Complex Networks with Node Intrinsic Fitness: On ... · The chapter closes with a survey of various types of stochastic processes that can be embedded on complex networks. 1 Networks

Brunel University London

Department of Mathematics

Complex Networks with NodeIntrinsic Fitness:

On Structural Properties andContagious Phenomena

Doctoral Dissertation of:Konrad Hoppe

2014

• i

Abstract

Complex networks is a vibrant research field and has received much attention over thelast decade. Central to this area is the question of how networks around us are con-structed. The essential notion of network research is that these systems are assembledin a decentralised way, thus no central agent is planning the network beforehand. De-spite this lack of central coordination, many networks present intriguing universalities,such as broad degree distributions, in the form of power-laws. The subject of study inthis thesis is a class of networks that are constructed by a node intrinsic variable, calledfitness. The way these networks grow could be called a rich-get-richer mechanism. Thefitter a node is, the more likely it is to acquire new connections inside the network. Sev-eral aspects that are directly connected to these networks are explored in this thesis. Inthe first part, the properties of growing networks that are driven by fitness are investi-gated and it is shown that the introduction of growth leads to a topological structure thatis different from its static counterpart. In the subsequent chapter, percolation on fit-ness driven networks is studied. The results give insights into possible mechanisms thatcan stabilise systems. Furthermore, the theory can be used to identify vulnerable struc-tures around us. In the following chapter, the world trade network is discussed. Thisnumerical investigation highlights possible improvements to the methodology to makestatistical analysis more robust. That chapter is followed by an analysis of time-varyingnetworks. Time-varying networks represent an interesting construct that allows a for-mulation of stochastic processes on the same time-scale as the evolution of the networkitself. This possibility is highly relevant to the investigation of epidemics, for instance.In the last chapter, a study of a system of clusters and their self-organised formation ispresented.

CONTENTSCONTENTS

CHAPTER 1 PREFACE PAGE V

1 Bibliographical Note v

2 Acknowledgement vi

CHAPTER 2 INTRODUCTION PAGE 1

1 Networks - The Lingua Franca of Science? 1

2 Examples of Networks 3

Seven Bridges of Königsberg — 3 • The Medici — 4 • The 9/11 Terrorists’ Network —6

3 Universal Structures of Complex Networks 7

4 Models of Networks 10

5 Stochastic Processes on Networks 15

Targeted Attacks and Random Failures — 16 • Epidemic Spreading — 16 • OpinionFormation — 19 • Percolation Phenomena on Complex Networks — 20

CHAPTER 3 A NETWORK GROWTH MODEL WITH NODE INTRINSIC FITNESS PAGE 23

1 Introduction 24

2 The Model 26

3 The Design Problem 30

4 Scale-free Networks 33

ii

• iii

5 Fitness Correlations 36

6 Conclusion 38

CHAPTER 4 PERCOLATION OF FAILURES ON COMPLEX NETWORKS PAGE 40

1 Introduction 41

2 Models 43

Static Model — 44 •Dynamic Model — 49

3 Percolation 52

4 Results 56

5 Conclusions 59

CHAPTER 5 THE WORLD TRADE NETWORK PAGE 60

1 Introduction 61

2 The Static Fitness Model 63

3 The Choice of a Data Source 66

4 Definition of Fitness 68

5 Intertemporal Structure of the WTN 70

6 Static Structure of the WTN 73

7 Conclusion 78

8 Countries Covered in this Study 80

CHAPTER 6 MUTUAL SELECTION IN TIME-VARYING NETWORKS PAGE 81

1 Introduction 82

2 Model 85

Degree Distribution — 85 • Epidemic Spreading — 88 • Consensus Formation — 93

3 Discussion 95

• iv

CHAPTER 7 CLUSTER FORMATION DYNAMICS BASED ON FITNESS PAGE 96

1 Introduction 97

2 Model 99

3 Results 103

Results for a Null Model without Fitness — 103 • Results of Fitness Based Cluster Dy-namics — 104

4 Concluding Remarks 110

CHAPTER A COMPUTATIONAL METHODOLOGY PAGE 114

Chapter 1Preface

1 Bibliographical Note

Some aspects of this work have been published in:

I. E. Smolyarenko, K. Hoppe, G. J. Rodgers. Network growth model with intrinsic vertexfitness. Physical Review E 88, 012805 (2013).

K. Hoppe, G. J. Rodgers. Mutual selection in time-varying networks. Physical Review E88, 042804 (2013).

K. Hoppe, G. J. Rodgers. Percolation on fitness-dependent networks with heterogeneousresilience. Physical Review E 90, 012815 (2014).

K. Hoppe, G. J. Rodgers. A microscopic study of the fitness-dependent topology of theworld trade network. Physica A: Statistical Mechanics and its Applications 419(1): 64-74(2015).

v

Preface 2. Acknowledgement • vi

2 Acknowledgement

Undertaking this PhD was a great experience and I have enjoyed learning so many newaspects during this time. There are many people without whom this would not have beenpossible.

First, I would like to thank my supervisor Professor Geoff Rodgers. He has been of sucha great support during my entire time of study and always stood behind me during theadverse periods of my PhD curriculum. Moreover, I am very thankful for the great op-portunities with which he has provided me. I am very proud to be a student of such adistinguished researcher and I look up to his poise with which he guides any audienceand situation.

I also want to thank my parents, Heidelotte Hoppe and Manfred Hoppe who have alwaysloved and supported me. This work would not be possible without them. The processof obtaining a doctorate degree is certainly longer than the core time of study and myparents have prepared me excellently to go this entire path. I am infinitely thankful forall their devotion.

A special acknowledgement goes to my girlfriend Emma Haddi, who was always patientwith me and very sensitive to the special circumstances of having a PhD student for apartner (which are plenty). Her love and support made me stay focussed and helped meto refuel during our times together. I love you, Emma.

My thanks also go out to the Department of Mathematics at Brunel that provided me witha generous stipend for the time of my doctoral research. In particular, I want to mentionhere Kay Lawrence. Her kind and helpful way always made me feel very welcome.

I also want to acknowledge the kind hospitality of Professor Bosiljka Tadic, whom I visitedin Ljubljana during my PhD studies. She has given me many interesting insights into thefield of dynamical systems and I very much enjoyed our conversations.

Also the staff members of the Graduate School at Brunel University should be mentionedhere. I am very thankful for their recognition and support during my time as a doctoralresearcher. They have made the experience at Brunel very joyful. I would like to thank es-pecially Diane Smith. She was always very welcoming and helpful and has kindly offeredto proofread this work, which is so much appreciated because of my persistent resilienceagainst British rules of punctuation.

Chapter 2Introduction

In this first chapter, the general concept of networks is introduced. The historical back-

ground of network research is illustrated by several examples that are found in the real

world. Despite the apparent complex and chaotic structure of many networks, some uni-

versal aspects of complex networks are highlighted, followed by various models that can

give rise to these universal patterns. The chapter closes with a survey of various types of

stochastic processes that can be embedded on complex networks.

1 Networks - The Lingua Franca of Science?

Complex networks is a research field that has drawn knowledge from two directions: the

mathematical treatment of graphs and the many particle systems in physics. The social

sciences have identified complex networks theory as a viable tool to understand many

phenomena around us, such as the spread of rumours, the cascading failure of institu-

tions and the contagion of diseases. Networks are commonly described by a set of nodes

and a set of edges that connect the nodes. These sets are also the main objects of study

in graph theory. While classical graph theory investigates various properties of graphs

that could be relatively small, such as the possibility of different colourings [7], ways of

traversing graphs during search [75], or forming round-tours [48], the study of complex

1

Introduction 1. Networks - The Lingua Franca of Science? • 2

networks that is rooted in physics aims to understand phenomena on almost exclusively

very large, and heterogeneous structures. Problems in this area are more probabilis-

tic and need different approaches compared to problems of classical graph theory. For

instance, while graph theoretical problems can often be cast in terms of the adjacency

matrix of a network, the structure of complex networks is often understood in terms of

sets of differential equations. The particular strength of complex networks research lies

in the ability to describe local interactions in a much richer way than is possible by us-

ing lattices or cellular automata. In particular, the strong heterogeneity of connectivity

patterns that appears to be omnipresent in the networks that surround us can easily be

modelled using networks, rather than lattices. Also other features, such as the tendency

of clustering in social networks, can easily be embedded into networks. In section 4

of this chapter, several constructive models that give rise to large scale heterogeneous

structures are presented.

The great interest in complex networks does not merely arise from these points, but

rather from the fact that almost everything that surrounds us can be described in terms

of a network. Obvious examples are the World Wide Web [4, 70] that is spanned by web-

pages, connected with hyperlinks. Also the physical internet that is spanned by routers

and cables has been investigated [52]. Other examples include the network of movie

actors [8], where nodes are actors and edges connect those that have acted in a movie

together. Similarly, scientific co-authorship networks have been investigated [8, 9, 90],

where nodes represent individual authors and edges connect academics that have co-

authored papers together. Other examples are transportation networks [12, 35, 63] and

the power-grid [2, 36]. More interesting than the mere description of the networks is

the analysis of processes that unravel on these. For instance, the prediction of disease

epidemics [35, 83, 120], and the spread of memes and information diffusion in social

networks [29, 126]. Furthermore, biological networks such as the structure and function

of the brain have been investigated [23] as well as protein encoding networks to better

understand the microscopic structure of our bodies and human diseases [118]. This in-

complete list of possible applications of complex networks research provides a glance

of the strength of this academic discipline. However, the application of networks is not

Introduction 2. Examples of Networks • 3

limited to these large scale structures. As initially mentioned, mathematical graph theory

can be utilised to solve a variety of problems, such as colouring and routing. Other appli-

cations of networks can be found in machine learning where artificial neural networks

are employed to mimic the human brain and understand complex data [106]. Finally,

computers perform elementary algebra by traversing trees which are a special class of

networks. The same structure is used for algorithms in mathematical logic. Thus, even

theorems can be proved using networks.

In summary, networks appear in a multitude of scientific disciplines and many problems

can be translated into graphs. The more recent history of complex networks, which are

classical graphs with the additional feature of being large and heterogeneous, has proven

extremely successful in tackling problems of the present century [65].

2 Examples of Networks

2.1 Seven Bridges of Königsberg

Problems related to graphs were explored early in the history of mathematics. A very fa-

mous problem is the Seven Bridges of Königsberg. Fig. 2.1a shows a map of Königsberg.

The task is to find a round tour through the city that visits every bridge exactly once.

Leonhard Euler proved in 1736 that this path does not exist [48]. Although the problem

is embedded in a geographic setting, its spatial dimension is irrelevant, because the po-

sition of the bridges does not matter. The only aspects that matter are the connecting

bridges. Therefore the problem can be represented as a graph, with the bridges as edges.

Fig. 2.1b illustrates this representation. Euler’s article is regarded as one of the earliest in

the field of graph theory and, the problem of finding an Eulerian tour has become a part

of standard theory.


(a)

#

#

#

#

(b)

Figure 2.1: (a)An ancient map of Königsberg with the river system in blue and the seven bridges markedin green. (b) Graphical representation of the bridges.

2.2 The Medici

The House of Medici was a very influential family in Florence that brought forth two

queens during the 15th and 16th centuries, as well as four popes in the course of the

15th-17th centuries. The family came to fame and political power for the first time un-

der Cosimo de’ Medici during the 14th century in Florence. The key to their sudden

prosperity was not their political dominance or wealth at that time, but rather their cen-

tral position in the network of marriages between different families [95]. Marriages be-

tween families have played a key role in arranging business deals and forming political

allies. Fig. 2.2 illustrates this network of inter-marriages. It is obvious from the figure

that the Medici inhabit a very central position in this network. This central position was

their advantage: many consolidations – commercial or political – between the families

of the Italian Renaissance were intermediated by the Medici. This graphical indicator

of importance in the network can also be quantified. Betweenness centrality of a node

measures the fraction of all shortest paths from all vertices to all others that pass through

that node. To make this more clear, define s i j (k ) as the number of shortest paths that go

from node i to node j through node k and s i j as the total number of shortest paths from

node i to j . For instance, the number of shortest paths from Guadagni to Salviati is 2,

and so is the number of paths that route via the Medici. Thus sGuadagni, Salviati = 2, and

sGuadagni, Salviati(Medici) = 2, while sGuadagni, Salviati(Albizzi) = sGuadagni, Salviati(Tornabuon) =


#

Acciaiuol

Medici

# Tornabuon

# Albizzi

# Ginori

#Peruzzi

#Barbadori

#Castellan

# Strozzi

#Ridolfi

#Pucci#

Bischeri

#Guadagni

#Salviati

#Pazzi

#Lambertes

Figure 2.2: Network of Florentine marriages in the 15th century. Nodes mark the families and edgesmarriages between the members of them.

1. The overall betweenness of a node is then

c (k ) =1

(N −1)(N −2)/2

∑

i 6=k 6=j

s i j (k )s i j

. (2.1)

The factor 2/(N −1)(N −2) normalises c (k ) to the unit interval. Interestingly the Medici

score the highest in terms of betweenness centrality with c (k ) = 0.522, while the next

highest centrality is found for the Guadagni with c (k ) = 0.255.

Another measure of importance is the number of adjacent edges of a node. This number

is called the node degree. Also in terms of the node degree, the Medici score top with six

adjacent edges, followed by the Strozzi and Guadagni with four adjacent edges each.

Given the subsequent rise of the Medici’s power in the 15th and 16th centuries, it is clear

that an investigation of the network structure of Florentine marriages provides viable in-

sights into future outcomes that lay beyond plain marriage patterns. The next subsection

will give some insights into a more recent network of interest, the network of terrorists

who were involved in the 9/11 attacks. Also in that case, simple ”back of the envelope

calculations” give useful insights into aspects that have later been identified by more

thorough investigations.


2.3 The 9/11 Terrorists’ Network

Shortly after the 9/11 attacks on the USA, information on the social ties of the terrorists

became publicly available. Nowadays, it is commonly known that Mohamed Atta was

one of the central planners of the attack. Fig. 2.3 illustrates the network of acquaintances

between the plane hijackers of 9/11. From that figure, it is clear that Mohammed Atta not

#

Wail Alshehri

#Wail Satam Suqami# Waleed Alshehri

# Abdul Aziz Al-Omari

2Marwan Al-Sheshhi

# Mohamed Atta2Fayez-Ahmed

2Mohand Alshehri

2Hamza Alghamdi

2Ahmed Alghamdi

4 Hani Hanjour

4 Khalid Al-Mihd4

Nawaf-Alhazmi

4 Salem Alhazmi

Ï Ziad Jarrah

ÏAhmed Al Haznawi

Ï

Saeed Alghamdi

Ï Ahmed Alhami

Figure 2.3: Network of terrorists involved in the 9/11 attacks. #: Flight AA #11 Crashed into the WorldTrade Centre. 2: Flight UA#175 Crashed into WTC. 4: Flight AA #77 Crashed into Pentagon. Ï: Flight UA#93 crashed in Pennsylvania.

only has a very central position within the network, but also serves as a intermediator

between the different teams that hijacked the four airplanes. Krebs [76] shows a larger

snapshot of this network with links also to other accomplices who provided money and

logistics. Considering this larger network, Mohamed Atta reserves a central role with

a betweenness centrality of 0.588 followed by Essid Sami Ben Khemais with a value of

0.252.

Introduction 3. Universal Structures of Complex Networks • 7

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Figure 2.4: Snapshot of the network of friendships on Facebook (source: Stanford Large Network DatasetCollection)

This is therefore another example of the importance of network metrics that shed light

on aspects far beyond the scope of topological insights. The network of terrorists and

further consequences of its topology have also been investigated in [99]. In particular, it

is important to note that an understanding of the underlying network/hierarchy struc-

ture is essential for a successful anti-terror operation. The impact of different network

structures on deliberate attempts to disturb the function of a network is discussed later

in this chapter.

3 Universal Structures of Complex Networks

The networks around us are of astonishing size and complexity, see Fig. 2.4 for an exam-

ple of a snapshot of Facebook. However, there are also a number of interesting regular-

ities in these structures. To illustrate this concept, some features of three different net-

works are investigated in this section: a graph of the world wide web provided by Google,

the collaboration network in the area of condensed matter on arXiv1, and the network of

publicly shared circles on Google+2. All data has been obtained from the Stanford Large

Network Dataset Collection3.

Many networks are so called small-world networks. These networks are characterised in

1www.arxiv.org2https://plus.google.com3https://snap.stanford.edu/data/

www.arxiv.orghttps://plus.google.comhttps://snap.stanford.edu/data/


terms of their average path length L. That is the number of steps it takes on average to

walk from one node to another, whereby steps can only be made along edges. A network

is identified as a small world if the average path length growths logarithmically with its

size [125], thus

L ∝ log N , (2.2)

where N is the number of nodes. The small-world property is sometimes also known as

the six degrees of separation, coined by Guare [62]. Another property of interest is the

local clustering coefficient c i of a node. The clustering coefficient measures how close

the neighbours of a node are to being a clique. A clique is a set of nodes that is fully

connected, such that every possible edge exists. Formally, the local clustering coefficient

is defined as

c i =1

k i (k i −1)

�

�

�

¦

e j k : v j , vk ∈Ni , e j k ∈ E©

�

�

� (2.3)

whereNi is the set of vertices in the direct neighbourhood of node i . Furthermore, E =

{e i ,j : i , j = 1, . . . , N } denotes the set of all edges and V the set of all nodes with V =

{vi : i = 1, . . . , N }. For an undirected network, the concept can be accordingly defined,

but needs renormalisation by factor 2. Fig. 2.5 illustrates the local clustering coefficient

graphically. The average clustering coefficient is then given by

i

#

# #

(a)

i

#

# #

(b)

i

#

# #

(c)

Figure 2.5: Example configurations of a node’s neighbourhood. Solid edges mark present edges, dashededges possible ones. The local clustering coefficient is for the different configurations: (a) c i = 0, (b) c i = 1,and (c) c i = 1/3.

〈c 〉=N∑

i=1

c iN

. (2.4)

Real-world networks show significantly more clustering than random networks. The

most simple random network model was proposed by Erdős and Rényi [46] and serves


as a null model for many network concepts. The model is described by a fixed number of

nodes and edges. Each edge exists with probability p , such that the existence of edges are

statistically independent from each other. In this case the clustering coefficient is sim-

ply 〈c 〉= p , since the probability that pairs of edges exist in the neighbourhood does not

depend on any neighbourhood linked property. Clustering describes the cliquishness of

a network. In terms of friendship networks, high clustering implies that the friends of

someone are themselves friends of each other. This transitivity of friendships is likely

in social networks and thus, real-world networks exhibit much higher clustering than

pure random graphs. Tab. 2.1 summarises these network measures for three different

networks. Another feature that can be observed in real-world networks is that they have

N M 〈c 〉 DiameterGoogle web graph 875, 713 5, 105, 039 0.5413 21arXiv cond-mat 23, 133 93497 0.6334 14Google+ circles 107, 614 13673453 0.4901 6

Table 2.1: Central properties of the three networks under investigation, with N as the number of nodes,M the number of edges, 〈c 〉 the average clustering coefficient, and the diameter as the longest shortest pathin the network.

broad degree distributions, to be more precise: power-law degree distributions [31]. The

degree distribution p (k ) is the probability that a randomly chosen node has k neigh-

bours. In the case of a power-law, p (k ) has the form

p (k )∝ k−a . (2.5)

Power-laws appear in many areas of interaction, also in the three networks that serve as

examples in this section; see Fig. 2.6. It is not entirely clear why interaction leads regu-

larly to power-law distributed properties. In fact there is an entire branch of literature

investigating this phenomenon [84]. For networks, one mechanism that is believed to

drive networks into the observed power-law degree distributions is preferential attach-

ment [8]. The notion of preferential attachment is that interactors with more existing

interactions are more likely to engage in new interactions. In terms of the network of

friendships, a person with many friends finds it easier to acquire new relationships, than

somebody who is entirely isolated. More details on this model can be found in the fol-

lowing section.

Introduction 4. Models of Networks • 10

1 10 100 100010-6

10-5

10-4

0.001

0.01

0.1

k

PHkL

(a)

2 5 10 20 50 1002001´10

-4

5´10-4

0.001

0.005

0.010

0.050

0.100

k

PHkL

(b)

1 10 100 1000 104

10-5

10-4

0.001

0.01

0.1

k

PHkL

(c)

Figure 2.6: Degree distributions of (a) the web graph (b) the collaboration network in the condensedmatter section on arXiv (c) the friendship network on Google+.

4 Models of Networks

In section 2, some examples of the usefulness of network studies have been illustrated.

While the focus was put on the existence of various observable artefacts, in this section

a number of models that lead to certain macroscopic patterns will be reviewed.

One of the first and simplest network models is the Erdös-Rényi (ER) Model [46]. Two

variants of this model exist: the G (N , M )model, that describes an undirected graph with

N nodes and M edges, and the G (N , p ) model, which elucidates a graph with N nodes

whereby each pair of nodes shares an edge with probability p . In both cases the number

of nodes is static and edges are deployed randomly. The resulting degree distribution is

binomial and can be approximated in the N →∞ regime as a Poisson distribution

PER(k ) =(N p )k e−N p

k !(2.6)

for the G (N , p )model. The relationship to the G (N , M )model is then described by M =�N

2

�

p .

The ER model is a useful null model to study networks, but it lacks several features that

are observed in real-world networks. One such feature is clustering. Networks around

us are highly clustered and have short average path lengths. A model that resembles this

feature is the Watts-Strogatz (WS) model [125]. The network consists of a fixed number

of nodes. All nodes are organised on a ring lattice. The central parameter of the model


are N , k ∗ and p . N is the number of nodes, k ∗ the number of nearest neighbours on the

ring lattice that each node is connected to and p is the probability that an existing edge

gets rewired. p drives the structure of the network from a regular ring lattice for p = 0 via

a small-world network for 0 < p < 1 towards a random network at p = 1. The rewiring

takes place in a sequential process that considers each edge exactly once. More details

can be found in [125]. The degree distribution depends on the value of p . For p = 0, p (k )

is peaked at k ∗: p (k ) = δ(k − k ∗) where δ(x ) is the Dirac delta function. For p = 1, the

degree distribution is Poissonian, as in the ER model. Also for values of 0 < p < 1, the

degree distribution can be computed in terms of a series representation [14].

A further generalisation of the Erdös-Rényi model has been introduced in [24]. The

model by Caldarelli et. al. extends the ER model by introducing a vertex intrinsic variable

and an attachment kernel f (x , y ), which is defined as the probability that a pair of nodes

with intrinsic variables x and y share an edge in a network with N nodes. The hidden

variable is sometimes called fitness, which refers to its ability to attract edges inside the

network. The degree distribution p (k ) can be found modelling a more detailed quan-

tity, namely p (k |x ), the fitness conditional degree distribution. One possible approach

is to expand p (k |x ) into its contributions [21, Eq. (15)]. Another approach is to derive it

from first principles, noticing that the static nature of the network can be ignored for the

sake of the argument and edges deployed sequentially. In this case the evolution of the

fitness conditional degree distribution in a network with M edges and N nodes can be

described by

pM+1,N (k |x ) = pM ,N (k |x )[1−λ(x , N )]+pM ,N (k −1|x )λ(x , N ), (2.7)

whereλ(x , N ) is the probability that a node with fitness x receives an edge during an edge

deployment step. The details of this derivation are discussed later in Chapter 5, following

Eq. (5.5). The resulting fitness dependent node degree distribution is found equivalent

to [21] to be Poissonian:

pM ,N (k |x ) =e−k̄ (x )[k̄ (x )]k

Γ(k +1), (2.8)

where k̄ (x ) is the fitness conditional degree expectation. The exact form of the degree


distribution of the resulting network depends then on the probability distribution of fit-

ness and the attachment kernel f (x , y ). The fitness model has also been analysed in

[111], where the inverse and direct problems have been addressed. That is the problem

of finding analytical forms of f (x , y ) and the fitness distribution ρ(x ), such that the re-

sulting network exhibits a given degree distribution. The solutions to these problems are

of high importance when the internal mechanisms of the network need to be understood

in more detail. The spreading of diseases can usually be expressed in terms of f (x , y ) and

ρ(x ), and solutions to the inverse and direct problem can give rise to these quantities.

In Chapter 3, a further extension of the fitness model, introducing growth is studied.

Growth has been proven to be an important ingredient for many real-world models [42].

The fitness conditional degree distribution can be found with the same technique as

outlined above for the static case. This distribution is given by

p (k |x ) =1

1+λ(x )

�

λ(x )1+λ(x )

�k−1. (2.9)

Wherebyλ(x ) is similarly defined to that in the previous paragraph on static fitness driven

networks. More details of this model can be found in Chapter 3.

Another class of network models that have attracted significant amount of attention in

the complex networks literature are degree based models. Early work on this class was

motivated by bibliographic studies of cross-referencing between different publications

[102]. The cumulative advantage of having more citations in order to receive more fol-

lowing citations in the future was studied in more detail in [103]. However, Price’s inves-

tigations were not widely recognised until 1999. Barabási and Albert [8] reinvestigated

what is nowadays called the Barabási Albert (BA) model. The BA model is a growing net-

work model, that models the cumulative advantage by imposing that each new node that

arrives to the network connects preferentially to nodes with high degree. More formally,

define the probability that a new node connects to an existing node i as

πnew→i =k i

∑

j k j=

k i2t

. (2.10)


k i is the degree of node i . Thus, the rate at which the degree of a node changes is

d k id t= k i /2t . (2.11)

Therefore, the final degree distribution is then found to be

p (k ) = 2k−3. (2.12)

This finding is the reason for the success of the BA model. Many real-world networks

exhibit power-law degree distributions [3]. Interestingly both features, growth and pref-

erential attachment, have to be present. Growth without preferential attachment leads

to an exponential distribution, while preferential attachment without growth leads to a

non-stationary distribution, ultimately resembling a complete graph, that is a network

with all possible edges present [8].

Next to the two classes of models that are built either exclusively by a hidden variable

or exclusively by a degree based mechanism, there exists also a third, a hybrid type.

Bianconi and Barabási [19] introduce a model which is motivated by the finding that

in the classical BA model old nodes had more chances to connect than young nodes

and will therefore have on average a higher degree. In real-world networks one can fre-

quently find nodes that arrive new to the network and are able to acquire large numbers

of new links in a short time. These out-performers must have some intrinsic quality

which drives this rapid growth. To account for this, a generalised attachment kernel that

considers this increased ability is introduced:

πnew→ik i x i

∑N (t )j=1 k j x j

, (2.13)

where x i is the fitness of node i and k i its degree. A generalisation of this model with

other approaches to link degree and fitness was introduced by Ergün and Rodgers [47].

Another form of combining the fitness model and the degree preferential attachment has

been investigated in [16]. Bedogne and Rodgers [16] introduce a model where one of two

things can happen at each time step. Either a new node with one link is added, whereby


the target node is chosen degree preferentially as in the BA model, or a new edge between

two existing nodes is deployed, depending on the nodes’ fitness. The resulting degree

distribution is a pure power-law. The exponent depends only on the relative frequency

of node additions compared to edge addition steps.

Perra et al. [100] present a model that is characterised by dynamics on a much shorter

time scale than in the previously discussed models. The preceding models are either en-

tirely static or growth occurs gradually, one update at a time. The time-varying network

model in [100] is entirely different. Time varying networks are described by a fixed num-

ber of nodes and a varying number of edges. If the model is described in discrete time,

then at each time step each node becomes active with a node intrinsic probability and

connects to a fixed number of existing nodes inside the network. The motivation behind

this class of models is, for example, in the mobile phone network, where nodes represent

mobile phones and edges phone calls. If a phone initiates a call to another, then these

two nodes share an edge during that particular instance. Other examples include human

contact to spread diseases or even rumours. The particular appeal of a time-varying net-

work model lays in the possibility to formulate the evolution of the network on the same

timescale as the unfolding of stochastic processes on top of it. More details about this

class of networks can be found in Chapter 6.

The preceding network models are flat, in the sense that edges are just of one type and

binary. However, there exist a number of systems for which these assumptions do not

hold. Investigating the structure of link weights is important in a number of applica-

tions. Consider for instance the strength of a social tie, distinguishing long time friends

from random encounters. Another example forms the air-route network, where nodes

are airports and edges the flight routes between them. The weight of a link might de-

scribe for instance the number of seats that are available during a day on a given route.

This knowledge can be used to comprehend the contagion of infectious diseases or the

severity of service interruptions. Additionally, edges cannot just have different weights,

but can also be of different types. Continuing the airline network example from above,

cities are not only connected by air-routes, but also by streets and the railway system.

These different layers can either be represented as separate networks or as different types

Introduction 5. Stochastic Processes on Networks • 15

of links. Fig. 2.7 illustrates this notion. Also the network of international trade forms an

#N2

#N3

#N1

#N4

(a)

#N2

#N3

#N1

#N4

#N2

#N3

#N1

#N4

1st Layer

2nd Layer

(b)

Figure 2.7: Schematic illustration of multiplex networks. (a) Representation of multiplicity as differentedge types. (b) Representation of multiplicity as different network layers.

example of a multiplex network. Each node represents a country and the different edge

types indicate the traded amount of single product classes between pairs of countries.

Another example is the power supply network, that consists of different levels, such as

gas pipelines, high voltage power lines, water pipes, etc. Apart from merely describing

these structures, it is of central interest to understand how to prevent failures of these

networks and protect them against deliberate attacks. The study of these phenomena is

the subject of the following section.

5 Stochastic Processes on Networks

Many real-world phenomena can be described in terms of networks, such as trade rela-

tions, human contacts, collaborations, etc. Beyond a detailed description of these net-

works, it is of great importance to understand how these networks behave in the pres-

ence of adverse events and to what extent their function is compromised. In this section,

a number of stochastic processes that occur on networks are discussed.


5.1 Targeted Attacks and Random Failures

Random networks described in the ER model and scale-free networks produced by the

BA model are very different with respect to their topological quantities. These differences

prove to be crucial when the networks are under targeted attack, or subject to random

failures. Albert, Jeong, and Barabási [5] present a numerical investigation of the conse-

quences for both type of networks. The scale-free (SF) network is very resilient against

random failures compared to the random network. The reason is the heterogeneous

contact pattern of the SF network. There is just a small number of hubs, that connect

large proportions of the network. Random failures are unlikely to occur at these hubs,

so a large fraction of the network can fail before the network’s function is affected. The

behaviour of the random network is entirely different. Most nodes have degree k ∼ 〈k 〉,

thus each node is equally important for the function of the network, and hence removing

nodes increases the diameter of the network linearly.

While the SF network is extremely resilient against random failures, it is prone to fail

quickly under targeted attack. An attacker who knows the hubs of the network that con-

nect large proportions can destroy the network’s function after a small number of re-

movals. In this case, the random network has the advantage. Since every node connects

on average the same fraction in the network, the additional knowledge of the attacker

does not contain any valuable information and thus the diameter increases in the same

way as in the case of random failures.

5.2 Epidemic Spreading

The study of epidemics has received attention from the mathematical community for a

long time. One of the earliest contributions has been made by the Swiss mathematician

Daniel Bernoulli. He investigated a mathematical model of smallpox in 1760 [17]. A com-

prehensive review of the coverage of infectious diseases in mathematics can be found for

instance in [66].


Before an underlying network structure was considered, the diseased population was

assumed to be perfectly mixed in the sense that an individual has equal probabilities

to connect to any other individual in the population to pass on or to receive the dis-

ease. This assumption allows one to describe the evolution of an epidemic in terms

of simple linear differential equations. A simple compartmental epidemic model is the

susceptible-infected-susceptible (SIS) model. In this model each node belongs to one of

the two compartments: infected I or susceptible S. If the population is assumed to be

perfectly mixed, then the evolution of the number of individuals in each compartment

is governed by

d s

d t=−βs i +γi and (2.14)

d i

d t=βs i −γi . (2.15)

where s = S/N and i = I /N are the fractions of susceptible, respectively infected indi-

viduals. β is the probability that a susceptible individual gets infected in a contact with

an infected individual and 1/γ is the time it takes to recover from the disease. Fig. 2.8

illustrates the development of the size of these two groups.

sHtL

iHtL

100 200 300 400 500t

0.2

0.4

0.6

0.8

1.0

Figure 2.8: Size of the two compartments: susceptible (S) and infected (I) in the SIS model with β = 1/10and γ= 9/100.

The figure shows that the number of individuals in each compartment reaches an equi-

librium value after a relatively short burn-in phase. A central quantity of epidemic re-

search is the reproduction number that measures the number of secondary infections

following a single infectious individual being introduced to the system. This number is


defined as

R0 =β/γ. (2.16)

R0 describes the potential of the disease. In particular, if R0 < 1, the infectious individual

cannot generate an epidemic, since there are not enough secondary cases in order to

sustain the disease. If however R0 > 1, then the diseased population grows ad infinitum.

If R0 = 1, the disease will stay forever in the population, since every single case produces

one successive case. R0 = 1 marks the epidemic threshold that separates the two phases

of infinite growth of the infectious fraction and instantaneous immunity.

The preceding technology can easily be expanded to account for networks in which the

degree distribution is peaked at 〈k 〉 and decays exponentially [13]. For heterogeneous

networks, this is not the case. A typical node degree does not exist as it is possible to

find many nodes with k � 〈k 〉. It is therefore necessary to consider the different degree

classes separately, to compute the evolution of the epidemics. The variables of interest

are the fraction of infected nodes with degree k : i k (t ) = Ik /Nk , and the fraction of sus-

ceptible individuals in degree class k sk (t ) = Sk /Nk . The evolution of the SIS model is

then described by

d i kd t=βk skΘk (t )−γi k and (2.17)

sk = 1− i k . (2.18)

Θk is the density of infected neighbours around a node with degree k :

Θk (t ) =∑

k ′P(k ′|k )i k ′ (t ) (2.19)

In the absence of degree correlations, this quality becomes

Θk (t ) =Θ(t ) =1

〈k 〉

∑

k ′k ′P(k ′)i k ′ (t ). (2.20)

Notice, that the k independence holds only in the absence of degree correlations. With-

out degree correlations the conditional probability to find a node with degree k ′ in the


neighbourhood of a node with degree k is given by

p (k |k ′) =k P(k )〈k 〉

. (2.21)

The epidemic threshold is found to be

β

γ≥〈k 〉〈k 2〉

. (2.22)

It has been illustrated that for many real-world networks that the degree distribution

follows a power-law p (k ) ∝ k−a with a typical exponent 2 < a < 3. Thus in the N →∞

limit, 〈k 2〉→∞ and therefore an epidemic threshold of zero is found, which implies that

a disease can never die out on a heterogeneous network. Real-world networks have finite

size and therefore a non-zero epidemic threshold can be computed, but nevertheless this

result shows that epidemics can spread over a heterogeneous network easily even with

very low transmission rates.

5.3 Opinion Formation

Opinion or consensus formation is a widely studied phenomenon in social sciences and

has received attention in the complex networks literature as well. The simplest models

are the voter model (VM), the invasion model (IM) and the link dynamics model (LD). All

these models comprise a set of agents that are characterised by a discrete, usually binary

variable s i =±1 that reflects their opinion on a certain topic, for instance Republican vs.

Democrat. Independent of the embedding geometry, the evolution of these three mod-

els is usually described in discrete time. In the voter model, at each time step a node is

chosen randomly and adopts the opinion of a randomly chosen neighbour. In the inva-

sion model, the randomly picked neighbour adopts the opinion of the initially chosen

node. In the LD model, an edge is chosen randomly and a randomly chosen node ad-

jacent to this edge adopts the opinion of the node on the other side of that edge. Early

work on voting and invasion can be found in [32]. If the community of agents is embed-

ded on a d -dimensional lattice, then the community reaches an ordered state, that is a


situation where each agent has the same opinion, for d = 1, 2. Castellano, Vilone, and

Vespignani [26] derive results for voting on the small-world network (WS model) [125].

Interestingly, long range interaction induced by the rewiring in the WS model leads to

a faster ordering/agreement of the system only in finitely sized networks. In the ther-

modynamic limit of N →∞, the small-world network reaches a non-trivial equilibrium,

in which both opinions can coexist. Sood and Redner [113] investigate the time it takes

until a consensus on a heterogeneous network can be reached. This time will be called

consensus time TN . The consensus time on a regular d -dimensional lattice scales with

as TN ∝N 2 for d = 1 and TN ∝N ln N for d = 2 and TN ∝N for d > 2 [113]. This scaling

behaviour is found to be different on heterogeneous graphs. For power-law networks for

example, the scaling is found to be for p (k )∝ k−a as TN ∝N ln N for a = 3, N (2a−4)/(a−1)

for 2< a < 3 and TN ∝ (ln N )2 for a = 2 [113]. More results on the invasion model and the

link dynamics model can be found in [112].

5.4 Percolation Phenomena on Complex Networks

Percolation theory on complex networks is concerned with topological properties of oc-

cupying clusters on top of a network. Occupation of a single node can be understood as

a diseased state and percolation theory finds therefore applications in epidemic mod-

elling [91]. The topological properties of the percolating component that is described

by occupied sites and links between those can be understood in three distinct phases.

These phases are separated by the percolation threshold pc . For low occupation prob-

abilities p < pc , many small clusters exist. If the occupation probability is greater than

the threshold, i.e. p > pc , then the network is occupied by one giant component that

is globally connected. In terms of an epidemic, this state describes the situation when

a disease has broken out and a major part of the population is infected. At p = pc , the

system is critical and occupied clusters of all sizes exist.

To compute the properties of the giant component, define q as the probability that a

randomly chosen edge does not lead to the percolating component. This probability

q can be calculated self consistently. Consider randomly picking a node i 0 and follow


one of its links randomly toward a node i 1. The link between i 0 and i 1 is only not part

of the percolating component if none of the other links attached to i 1 are not part of

the component. The overall probability for this event can be split into two parts: the

probability that the randomly picked link ends at a node with degree k and that none of

the other k −1 links belongs to the percolating component. In an uncorrelated network,

p (k |k ′) is independent of k ′ and the probability that none of the k − 1 links belongs to

the percolating component is just q k−1. Therefore q can be computed by

q =∑

k

k p (k )〈k 〉

q k−1. (2.23)

The construction above appears cumbersome at first glance, because of its construction

in terms of probabilities that a event does not occur. However, the readers can convince

themselves that the positive formulation would demand a more detailed analytic expres-

sion.

Define now the probability that a node belongs to the giant component as pG . Obviously,

1−pG =∑

k

p (k )q k . (2.24)

This probability can be computed by solving Eq. (2.23) and substituting for q . A trivial

solution to Eq. (2.23) is q = 1. Substituting q = 1 into Eq. (2.24) leads to pG = 0. There

exists however maximally one more non-trivial solution to Eq. (2.23). Define G (q ) =∑

kk p (k )〈k 〉 q

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