multiplex networks: structure and dynamics

Multiplex Networks: Structure and Dynamics

Emanuele Cozzo

Tesis doctoralDirector: Yamir MorenoUniversidad de Zaragoza

February 2, 2016

The complex networks view of complex systems

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In the beginning were networks, and networks wereeverywhere

Structural approachshift

Metaphor =⇒ substantial notion⇓

Contemporary Complex Networks Science

Science of Complex Networks

Interdisciplinary point of view on complex systems → unifying languageAbstraction from the details of a systemFocus on the structure of interactions.

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In the beginning were networks, and networks wereeverywhere

Structural approachshift

Metaphor =⇒ substantial notion⇓

Contemporary Complex Networks Science

Science of Complex Networks

Interdisciplinary point of view on complex systems → unifying languageAbstraction from the details of a systemFocus on the structure of interactions.

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Hypothesis

Structure and Function are intimately related

Abastraction =⇒ Graph Model of the System

Paraphrasing Wellman: It is a comprehensive paradigmatic way of taking structure seriously by studying directly howpatterns of ties determine the functioning of a system.

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A physicist point of view

Complex networks are systems that display a strong disorder withlarge fluctuations of the structural characteristics

Four steps:

Step 1: formal representation

Step 2: topological characterization

Step 3: statistical characterization

Step 4: functional characterization.

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From simple networks to multiplex networks

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The concept of multiplex network has been around for many decades:1962 Max Gluckman (antropology) - 1969 Kapferer (sociology of

work)

Concept of multiplex networks• communication media

• multiplicity of roles and milieux

communication media constituents continuously switch amonga variety of media

roles interactions are always context dependent

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Contemporary debate

Internet and mobile communications ↔ social and technologicalrevolution⇓

new steam for the formal and quantitative study on multiplexnetworks

Botler and Gusin media

Rainie and Wellman roles

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Not only social...

Biology integration of multiple set of omic data

Transportation different modes

Engineering interdependence of different lifelines

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Not only social...

Biology integration of multiple set of omic data

Transportation different modes

Engineering interdependence of different lifelines

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Not only social...

Biology integration of multiple set of omic data

Transportation different modes

Engineering interdependence of different lifelines

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Not only social...

Biology integration of multiple set of omic data

Transportation different modes

Engineering interdependence of different lifelines

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Basic Definitions and Formalism

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Multiplex networks as a primary object

• We propose a formal language intended to be general andcomplete enough

A rigorous algebraicformalism →

further morecomplex reasonings

design datastructures andalgorithms

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Graph Model

Network→

modelGraph: G (V ,E )

The notion of layer must beintroduced

Layer:

An index that represents aparticular type of interaction orrelation

L = {1, ...,m} index set| L |= m the number of layers

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Graph Model

Network→

modelGraph: G (V ,E )

The notion of layer must beintroduced

Layer:

An index that represents aparticular type of interaction orrelation

L = {1, ...,m} index set| L |= m the number of layers

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Nodes and node-layer pairs

Participation Graph:

• the set of nodes V ,GP = (V , L,P): binary relation,where P ⊆ V × L

Representative of node u in layerα

(u, α) ∈ P, with u ∈ V , andα ∈ L, is read node u participatesin layer α

define: node-layer pairs• | P |= N number of node-layer pairs, | V |= n

number of nodes

(u,1)

(u,2)

(v,1)

(v,2)

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node-aligned multiplex networks

If each node u ∈ V has a representative in each layer we call themultiplex a node-aligned multiplex and | P |= nm

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Layer-graphs

Each system of relations or interactions of different kind is naturallyrepresented by a graph

Gβ(Vβ,Eβ)

• Vβ ∈ P, Vβ = {(u, α) ∈ P | α = β} the set of allthe representatives of the node set in a particular layer

• | Vβ |= nβ the number of node-layer pairs in layer β

• Node-aligned multiplex networks: nα = n ∀α ∈ L.

• Eβ ⊆ Vβ × Vβ the set of edges. Interactions orrelations of a particular type

G1

G2

G3 G4

M = {Gα}α∈L, the set of all layer-graphs

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The Coupling Graph

GC (P,EC ) on P

EC = {((u, α), (v , β)) ⇐⇒ u =v)}

Formed by n =| P | disconnectedcomponents

(complete graphs or disconnectednodes)

⇒ supra-nodes

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Multiplex Network Representation

A multiplex network is represented by :

M = (V , L,P,M):

• the node set V represents the components of the system

• the layer set L represents different types of relations or interactionsin the system

• the participation graph GP encodes the information about whatnode takes part in a particular type of relation and defines therepresentative of each component in each type of relation, i.e., thenode-layer pair

• the layer-graphs M represent the networks of interactions of aparticular type between the components, i.e., the networks ofrepresentatives of the components of the system.

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Synthetic Representation

The union of all the layer-graphs:

The intra-layer graph

Gl =⋃α Gα

Define

The supra-graph

GM = Gl ∪ GC

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Adjacencies Matrices

Adjacency matrix

G (V ,E )→ A, auv = 1u∼v

Layer adjacency matrix

Layer graph Gα → Aα, nα × nα symmetric matrix , with aαij = 1 iffthere is an edge between i and j in Gα

Coupling matrix

Coupling graph GC → C = {cij}, an N × N matrix , with cij = 1 iffthey are representatives of the same node in different layers

Standard labelling → C: block-matrix

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Supra-Adjacency Matrix

A =⊕α

Aα + C = A+ C

By definition A is the adjacency matrix of Gl . A the adjacencymatrix of GM

Node-aligned multiplex networks

A = A+ Km ⊗ In

Identical layer-graphs

A = Im ⊗ A + Km ⊗ In,

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0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0

1 2 3 4 5

1 2 3 4 5

A = =A1

A2

C12

C21

0

0

C12

C21

=A

1

A2

00

A =

1 2

3

4 5

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Supra-Laplacian

L = D − A

By definition

L =⊕α

Lα + LC .

Node-aligned multiplex network

L =⊕α

(Lα+(m−1)IN)−Km⊗ In

Identical layer-graphs

L = Im⊗ (L + (m−1)In)−Km⊗ In

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Multiplex Walk Matrices

A walk on a graph is a sequence of adjacent vertices. The length of awalk is its number of edges.

Nij(k) = (Ak)ij

Multiplex networks contain walks that can traverse different additional layers

Define

a supra-walk is a walk on a multiplex network in which, either beforeor after each intra-layer step, a walk can either continue on the samelayer or change to an adjacent layer

C = αI + βC (1)

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• AC encodes the steps in which after each intra-layer step a walkcan change layer

• CA encodes the steps in which before each intra-layer step a walkcan change layer.

adjacency matrix of a directed (possible weighted) graph

Define: Auxiliary supra-graph GM whose adjacency matrix isM =M(A, C)

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• AC encodes the steps in which after each intra-layer step a walkcan change layer

• CA encodes the steps in which before each intra-layer step a walkcan change layer.

adjacency matrix of a directed (possible weighted) graph

Define: Auxiliary supra-graph GM whose adjacency matrix isM =M(A, C)

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Quotient graphs

It is natural to try to aggregate the interaction pattern of each layerin a single network somehow

(a) (b)

(c)

The natural definition of an aggregate network is given by the notionof quotient network

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Quotient graphs

Suppose that {V1, . . . ,Vm} is a partition of the node set of a graph G withadjacency matrix A(G )ni =| Vi |The quotient graph Q(G ) is a coarsening of the network with respect tothat partition.

It has one node per cluster Vi , and an edge from Vi to Vj weighted by an

average connectivity from Vi to Vj

Exact results relate the adjacency and laplacian spectrum of thequotient graph to the adjacency and laplacian spectrum of the parentgraph, respectivelyThe Laplacian of the quotient must be defined carefully

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Coarsening a Multiplex

Two natural partitions: supra-nodes and layers.Define

• aggregate network: quotientgraph of the parent multiplex.Partition according tosupra-nodes.

• network of layers: quotientgraph of the parent multiplex.Partition according to layers.

(a) (b)

(c)

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Aggregate network

A = Λ−1STn ASn, (2)

• Sn = (siu) characteristic matrix

• Λ = diag{κ1, . . . , κn} the multiplexity degree matrix.

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Network of layers

The network of layers has adjacency matrix given by

Al = Λ−1STl ASl , (3)

• Sl = {siα} characteristic matrix

• Λ = diag{n1, . . . , nm} layer size matrix

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Supra-walk and Coarse-graining

we have a relation between the number of supra-walks in a multiplexnetwork and the weight of weighted walks in its aggregate networkwhen the multiplex is node-aligned and switching layer has no cost

STn (AC )lSn = ml+1Wl = mlWl . (4)

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Structural Metrics

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Structural Metric

Structural metric

Is a measure of some property directly dependent on the system ofrelations between the components of the network: a measure of aproperty that depends on the edge set

Graph ←→ Adjacency matrix⇓

can be expressed as a function of the adjacency matrix

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How to, properly, generalize structural metrics tomultiplex networks?

We propose that a structural metric for multiplex networks should

• reduce to the ordinary single-layer metric (if defined) when layersreduce to one

• be defined for node-layer pairs• be defined for non-node-aligned multiplex networks

An additional requirement for intensive metrics:• For a multiplex of identical layers when changing layer has no cost,

an intensive structural metric should take the same value whenmeasured on the multiplex network and on one layer taken as anisolated network.

Start from first principles

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How to, properly, generalize structural metrics tomultiplex networks?

We propose that a structural metric for multiplex networks should

• reduce to the ordinary single-layer metric (if defined) when layersreduce to one

• be defined for node-layer pairs• be defined for non-node-aligned multiplex networks

An additional requirement for intensive metrics:• For a multiplex of identical layers when changing layer has no cost,

an intensive structural metric should take the same value whenmeasured on the multiplex network and on one layer taken as anisolated network.

Start from first principles

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How to, properly, generalize structural metrics tomultiplex networks?

We propose that a structural metric for multiplex networks should

• reduce to the ordinary single-layer metric (if defined) when layersreduce to one

• be defined for node-layer pairs• be defined for non-node-aligned multiplex networks

An additional requirement for intensive metrics:• For a multiplex of identical layers when changing layer has no cost,

an intensive structural metric should take the same value whenmeasured on the multiplex network and on one layer taken as anisolated network.

Start from first principles36 of 70

Structure of triadic relations in multiplex networks

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Walks as first principles

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In a monoplex network:

define

the local clustering coefficient Cu as the number of 3-cycles(triangles) tu that start and end at the focal node u divided by thenumber of 3-cycles du such that the second step of the cycle occursin a complete graph

tu = (A3)uu, du = (AFA)uu (5)

local clustering coefficient

Cu =tudu

(6)

global clustering coefficient

C =

∑u tu∑u du

(7)

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Multiplex networks contain cycles that can traverse different additional

layers but still have 3 intra-layer steps.

A supra-step consists either of only a single intra-layer step or of astep that includes both an intra-layer step changing from one layer toanother (either before or after having an intra-layer step)

tM,i = [(AC)3 + (CA)3]ii = 2[(AC)3]ii (8)

dM,i = 2[ACFCAC]ii (9)

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Local and Global clustering coefficient for MultiplexNetworks

We can calculate a natural multiplex analog to the usual monoplex localclustering coefficient for any node i of the supra-graph.A node u allows an intermediate description for clustering between local(node-layer pair) and the global (system level) clustering coefficients

c∗,i =t∗,id∗,i

, (10)

C∗,u =

∑i∈l(u) t∗,i∑i∈l(u) d∗,i

, (11)

C∗ =

∑i t∗,i∑i d∗,i

, (12)

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Layer-decomposed clustering coefficients

Our definition allows to decompose the previous expressions in termsof the contributions from cycles that traverse exactly one, two, andthree layers (i.e., for m = 1, 2, 3) to give

t∗,ı = t∗,1,iα3 + t∗,2,iαβ

2 + t∗,3,iβ3 , (13)

d∗,i = d∗,1,iα3 + d∗,2,iαβ

2 + d∗,3,iβ3 , (14)

C(m)∗ =

∑i t∗,m,i∑i d∗,m,i

. (15)

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Clustering Coefficients in Erdos-Renyi (ER) MultiplexNetworks

0.2

0.4

0.6

0.8

⟨ C ∗⟩AC

(1)M

C(2)M

C(3)M

p

B C

0.2 0.4 0.6 0.8x

0.2

0.4

0.6

0.8

⟨ c ∗⟩

DcAAAcAACACcACAACcACACAcACACACp

0.2 0.4 0.6 0.8x

E

0.2 0.4 0.6 0.8x

F

(A, B, C) Global and (D, E, F) local multiplex clustering coefficients in multiplex networks that consist of ER layers.The markers give the results of simulations of 100-node ER node-aligned multiplex networks that we average over 10

realizations. The solid curves are theoretical approximations. Panels (A, C, D, F) show the results for three-layernetworks, and panels (B, E) show the results for six-layer networks. The ER edge probabilities of the layers are (A, D)

{0.1, 0.1, x}, (B, E) {0.1, 0.1, 0.1, 0.1, x, x}, and (C, F) {0.1, x, 1− x}

Structure of triadic relations in multiplex networks EC, et al.- New Journal of Physics 2015

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Clustering Coefficient in Social Network is ContextDependent

For each social network we analysed

CM < C(1)M and C

(1)M > C

(2)M > C

(3)M

The primary contribution to the triadic structure in multiplex socialnetworks arises from 3-cycles that stay within a given layer.

Tailor Shop Management Families Bank Tube Airline

CMorig. 0.319** 0.206** 0.223’ 0.293** 0.056 0.101**ER 0.186 ± 0.003 0.124 ± 0.001 0.138 ± 0.035 0.195 ± 0.009 0.053 ± 0.011 0.038 ± 0.000

C(1)M

orig. 0.406** 0.436** 0.289’ 0.537** 0.013” 0.100**ER 0.244 ± 0.010 0.196 ± 0.015 0.135 ± 0.066 0.227 ± 0.038 0.053 ± 0.013 0.064 ± 0.001

C(2)M

orig. 0.327** 0.273** 0.198 0.349** 0.043* 0.150**ER 0.191 ± 0.004 0.147 ± 0.002 0.138 ± 0.040 0.203 ± 0.011 0.053 ± 0.020 0.041 ± 0.000

C(3)M

orig. 0.288** 0.192** - 0.227** 0.314** 0.086**ER 0.165 ± 0.004 0.120 ± 0.001 - 0.186 ± 0.010 0.051 ± 0.043 0.037 ± 0.000

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Context Matter

Triadic-closure mechanisms in social networks cannot be consideredpurely at the aggregated network level.These mechanisms appear to be more effective inside of layers thanbetween layers.

0 0.4 0.8 1cx

0.0

0.2

0.4

0.6

0.8

1.0

c y

c(1)

M,i / c (2)

M,i

c(2)

M,i / c (3)

M,i

c(1)

M,i / c (3)

M,i

0.5 0.0 0.5cx− cx

0.5

0.0

0.5

c y−

c y

A B

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• Existing definitions of multiplex clustering coefficients are mostlyad hoc:difficult to interpret

• Starting from the basic concepts of walks and cycles →transparent and general definition of transitivity.

• Clustering coefficients always properly normalized

• Reduces to a weighted clustering coefficient of an aggregatednetwork for particular values of the parameters

• Multiplex clustering coefficients decomposable by construction

• Do not require every node to be present in all layers

It is insufficient to generalize existing diagnostics in a naıve manner.One must instead construct their generalizations from first principles

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Spectra

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Important information on the topological properties can be extractedfrom the eigenvalues of one of its associated matrix

like spectroscopy for condensed matter physics, graph spectra arecentral in the study of the structural properties of a complex network

Eigendecomposition

A = XΛXT

Eigendecomposition⇓

Topology ⇔ Dynamics (critical phenomena)

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The largest eigenvalue of the supra-adjacencymatrix

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Largest eigenvalue of theadjacency matrix associatedto a network

⇒

• a variety of differentdynamical processes

• a variety of structuralproperties (the entropydensity per step of theensemble of walks in anetwork)

Perturbative approach

A as a perturbed version of A, C being the perturbation|| C ||<|| A ||

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Dominant Layer

λ = λ+ ∆λ

Call the layer δ for which λδ = λ the dominant layerApproximation

∆λ ≈ φTCφφTφ

+1

λ

φTC2φ

φTφ

φTCφφTφ

= 0

Effective multiplexity

z =∑i

ci(φ)2

i

φTφ

∆λ ≈ z

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Structural and Dynamical consequences

The entropy production rate of the ensemble of paths {πij(l)} forlarge length l depends only on the dominant layer and the effectivemultiplexity

h = ln λN ∼ ln(λ+z

λ)

Large walks on a multiplex are dominated by walks on the dominantlayer

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Structural and Dynamical consequences

0 0.2 0.4 0.6 0.8 1 β/µ

0

0.2

0.4

0.6

0.8

1

ρ

η = 0.25 η = 0.5 η = 1.0 η = 2.0 η = 3.0

0 0.1 0.2 0.3 0.4 0.5 β/µ

0

0.2

0.4

0.6 ρ

η = 0.0

1/Λ1

1/Λ2

0 0.2 0.4 0.6 0.8 1 β/µ

0

0.2

0.4

0.6

0.8

1

ρ1,ρ

2

Layer 1

Layer 2

0 0.1 0.2 β/µ

0

0.2

0.4

ρ1,ρ

2

1/Λ2

1/Λ1

η =2.0 b)

a)

Contact-based social contagion in multiplex networks EC,R.A. Banos, S. Meloni, Y. Moreno - Physical Review E,2013

The dominant layer sets thecritical point for a contact-basedsocial contagion process on the

multiplex network

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Dimensionality reduction and spectral properties

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Interlacing results

Theorem The adjacency eigenvalues of a quotient network interlace the

adjacency eigenvalues of the parent network. The same result applies for

Laplacian eigenvalues.

µi ≤ µi ≤ µi+(N−n) (16)

µi ≤ µ(l)i ≤ µi+(N−m) (17)

An inclusion relation holds for equitable partition.It holds for the network of layer in the case of node-aligned multiplexnetwork.The spectrum of the network of layers IS INCLUDED in the spectrumof the whole multiplex network

Dimensionality reduction and spectral properties of multilayer networksR.J. Sanchez-Garcıa, E. Cozzo, Y. Moreno - Physical Review E, 2014

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The algebraic connectivity

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The algebraic connectivity

The algebraic connectivity of a graph G is the second-smallesteigenvalue of the Laplacian matrix of G

Define the algebraic connectivity of a multiplex as the second-smallesteigenvalue of its supra-Laplacian matrix

From the interlacing result we know that

µ2 ≤ µ(a)2 (18)

µ2 ≤ m (19)

and

m is always an eigenvalue of the supra-Laplacian

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Look for the condition under which µ2 = m holds

Conditions

if µ(a)2 < m or µ2 > 1 then µ2 6= m,

This result points to a mechanism which can trigger a structuraltransition of a multiplex network

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Structural organization and transitions

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theoretical question: Will critical phenomena behave differently onmultiplex networks with respect to traditional networks?

So farTheoretical indication that such differences in the critical behaviours

indeed exists

Three different topological scales in a multiplex:

• the individual layers

• the network of layers

• the aggregate network

Quotient graphs give the connection in terms of spectral properties

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Eigengap

Gaps in the Laplacian spectrum are known to unveil a number ofstructural and dynamical properties of the network related to the

presence of different topological scales in it

Introduce a weight parameter p for the coupling→tune the relative strength of the coupling with respect to intra-layer

connectivity

L =⊕α

Lα + pLC

if node-aligned

L =⊕α

(L(α) + p(m − 1)In)− pKm ⊗ In

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gap

non-bounded

bounded

From the interlacing result we know:

• n bounded eigenvalues

• mp is always an eigenvalue of the system

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Scales separation

gk =µk+1 − µkµk+1

(20)

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Aggregate Equivalent MultiplexDefine Aggregate EquivalentMultiplex: A multiplex with thesame number of layers of theoriginal one with the aggregatenetwork in each layer.

{µAEMk } = {µi + µ(l)j } (21)

Very smooth transition

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• before p∗ structurally dominated by the network of layers

• after p� structurally dominated by the aggregate network

• between those two points the system is in an effective multiplexstate

• VN-entropy shows a peak in the central region

• the relative entropy between the parent multiplex and its AEMvaries smoothly with p

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Conclusions

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• we have introduced the basic formalism to describe multiplexnetworks in terms of graphs and associated matrices

• well defined structural metrics that unveils the functioning of thesystem and its context dependent nature

• the effect of the coupling on the dynamical and topologicalproperties of the system

• we have introduced a coarse-grained representation of multiplexnetworks in terms of quotient graphs

• exact results on the spectra unveil the interplay between differenttopological scales in the system and associated structuraltransitions

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Multiplex networks:

a challenge and an opportunity of innovation for the science ofcomplex networks

First challenge:

The need of a common formal language to represent them

An opportunity:

The necessity to reconsider the very foundations of the discipline

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A key step for the structure and function hypothesis⇓

Natural evolution of complex networks science as a mature discipline

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The future

Different possibilities:

• the statistical characterization of the Laplacian and adjacencyspectra

• the generalization of more structural metrics in the commonframework settled up by the walk matrix representation

• a deeper understanding of structural transitions in multiplexnetworksespecially with regard to the role played by symmetries andcorrelations among and across layers

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Related Publications

• Stability of Boolean multilevel networks E. Cozzo, A. Arenas, Y. Moreno - Physical Review E, 2012

• Contact-based social contagion in multiplex networks E. Cozzo, R.A. Banos, S. Meloni, Y. Moreno - PhysicalReview E, 2013

• Mathematical formulation of multilayer networks Manlio De Domenico, Albert Sole-Ribalta, Emanuele Cozzo,Mikko Kivela, Yamir Moreno, Mason A Porter, Sergio Go mez, Alex Arenas - Physical Review X, 2013

• Dimensionality reduction and spectral properties of multilayer networks R.J. Sanchez-Garcıa, E. Cozzo, Y. Moreno -Physical Review E, 2014

• Multilayer networks: metrics and spectral properties E. Cozzo, G.F. de Arruda, F.A. Rodrigues, Y. Moreno - arXivpreprint arXiv:1504.05567, 2015 (in press)

• Structure of triadic relations in multiplex networks E. Cozzo, M. Kivela, M. De Domenico, A. Sole-Ribalta, A.Arenas, S. Gomez, M. A. Porter and Y. Moreno - New Journal of Physics 2015

• On degree-degree correlations in multilayer networks G.F. de Arruda, E. Cozzo, Y. Moreno, F.A. Rodrigues -Physica D (in press)

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