1st ed., Cambridge: Cambridge University Press. Available at: http://portal.acm.org/citation.cfm?id=1521587.

Bharathi, S., Kempe, D. & Salek, M., 2007. Competitive influence maximization in social networks. Internet and Network Economics, pp.306–311.

Bogomolnaia, A. & Jackson, M.O., 2002. The Stability of Hedonic Coalition Structures. Games and Economic Behavior, 38(2), pp.201–230. Available at: http://www.sciencedirect.com/science/article/pii/S0899825601908772 [Accessed March 24, 2014].

Borgatti, S.P., 2005. Centrality and network flow. Social Networks, 27, pp.55–71.

Borgatti, S.P. & Foster, P.C., 2003. The network paradigm in organizational research: A review and typology. Journal of Management, 29, pp.991–1013.

Brandes, U., 2001. A faster algorithm for betweenness centrality*. The Journal of Mathematical Sociology, 25, pp.163–177.

Brandes, U. & Wagner, D., 2003. Visone: Analysis and Visualization of Social Networks. Graph drawing software, pp.321–340. Available at: file:///D:/_Temp/Analysis and Visualization of Social Networks.pdf.

Brautbar, M. & Kearns, M.J., 2010. Local algorithms for finding interesting individuals in large networks. University of Pennsylvania Scholarly Commons. Available at: http://repository.upenn.edu/cgi/viewcontent.cgi?article=1694&context=cis_papers [Accessed March 4, 2014].

Van den Brink, R. & Gilles, R.P., 2000. Measuring domination in directed networks. Social Networks, 22(2), pp.141–157. Available at: http://www.sciencedirect.com/science/article/pii/S0378873300000198 [Accessed May 13, 2014].

Castro, J., Gómez, D. & Tejada, J., 2009. Polynomial calculation of the Shapley value based on sampling. Computers and Operations Research, 36, pp.1726–1730.

37

Chasparis, G.C. & Shamma, J.S., 2010. Control of preferences in social networks. In Proceedings of the IEEE Conference on Decision and Control. pp. 6651–6656.

Cormen, T.H., Leiserson, C.E. & Rivest, R.L., 2001. Introduction to Algorithms , Second Edition, Available at: http://irkutsk.openet.ru/handle/123456789/144.

Davis, A.B., Gardner, B. & Gardner, M.R., 1941. Deep South, Chicago: The University of Chicago Press.

Dawande, M. et al., 2012. Structural Search and Optimization in Social Networks. INFORMS Journal on Computing, 24(4), pp.611–623. Available at: http://pubsonline.informs.org/doi/abs/10.1287/ijoc.1110.0473 [Accessed May 13, 2014].

Edelman, 2014. Edelman Trust Barometer Executive Summary, Available at: http://www.edelman.com/insights/intellectual-property/2014-edelman-trust-barometer/about-trust/executive-summary/.

Ehrgott, M., 2005. Multicriteria Optimization 2nd ed., Springer. Available at: http://www.ncbi.nlm.nih.gov/pubmed/24694125.

Eusébio, A. & Figueira, J.R., 2009. Finding non-dominated solutions in bi-objective integer network flow problems. Computers & Operations Research, 36(9), pp.2554–2564. Available at: http://www.sciencedirect.com/science/article/pii/S0305054808002190 [Accessed March 27, 2014].

Eusébio, A., Figueira, J.R. & Ehrgott, M., 2014. On finding representative non-dominated points for bi-objective integer network flow problems. Computers & Operations Research, 48, pp.1–10. Available at: http://www.sciencedirect.com/science/article/pii/S0305054814000446 [Accessed May 13, 2014].

Everett, M.G. et al., 2005. Extending Centrality. In Models and Methods in Social Network Analysis. pp. 57 – 76.

Everett, M.G. & Borgatti, S.P., 2010. Induced, endogenous and exogenous centrality. Social Networks, 32(4), pp.339–344. Available at: http://www.sciencedirect.com/science/article/pii/S0378873310000341 [Accessed May 1, 2014].

Everett, M.G. & Borgatti, S.P., 1999a. The Centrality of Groups and Classes. Journal of Mathematical Sociology, 23(3), pp.181 – 201.

Everett, M.G. & Borgatti, S.P., 1999b. The centrality of groups and classes. The Journal of Mathematical Sociology, 23, pp.181–201.

Festinger et al., 1950. Social pressures in informal groups, New York, USA: Harper & Bros.

Freeman, L.C., 1978. Centrality in Social Networks Conceptual Clarification. Social Networks, 1(3), pp.215–239.

Freeman, L.C., 2004. The development of social network analysis. aris.ss.uci.edu. Available at: http://aris.ss.uci.edu/~lin/book.pdf\nfile://localhost/Users/enrico/Documents/Papers/2004/Freeman/aris.ss.uci.edu 2004 Freeman.pdf\npapers://e411e0f6-a7ed-4df0-b2a1-432575887559/Paper/p7513.

Freeman, L.C., Borgatti, S.P. & White, D.R., 1991. Centrality in valued graphs: A measure of betweenness based on network flow. Social Networks, 13, pp.141–154.

38

Fudenberg, D. & Tirole, J., 1984. Game Theory, 2455 Teller Road, Newbury Park California 91320 United States of America: MIT Pess Books. Available at: http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0262061414 [Accessed May 20, 2014].

Galeotti, A., Goyal, S. & Kamphorst, J., 2006. Network formation with heterogeneous players. Games and Economic Behavior, 54(2), pp.353–372. Available at: http://www.sciencedirect.com/science/article/pii/S0899825605000229 [Accessed May 13, 2014].

Goluch, T., 2012. The use of Game Theory in Small Business. Gdansk University of Technology. Available at: http://www.kaims.pl/~goluch/doc/zie_MSc-two-sided.pdf [Accessed May 11, 2014].

Gomez, D. et al., 2003. Centrality and power in social networks: a game theoretic approach. Mathematical Social Sciences, 46, pp.27–54.

Gómez, D., Figueira, J.R. & Eusébio, A., 2013. Modeling centrality measures in social network analysis using bi-criteria network flow optimization problems. European Journal of Operational Research, 226(2), pp.354–365. Available at: http://www.sciencedirect.com/science/article/pii/S0377221712008752 [Accessed May 5, 2014].

Goyal, S. & Kearns, M.J., 2012. Competitive contagion in networks. STOC 2012, pp.759–774. Available at: http://arxiv.org/abs/1110.6372 [Accessed April 29, 2014].

Granovetter, M.S., 1973. The Strength of Weak Ties. American Journal of Sociology, 78, p.1360.

Grossmann, M. & Dominguez, C.B.K., 2009. Party Coalitions and Interest Group Networks. American Politics Research, 37(5), pp.767–800. Available at: http://apr.sagepub.com/cgi/doi/10.1177/1532673X08329464 [Accessed May 1, 2014].

Hernández, P., Muñoz-Herrera, M. & Sánchez, Á., 2013. Heterogeneous network games: Conflicting preferences. Games and Economic Behavior, 79, pp.56–66. Available at: http://linkinghub.elsevier.com/retrieve/pii/S0899825613000158 [Accessed May 13, 2014].

Hillier, F.S. & Lieberman, G.J., 2010. Introduction to Operations Research 9th ed. K. E. Case & P. M. Wolfe, eds., New York, USA: McGraw-Hill.

INFORMS, 2014. What is Operations Research? Available at: https://www.informs.org/About-INFORMS/What-is-Operations-Research [Accessed May 12, 2014].

Jackson, M.O., 2011. A Brief Introduction to the Basics of Game Theory, Available at: http://ssrn.com/abstract=1968579.

Jackson, M.O., 2008. Social and economic networks 1st ed., New Jersey: Princeton University Press. Available at: http://books.google.com/books?hl=en&lr=&id=rFzHinVAq7gC&oi=fnd&pg=PR11&dq=Social+and+Economic+Networks&ots=v_g8GW2RjZ&sig=6SlcV7ivM-WvLATH4R0MDZGU_gQ [Accessed May 13, 2014].

Jackson, M.O. & Wolinsky, A., 1996. A strategic model of social and economic networks. Journal of Economic Theory, 71, pp.44–74. Available at: http://www.sciencedirect.com/science/article/pii/S0022053196901088 [Accessed March 4, 2014].

Keynes, J.M., 1936. The State of Long-term Expectation. In The General Theory of Employment, Interest and Money. Palgrave Macmillan.

39

Kleinberg, J. & Ligett, K., 2013. Information-sharing in social networks. Games and Economic Behavior, 82, pp.702–716. Available at: http://www.sciencedirect.com/science/article/pii/S0899825613001401 [Accessed May 13, 2014].

Klingman, D., Napier, A. & Stutz, J., 1974. NETGEN: A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems. Management Science, 20, pp.814–821.

Kolaczyk, E.D., Chua, D.B. & Barthélemy, M., 2009. Group betweenness and co-betweenness: Inter-related notions of coalition centrality. Social Networks, 31(3), pp.190–203. Available at: http://linkinghub.elsevier.com/retrieve/pii/S0378873309000045 [Accessed May 13, 2014].

Leinhardt, S., 1977. Social Networks: A Developing Paradigm S. Leinhardt, ed., Academic Press, INC.

Lewin, Kurt & Lippitt, R., 1938. An experimental approach to the study of autocracy and democracy: a preliminary note. Sociometry, 1(3/4), pp.292 – 300.

Lindelauf, R.H.A., Hamers, H.J.M. & Husslage, B.G.M., 2013. Cooperative game theoretic centrality analysis of terrorist networks: The cases of Jemaah Islamiyah and Al Qaeda. European Journal of Operational Research, 229(1), pp.230–238. Available at: http://linkinghub.elsevier.com/retrieve/pii/S0377221713001653 [Accessed April 28, 2014].

Lloyd, W.W. & Lunt, P.S., 1941. The social life of a modern community, New Haven: Yale University Press,.

Lorrain, F. & White, H.C., 1971. Structural equivalence of individuals in social networks. Journal of Mathematical Sociology, 1, pp.49–80. Available at: http://www.tandfonline.com/doi/abs/10.1080/0022250X.1971.9989788.

Merida-Campos, C. & Willmott, S., 2007. Exploring Social Networks in Request for Proposal Dynamic Coalition formation Problems. Lectur Notes in Computer Science, 4696, pp.143–152.

Moreno, J.L., 1932. Application of the group method to classification,, New York, USA: National Committee on Prisons and Prison Labor.

Moreno, J.L., 1934. Who shall survive?, Washington DC: Nervous and Mental Disease Publishing Company.

Nash, J.F., 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36(1), pp.48–49. Available at: http://www.pnas.org/cgi/content/long/36/1/48 [Accessed May 6, 2014].

Newcomb, T.M., 1961. The acquaintance process, New York, USA: Holt, Rhinehart, and Winston.

Nowak, A.S. & Radzik, T., 1994. The Shapley Value for n-Person Games in Generalized Characteristic Function Form. Games and Economic Behavior, 6, pp.150–161. Available at: http://www.sciencedirect.com/science/article/pii/S0899825684710086.

Pollner, P. et al., 2008. Centrality properties of directed module members in social networks. Physica A: Statistical Mechanics and its Applications, 387, pp.4959–4966.

Del Pozo, M. et al., 2011. Centrality in directed social networks. A game theoretic approach. Social Networks, 33(3), pp.191–200. Available at: http://www.sciencedirect.com/science/article/pii/S0378873311000177 [Accessed May 13, 2014].

40

Ressler, S., 2006. Social Network Analysis as an Approach to Combat Terrorism  : Past , Present , and Future Research. Homeland Security Affairs, 2(2), pp.1 – 10. Available at: http://www.hsaj.org/?article=2.2.8.

Shapley, L.S., 1953. A Value for n-person Games. In H. W. Kuhn & A. W. Tucker, eds. Contributions to the Theory of Games, Volume II. Princeton University Press, pp. 307 – 317.

Smith, J.M. et al., 2014. Power in politically charged networks. Social Networks, 36(Special Issue on Political Networks), pp.162–176. Available at: http://linkinghub.elsevier.com/retrieve/pii/S0378873313000439 [Accessed April 29, 2014].

SPSS, 2009. SPSS Statistics 17.0 for Microsoft Windows.

Szczepański, P.L., Michalak, T. & Rahwan, T., 2012. A new approach to betweenness centrality based on the Shapley Value. In AAMAS ’12 Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems. pp. 239–246.

Thompson, A.A. et al., 2014. Crafting and Executing Strategy: The Quest for Competitive Advantage 19th ed., McGraw-Hill.

Travers, J. & Milgram, S., 1969. An Experimental Study of the Small World Problem. Sociometry, 32, pp.425–443.

Turocy, T.L. & Stengel, B. von, 2003. Encyclopedia of Information Systems, Elsevier. Available at: http://www.sciencedirect.com/science/article/pii/B0122272404000769 [Accessed May 11, 2014].

Tutzauer, F., 2007. Entropy as a measure of centrality in networks characterized by path-transfer flow. Social Networks, 29, pp.249–265.

Varoufakis, Y., 2001. Volume 1: Foundations. In Y. Varoufakis, ed. Game Theory: Critical Concepts in the Social Sciences. Routledge, pp. 1 – 21.

Watts, A., 2001. A Dynamic Model of Network Formation. Games and Economic Behavior, 34(2), pp.331–341. Available at: http://www.sciencedirect.com/science/article/pii/S0899825600908030 [Accessed May 13, 2014].

Watts, D.J., 1999. Networks, Dynamics, and the Small‐World Phenomenon. American Journal of Sociology, 105, pp.493–527.

Watts, D.J., 2004. The “New” Science of Networks. Annual Review of Sociology, 30, pp.243–270.

White, D.R. & Borgatti, S.P., 1994. Betweenness centrality measures for directed graphs. Social Networks, 16, pp.335–346.

White, H.C., Boorman, S.A. & Breiger, R.L., 1976. Social Structure from Multiple Networks. I. Blockmodels of Roles and Positions. American Journal of Sociology, 81, p.730.