Complex networks principles, methods and applications

Complex Networks
Principles, Methods and Applications

Networks constitute the backbone of complex systems, from the human brain to computer
communications, transport infrastructures to online social systems, metabolic reactions
to financial markets. Characterising their structure improves our understanding of the
physical, biological, economic and social phenomena that shape our world.
Rigorous and thorough, this textbook presents a detailed overview of the new theory
and methods of network science. Covering algorithms for graph exploration, node ranking
and network generation, among the others, the book allows students to experiment with
network models and real-world data sets, providing them with a deep understanding of the
basics of network theory and its practical applications. Systems of growing complexity are
examined in detail, challenging students to increase their level of skill. An engaging presentation of the important principles of network science makes this the perfect reference for
researchers and undergraduate and graduate students in physics, mathematics, engineering,
biology, neuroscience and social sciences.
Vito Latora is Professor of Applied Mathematics and Chair of Complex Systems at Queen
Mary University of London. Noted for his research in statistical physics and in complex
networks, his current interests include time-varying and multiplex networks, and their
applications to socio-economic systems and to the human brain.
Vincenzo Nicosia is Lecturer in Networks and Data Analysis at the School of Mathematical
Sciences at Queen Mary University of London. His research spans several aspects of network structure and dynamics, and his recent interests include multi-layer networks and
their applications to big data modelling.
Giovanni Russo is Professor of Numerical Analysis in the Department of Mathematics and
Computer Science at the University of Catania, Italy, focusing on numerical methods
for partial differential equations, with particular application to hyperbolic and kinetic
problems.

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Complex Networks
Principles, Methods and Applications

VITO LATOR A
Queen Mary University of London

VINCENZO NICOSIA
Queen Mary University of London

GIOVANNI RUSSO
University of Catania, Italy

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education, learning, and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781107103184
DOI: 10.1017/9781316216002
© Vito Latora, Vincenzo Nicosia and Giovanni Russo 2017
This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2017
Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Latora, Vito, author. | Nicosia, Vincenzo, author. | Russo, Giovanni, author.
Title: Complex networks : principles, methods and applications / Vito Latora,
Queen Mary University of London, Vincenzo Nicosia, Queen Mary University
of London, Giovanni Russo, Università degli Studi di Catania, Italy.
Description: Cambridge, United Kingdom ; New York, NY : Cambridge University
Press, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2017026029 | ISBN 9781107103184 (hardback)
Subjects: LCSH: Network analysis (Planning)
Classification: LCC T57.85 .L36 2017 | DDC 003/.72–dc23
LC record available at />ISBN 978-1-107-10318-4 Hardback
Additional resources for this publication at www.cambridge.org/9781107103184.
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.

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To Giusi, Francesca and Alessandra

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Contents

Preface
Introduction
The Backbone of a Complex System
Complex Networks Are All Around Us
Why Study Complex Networks?
Overview of the Book
Acknowledgements

1 Graphs and Graph Theory
1.1 What Is a Graph?
1.2 Directed, Weighted and Bipartite Graphs
1.3 Basic Definitions
1.4 Trees
1.5 Graph Theory and the Bridges of Königsberg
1.6 How to Represent a Graph
1.7 What We Have Learned and Further Readings
Problems

2 Centrality Measures
2.1 The Importance of Being Central
2.2 Connected Graphs and Irreducible Matrices
2.3 Degree and Eigenvector Centrality
2.4 Measures Based on Shortest Paths
2.5 Movie Actors

2.6 Group Centrality
2.7 What We Have Learned and Further Readings
Problems

3 Random Graphs
3.1
3.2
3.3
3.4
3.5
3.6

Erd˝os and Rényi (ER) Models
Degree Distribution
Trees, Cycles and Complete Subgraphs
Giant Connected Component
Scientific Collaboration Networks
Characteristic Path Length

page xi
xii
xii
xiv
xv
xvii
xx
1
1
9
13

17
19
23
28
28
31
31
34
39
47
56
62
64
65
69
69
76
79
84
90
94

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viii

3.7 What We Have Learned and Further Readings
Problems

103
104

4 Small-World Networks

107
107
112
116
127
135
144
148
148

4.1 Six Degrees of Separation
4.2 The Brain of a Worm
4.3 Clustering Coefficient
4.4 The Watts–Strogatz (WS) Model
4.5 Variations to the Theme
4.6 Navigating Small-World Networks
4.7 What We Have Learned and Further Readings
Problems

5 Generalised Random Graphs
5.1 The World Wide Web
5.2 Power-Law Degree Distributions

5.3 The Configuration Model
5.4 Random Graphs with Arbitrary Degree Distribution
5.5 Scale-Free Random Graphs
5.6 Probability Generating Functions
5.7 What We Have Learned and Further Readings
Problems

6 Models of Growing Graphs
6.1 Citation Networks and the Linear Preferential Attachment
6.2 The Barabási–Albert (BA) Model
6.3 The Importance of Being Preferential and Linear
6.4 Variations to the Theme
6.5 Can Latecomers Make It? The Fitness Model
6.6 Optimisation Models
6.7 What We Have Learned and Further Readings
Problems

7 Degree Correlations
7.1 The Internet and Other Correlated Networks
7.2 Dealing with Correlated Networks
7.3 Assortative and Disassortative Networks
7.4 Newman’s Correlation Coefficient
7.5 Models of Networks with Degree–Degree Correlations
7.6 What We Have Learned and Further Readings
Problems

8 Cycles and Motifs
8.1
8.2
8.3

Counting Cycles
Cycles in Scale-Free Networks
Spatial Networks of Urban Streets

151
151
161
171
178
184
188
202
204
206
206
215
224
230
241
248
252
253
257
257
262
268
275
285
290

291
294
294
303
307

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ix

8.4 Transcription Regulation Networks
8.5 Motif Analysis
8.6 What We Have Learned and Further Readings
Problems

316
324
329
330

9 Community Structure

332
332
336
342
349

354
357
365
369
371

9.1 Zachary’s Karate Club
9.2 The Spectral Bisection Method
9.3 Hierarchical Clustering
9.4 The Girvan–Newman Method
9.5 Computer Generated Benchmarks
9.6 The Modularity
9.7 A Local Method
9.8 What We Have Learned and Further Readings
Problems

10 Weighted Networks

374
374
381
387
393
401
407
408

10.1 Tuning the Interactions
10.2 Basic Measures
10.3 Motifs and Communities

10.4 Growing Weighted Networks
10.5 Networks of Stocks in a Financial Market
10.6 What We Have Learned and Further Readings
Problems

Appendices
A.1
A.2
A.3
A.4
A.5
A.6
A.7
A.8
A.9
A.10
A.11
A.12
A.13
A.14
A.15
A.16
A.17
A.18
A.19

Problems, Algorithms and Time Complexity
A Simple Introduction to Computational Complexity
Elementary Data Structures
Basic Operations with Sparse Matrices

Eigenvalue and Eigenvector Computation
Computation of Shortest Paths
Computation of Node Betweenness
Component Analysis
Random Sampling
Erd˝os and Rényi Random Graph Models
The Watts–Strogatz Small-World Model
The Configuration Model
Growing Unweighted Graphs
Random Graphs with Degree–Degree Correlations
Johnson’s Algorithm to Enumerate Cycles
Motifs Analysis
Girvan–Newman Algorithm
Greedy Modularity Optimisation
Label Propagation

410
410
420
425
440
444
452
462
467
474
485
489
492
499

506
508
511
515
519
524

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A.20 Kruskal’s Algorithm for Minimum Spanning Tree
A.21 Models for Weighted Networks
List of Programs
References
Author Index
Index

528
531
533
535
550
552

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Preface

Social systems, the human brain, the Internet and the World Wide Web are all examples
of complex networks, i.e. systems composed of a large number of units interconnected
through highly non-trivial patterns of interactions. This book is an introduction to the beautiful and multidisciplinary world of complex networks. The readers of the book will be
exposed to the fundamental principles, methods and applications of a novel discipline: network science. They will learn how to characterise the architecture of a network and model
its growth, and will uncover the principles common to networks from different fields.
The book covers a large variety of topics including elements of graph theory, social
networks and centrality measures, random graphs, small-world and scale-free networks,
models of growing graphs and degree–degree correlations, as well as more advanced topics
such as motif analysis, community structure and weighted networks. Each chapter presents
its main ideas together with the related mathematical definitions, models and algorithms,
and makes extensive use of network data sets to explore these ideas.
The book contains several practical applications that range from determining the role of
an individual in a social network or the importance of a player in a football team, to identifying the sub-areas of a nervous systems or understanding correlations between stocks in
a financial market.
Thanks to its colloquial style, the extensive use of examples and the accompanying software tools and network data sets, this book is the ideal university-level textbook for a
first module on complex networks. It can also be used as a comprehensive reference for
researchers in mathematics, physics, engineering, biology and social sciences, or as a historical introduction to the main findings of one of the most active interdisciplinary research
fields of the moment.
This book is fundamentally on the structure of complex networks, and we hope it will
be followed soon by a second book on the different types of dynamical processes that can
take place over a complex network.
Vito Latora
Vincenzo Nicosia
Giovanni Russo

xi

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Introduction

The Backbone of a Complex System
Imagine you are invited to a party; you observe what happens in the room when the other
guests arrive. They start to talk in small groups, usually of two people, then the groups grow
in size, they split, merge again, change shape. Some of the people move from one group
to another. Some of them know each other already, while others are introduced by mutual
friends at the party. Suppose you are also able to track all of the guests and their movements
in space; their head and body gestures, the content of their discussions. Each person is
different from the others. Some are more lively and act as the centre of the social gathering:
they tell good stories, attract the attention of the others and lead the group conversation.
Other individuals are more shy: they stay in smaller groups and prefer to listen to the
others. It is also interesting to notice how different genders and ages vary between groups.
For instance, there may be groups which are mostly male, others which are mostly female,
and groups with a similar proportion of both men and women. The topic of each discussion
might even depend on the group composition. Then, when food and beverages arrive, the
people move towards the main table. They organise into more or less regular queues, so
that the shape of the newly formed groups is different. The individuals rearrange again into
new groups sitting at the various tables. Old friends, but also those who have just met at
the party, will tend to sit at the same tables. Then, discussions will start again during the
dinner, on the same topics as before, or on some new topics. After dinner, when the music
begins, we again observe a change in the shape and size of the groups, with the formation
of couples and the emergence of collective motion as everybody starts to dance.
The social system we have just considered is a typical example of what is known today
as a complex system [16, 44]. The study of complex systems is a new science, and so a

commonly accepted formal definition of a complex system is still missing. We can roughly
say that a complex system is a system made by a large number of single units (individuals,
components or agents) interacting in such a way that the behaviour of the system is not
a simple combination of the behaviours of the single units. In particular, some collective
behaviours emerge without the need for any central control. This is exactly what we have
observed by monitoring the evolution of our party with the formation of social groups, and
the emergence of discussions on some particular topics. This kind of behaviour is what we
find in human societies at various levels, where the interactions of many individuals give
rise to the emergence of civilisation, urban forms, cultures and economies. Analogously,
animal societies such as, for instance, ant colonies, accomplish a variety of different tasks,
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Introduction

from nest maintenance to the organisation of food search, without the need for any central
control.
Let us consider another example of a complex system, certainly the most representative
and beautiful one: the human brain. With around 102 billion neurons, each connected by
synapses to several thousand other neurons, this is the most complicated organ in our body.
Neurons are cells which process and transmit information through electrochemical signals.
Although neurons are of different types and shapes, the “integrate-and-fire” mechanism
at the core of their dynamics is relatively simple. Each neuron receives synaptic signals,
which can be either excitatory or inhibitory, from other neurons. These signals are then

integrated and, provided the combined excitation received is larger than a certain threshold,
the neuron fires. This firing generates an electric signal, called an action potential, which
propagates through synapses to other neurons. Notwithstanding the extreme simplicity of
the interactions, the brain self-organises collective behaviours which are difficult to predict from our knowledge of the dynamics of its individual elements. From an avalanche of
simple integrate-and-fire interactions, the neurons of the brain are capable of organising a
large variety of wonderful emerging behaviours. For instance, sensory neurons coordinate
the response of the body to touch, light, sounds and other external stimuli. Motor neurons
are in charge of the body’s movement by controlling the contraction or relaxation of the
muscles. Neurons of the prefrontal cortex are responsible for reasoning and abstract thinking, while neurons of the limbic system are involved in processing social and emotional
information.
Over the years, the main focus of scientific research has been on the characteristics of the
individual components of a complex system and to understand the details of their interactions. We can now say that we have learnt a lot about the different types of nerve cells and
the ways they communicate with each other through electrochemical signals. Analogously,
we know how the individuals of a social group communicate through both spoken and body
language, and the basic rules through which they learn from one another and form or match
their opinions. We also understand the basic mechanisms of interactions in social animals;
we know that, for example, ants produce chemicals, known as pheromones, through which
they communicate, organise their work and mark the location of food. However, there is
another very important, and in no way trivial, aspect of complex systems which has been
explored less. This has to do with the structure of the interactions among the units of a
complex system: which unit is connected to which others. For instance, if we look at the
connections between the neurons in the brain and construct a similar network whose nodes
are neurons and the links are the synapses which connect them, we find that such a network has some special mathematical properties which are fundamental for the functioning
of the brain. For instance, it is always possible to move from one node to any other in a
small number of steps, and, particularly if the two nodes belong to the same brain area,
there are many alternative paths between them. Analogously, if we take snapshots of who
is talking to whom at our hypothetical party, we immediately see that the architecture of
the obtained networks, whose nodes represent individuals and links stand for interactions,
plays a crucial role in both the propagation of information and the emergence of collective
behaviours. Some sub-structures of a network propagate information faster than others;

this means that nodes occupying strategic positions will have better access to the resources

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Introduction

of the system. In practice, what also matters in a complex system, and it matters a lot, is
the backbone of the system, or, in other words, the architecture of the network of interactions. It is precisely on these complex networks, i.e. on the networks of the various complex
systems that populate our world, that we will be focusing in this book.

Complex Networks Are All Around Us
Networks permeate all aspects of our life and constitute the backbone of our modern world.
To understand this, think for a moment about what you might do in a typical day. When
you get up early in the morning and turn on the light in your bedroom, you are connected
to the electrical power grid, a network whose nodes are either power stations or users,
while links are copper cables which transport electric current. Then you meet the people of
your family. They are part of your social network whose nodes are people and links stand
for kinship, friendship or acquaintance. When you take a shower and cook your breakfast
you are respectively using a water distribution network, whose nodes are water stations,
reservoirs, pumping stations and homes, and links are pipes, and a gas distribution network.
If you go to work by car you are moving in the street network of your city, whose nodes
are intersections and links are streets. If you take the underground then you make use of a
transportation network, whose nodes are the stations and links are route segments.
When you arrive at your office you turn on your laptop, whose internal circuits form a
complicated microscopic network of logic gates, and connect it to the Internet, a worldwide

network of computers and routers linked by physical or logical connections. Then you
check your emails, which belong to an email communication network, whose nodes are
people and links indicate email exchanges among them. When you meet a colleague, you
and your colleague form part of a collaboration network, in which an edge exists between
two persons if they have collaborated on the same project or coauthored a paper. Your
colleagues tell you that your last paper has got its first hundred citations. Have you ever
thought of the fact that your papers belong to a citation network, where the nodes represent
papers, and links are citations?
At lunchtime you read the news on the website of your preferred newspaper: in doing
this you access the World Wide Web, a huge global information network whose nodes are
webpages and edges are clickable hyperlinks between pages. You will almost surely then
check your Facebook account, a typical example of an online social network, then maybe
have a look at the daily trending topics on Twitter, an information network whose nodes
are people and links are the “following” relations.
Your working day proceeds quietly, as usual. Around 4:00pm you receive a phone call
from your friend John, and you immediately think about the phone call network, where
two individuals are connected by a link if they have exchanged a phone call. John invites
you and your family for a weekend at his cottage near the lake. Lakes are home to a
variety of fishes, insects and animals which are part of a food web network, whose links
indicate predation among different species. And while John tells you about the beauty of
his cottage, an image of a mountain lake gradually forms in your mind, and you can see a

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Introduction

white waterfall cascading down a cliff, and a stream flowing quietly through a green valley.
There is no need to say that “lake”, “waterfall”, “white”, “stream”, “cliff”, “valley” and
“green” form a network of words associations, in which a link exists between two words
if these words are often associated with each other in our minds. Before leaving the office,
you book a flight to go to Prague for a conference. Obviously, also the air transportation
system is a network, whose nodes are airports and links are airline routes.
When you drive back home you feel a bit tired and you think of the various networks
in our body, from the network of blood vessels which transports blood to our organs to the
intricate set of relationships among genes and proteins which allow the perfect functioning
of the cells of our body. Examples of these genetic networks are the transcription regulation networks in which the nodes are genes and links represent transcription regulation of
a gene by the transcription factor produced by another gene, protein interaction networks
whose nodes are protein and there is a link between two proteins if they bind together to
perform complex cellular functions, and metabolic networks where nodes are chemicals,
and links represent chemical reactions.
During dinner you hear on the news that the total export for your country has decreased
by 2.3% this year; the system of commercial relationships among countries can be seen
as a network, in which links indicate import/export activities. Then you watch a movie on
your sofa: you can construct an actor collaboration network where nodes represent movie
actors and links are formed if two actors have appeared in the same movie. Exhausted, you
go to bed and fall asleep while images of networks of all kinds still twist and dance in your
mind, which is, after all, the marvellous combination of the activity of billions of neurons
and trillions of synapses in your brain network. Yet another network.

Why Study Complex Networks?
In the late 1990s two research papers radically changed our view on complex systems,
moving the attention of the scientific community to the study of the architecture of a complex system and creating an entire new research field known today as network science. The
first paper, authored by Duncan Watts and Steven Strogatz, was published in the journal
Nature in 1998 and was about small-world networks [311]. The second one, on scale-free
networks, appeared one year later in Science and was authored by Albert-László Barabási

and Réka Albert [19]. The two papers provided clear indications, from different angles,
that:
• the networks of real-world complex systems have non-trivial structures and are very
different from lattices or random graphs, which were instead the standard networks
commonly used in all the current models of a complex system.
• some structural properties are universal, i.e. are common to networks as diverse as those
of biological, social and man-made systems.
• the structure of the network plays a major role in the dynamics of a complex system and
characterises both the emergence and the properties of its collective behaviours.

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Introduction

xvi

Table 1 A list of the real-world complex networks that will be studied in this book. For each network, we
report the chapter of the book where the corresponding data set will be introduced and analysed.
Complex networks

Nodes

Links

Chapter

Elisa’s kindergarten

Actor collaboration networks
Co-authorship networks
Citation networks
Zachary’s karate club

Children
Movie actors
Scientists
Scientific papers
Club members

Friendships
Co-acting in a film
Co-authoring a paper
Citations
Friendships

1
2
3
6
9

C. elegans neural network
Transcription regulation networks

Neurons
Genes

Synapses

Transcription regulation

4
8

World Wide Web
Internet
Urban street networks
Air transport network
Financial markets

Web pages
Routers
Street crossings
Airports
Stocks

Hyperlinks
Optical fibre cables
Streets
Flights
Time correlations

5
7
8
10
10

Both works were motivated by the empirical analysis of real-world systems. Four networks were introduced and studied in these two papers. Namely, the neural system of

a few-millimetres-long worm known as the C. elegans, a social network describing how
actors collaborate in movies, and two man-made networks: the US electrical power grid and
a sample of the World Wide Web. During the last decade, new technologies and increasing
computing power have made new data available and stimulated the exploration of several
other complex networks from the real world. A long series of papers has followed, with
the analysis of new and ever larger networks, and the introduction of novel measures and
models to characterise and reproduce the structure of these real-world systems. Table 1
shows only a small sample of the networks that have appeared in the literature, namely
those that will be explicitly studied in this book, together with the chapter where they
will be considered. Notice that the table includes different types of networks. Namely,
five networks representing three different types of social interactions (namely friendships,
collaborations and citations), two biological systems (respectively a neural and a gene network) and five man-made networks (from transportation and communication systems to a
network of correlations among financial stocks).
The ubiquitousness of networks in nature, technology and society has been the principal
motivation behind the systematic quantitative study of their structure, their formation and
their evolution. And this is also the main reason why a student of any scientific discipline
should be interested in complex networks. In fact, if we want to master the interconnected
world we live in, we need to understand the structure of the networks around us. We have
to learn the basic principles governing the architecture of networks from different fields,
and study how to model their growth.
It is also important to mention the high interdisciplinarity of network science. Today,
research on complex networks involves scientists with expertise in areas such as mathematics, physics, computer science, biology, neuroscience and social science, often working

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Introduction

xvii

800

10000

600

6000

# papers

# citations

8000

WS
BA

4000

200

2000

t

Fig. 1

0

1995

400

2000

2005
year

2010

2015

2000

1995

2005
year

2010

2015

Left panel: number of citations received over the years by the 1998 Watts and Strogatz (WS) article on small-world
networks and by the 1999 Barabási and Albert (BA) article on scale-free networks. Right panel: number of papers on
complex networks that appeared each year in the public preprint archive arXiv.org.
side by side. Because of its interdisciplinary nature, the generality of the results obtained,
and the wide variety of possible applications, network science is considered today a
necessary ingredient in the background of any modern scientist.

Finally, it is not difficult to understand that complex networks have become one of the
hottest research fields in science. This is confirmed by the attention and the huge number
of citations received by Watts and Strogatz, and by Barabási and Albert, in the papers
mentioned above. The temporal profiles reported in the left panel of Figure 1 show the
exponential increase in the number of citations of these two papers since their publication.
The two papers have today about 10,000 citations each and, as already mentioned, have
opened a new research field stimulating interest for complex networks in the scientific
community and triggering an avalanche of scientific publications on related topics. The
right panel of Figure 1 reports the number of papers published each year after 1998 on the
well-known public preprint archive arXiv.org with the term “complex networks” in their
title or abstract. Notice that this number has gone up by a factor of 10 in the last ten years,
with almost a thousand papers on the topic published in the archive in the year 2013. The
explosion of interest in complex networks is not limited to the scientific community, but
has become a cultural phenomenon with the publications of various popular science books
on the subject.

Overview of the Book
This book is mainly intended as a textbook for an introductory course on complex networks
for students in physics, mathematics, engineering and computer science, and for the more
mathematically oriented students in biology and social sciences. The main purpose of the
book is to expose the readers to the fundamental ideas of network science, and to provide
them with the basic tools necessary to start exploring the world of complex networks. We
also hope that the book will be able to transmit to the reader our passion for this stimulating
new interdisciplinary subject.

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xviii

Introduction

The standard tools to study complex networks are a mixture of mathematical and computational methods. They require some basic knowledge of graph theory, probability,
differential equations, data structures and algorithms, which will be introduced in this
book from scratch and in a friendly way. Also, network theory has found many interesting applications in several different fields, including social sciences, biology, neuroscience
and technology. In the book we have therefore included a large variety of examples to
emphasise the power of network science. This book is essentially on the structure of complex networks, since we have decided that the detailed treatment of the different types of
dynamical processes that can take place over a complex network should be left to another
book, which will follow this one.
The book is organised into ten chapters. The first six chapters (Chapters 1–6) form the
core of the book. They introduce the main concepts of network science and the basic
measures and models used to characterise and reproduce the structure of various complex networks. The remaining four chapters (Chapters 7–10) cover more advanced topics
that could be skipped by a lecturer who wants to teach a short course based on the book.
In Chapter 1 we introduce some basic definitions from graph theory, setting up the language we will need for the remainder of the book. The aim of the chapter is to show
that complex network theory is deeply grounded in a much older mathematical discipline,
namely graph theory.
In Chapter 2 we focus on the concept of centrality, along with some of the related measures originally introduced in the context of social network analysis, which are today used
extensively in the identification of the key components of any complex system, not only
of social networks. We will see some of the measures at work, using them to quantify the
centrality of movie actors in the actor collaboration network.
Chapter 3 is where we first discuss network models. In this chapter we introduce the
classical random graph models proposed by Erd˝os and Rényi (ER) in the late 1950s, in
which the edges are randomly distributed among the nodes with a uniform probability.
This allows us to analytically derive some important properties such as, for instance, the
number and order of graph components in a random graph, and to use ER models as term
of comparison to investigate scientific collaboration networks. We will also show that the
average distance between two nodes in ER random graphs increases only logarithmically
with the number of nodes.

In Chapter 4 we see that in real-world systems, such as the neural network of the C. elegans or the movie actor collaboration network, the neighbours of a randomly chosen node
are directly linked to each other much more frequently than would occur in a purely random network, giving rise to the presence of many triangles. In order to quantify this, we
introduce the so-called clustering coefficient. We then discuss the Watts and Strogatz (WS)
small-world model to construct networks with both a small average distance between nodes
and a high clustering coefficient.
In Chapter 5 the focus is on how the degree k is distributed among the nodes of a network.
We start by considering the graph of the World Wide Web and by showing that it is a
scale-free network, i.e. it has a power–law degree distribution pk ∼ k−γ with an exponent
γ ∈ [2, 3]. This is a property shared by many other networks, while neither ER random
graphs nor the WS model can reproduce such a feature. Hence, we introduce the so-called

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Introduction

configuration model which generalises ER random graph models to incorporate arbitrary
degree distributions.
In Chapter 6 we show that real networks are not static, but grow over time with the
addition of new nodes and links. We illustrate this by studying the basic mechanisms of
growth in citation networks. We then consider whether it is possible to produce scale-free
degree distributions by modelling the dynamical evolution of the network. For this purpose
we introduce the Barabási–Albert model, in which newly arriving nodes select and link
existing nodes with a probability linearly proportional to their degree. We also consider
some extensions and modifications of this model.
In the last four chapters we cover more advanced topics on the structure of complex

networks.
Chapter 7 is about networks with degree–degree correlations, i.e. networks such that the
probability that an edge departing from a node of degree k arrives at a node of degree k
is a function both of k and of k. Degree–degree correlations are indeed present in realworld networks, such as the Internet, and can be either positive (assortative) or negative
(disassortative). In the first case, networks with small degree preferentially link to other
low-degree nodes, while in the second case they link preferentially to high-degree ones. In
this chapter we will learn how to take degree–degree correlations into account, and how to
model correlated networks.
In Chapter 8 we deal with the cycles and other small subgraphs known as motifs which
occur in most networks more frequently than they would in random graphs. We consider
two applications: firstly we count the number of short cycles in urban street networks of
different cities from all over the world; secondly we will perform a motif analysis of the
transcription network of the bacterium E. coli.
Chapter 9 is about network mesoscale structures known as community structures. Communities are groups of nodes that are more tightly connected to each other than to other
nodes. In this chapter we will discuss various methods to find meaningful divisions of
the nodes of a network into communities. As a benchmark we will use a real network, the
Zachary’s karate club, where communities are known a priori, and also models to construct
networks with a tunable presence of communities.
In Chapter 10 we deal with weighted networks, where each link carries a numerical value
quantifying the intensity of the connection. We will introduce the basic measures used to
characterise and classify weighted networks, and we will discuss some of the models of
weighted networks that reproduce empirically observed topology–weight correlations. We
will study in detail two weighted networks, namely the US air transport network and a
network of financial stocks.
Finally, the book’s Appendix contains a detailed description of all the main graph algorithms discussed in the various chapters of the book, from those to find shortest paths,
components or community structures in a graph, to those to generate random graphs or
scale-free networks. All the algorithms are presented in a C-like pseudocode format which
allows us to understand their basic structure without the unnecessary complication of a
programming language.
The organisation of this textbook is another reason why it is different from all the other

existing books on networks. We have in fact avoided the widely adopted separation of

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xx

Introduction

the material in theory and applications, or the division of the book into separate chapters respectively dealing with empirical studies of real-world networks, network measures,
models, processes and computer algorithms. Each chapter in our book discusses, at the
same time, real-world networks, measures, models and algorithms while, as said before,
we have left the study of processes on networks to an entire book, which will follow this
one. Each chapter of this book presents a new idea or network property: it introduces a
network data set, proposes a set of mathematical quantities to investigate such a network,
describes a series of network models to reproduce the observed properties, and also points
to the related algorithms. In this way, the presentation follows the same path of the current
research in the field, and we hope that it will result in a more logical and more entertaining
text. Although the main focus of this book is on the mathematical modelling of complex
networks, we also wanted the reader to have direct access to both the most famous data
sets of real-world networks and to the numerical algorithms to compute network properties and to construct networks. For this reason, the data sets of all the real-world networks
listed in Table 1 are introduced and illustrated in special DATA SET Boxes, usually one
for each chapter of the book, and can be downloaded from the book’s webpage at www.
complex-networks.net. On the same webpage the reader can also find an implementation in the C language of the graph algorithms illustrated in the Appendix (in C-like
pseudocode format). We are sure that the student will enjoy experimenting directly on realworld networks, and will benefit from the possibility of reproducing all of the numerical
results presented throughout the book.
The style of the book is informal and the ideas are illustrated with examples and applications drawn from the recent research literature and from different disciplines. Of course,
the problem with such examples is that no-one can simultaneously be an expert in social

sciences, biology and computer science, so in each of these cases we will set up the relative
background from scratch. We hope that it will be instructive, and also fun, to see the connections between different fields. Finally, all the mathematics is thoroughly explained, and
we have decided never to hide the details, difficulties and sometimes also the incoherences
of a science still in its infancy.

Acknowledgements
Writing this book has been a long process which started almost ten years ago. The book has
grown from the notes of various university courses, first taught at the Physics Department
of the University of Catania and at the Scuola Superiore di Catania in Italy, and more
recently to the students of the Masters in “Network Science” at Queen Mary University of
London.
The book would not have been the same without the interactions with the students we
have met at the different stages of the writing process, and their scientific curiosity. Special
thanks go to Alessio Cardillo, Roberta Sinatra, Salvatore Scellato and the other students
and alumni of Scuola Superiore, Salvatore Assenza, Leonardo Bellocchi, Filippo Caruso,
Paolo Crucitti, Manlio De Domenico, Beniamino Guerra, Ivano Lodato, Sandro Meloni,

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Introduction

Andrea Santoro and Federico Spada, and to the students of the Masters in “Network
Science”.
We acknowledge the great support of the members of the Laboratory of Complex
Systems at Scuola Superiore di Catania, Giuseppe Angilella, Vincenza Barresi, Arturo

Buscarino, Daniele Condorelli, Luigi Fortuna, Mattia Frasca, Jesús Gómez-Gardeñes and
Giovanni Piccitto; of our colleagues in the Complex Systems and Networks research
group at the School of Mathematical Sciences of Queen Mary University of London,
David Arrowsmith, Oscar Bandtlow, Christian Beck, Ginestra Bianconi, Leon Danon,
Lucas Lacasa, Rosemary Harris, Wolfram Just; and of the PhD students Federico Battiston, Moreno Bonaventura, Massimo Cavallaro, Valerio Ciotti, Iacopo Iacovacci, Iacopo
Iacopini, Daniele Petrone and Oliver Williams.
We are greatly indebted to our colleagues Elsa Arcaute, Alex Arenas, Domenico
Asprone, Tomaso Aste, Fabio Babiloni, Franco Bagnoli, Andrea Baronchelli, Marc
Barthélemy, Mike Batty, Armando Bazzani, Stefano Boccaletti, Marián Boguñá, Ed
Bullmore, Guido Caldarelli, Domenico Cantone, Gastone Castellani, Mario Chavez, Vittoria Colizza, Regino Criado, Fabrizio De Vico Fallani, Marina Diakonova, Albert Dí
az-Guilera, Tiziana Di Matteo, Ernesto Estrada, Tim Evans, Alfredo Ferro, Alessandro Fiasconaro, Alessandro Flammini, Santo Fortunato, Andrea Giansanti, Georg von
Graevenitz, Paolo Grigolini, Peter Grindrod, Des Higham, Giulia Iori, Henrik Jensen,
Renaud Lambiotte, Pietro Lió, Vittorio Loreto, Paolo de Los Rios, Fabrizio Lillo, Carmelo
Maccarrone, Athen Ma, Sabato Manfredi, Massimo Marchiori, Cecilia Mascolo, Rosario
Mantegna, Andrea Migliano, Raúl Mondragón, Yamir Moreno, Mirco Musolesi, Giuseppe
Nicosia, Pietro Panzarasa, Nicola Perra, Alessandro Pluchino, Giuseppe Politi, Sergio
Porta, Mason Porter, Giovanni Petri, Gaetano Quattrocchi, Daniele Quercia, Filippo Radicchi, Andrea Rapisarda, Daniel Remondini, Alberto Robledo, Miguel Romance, Vittorio
Rosato, Martin Rosvall, Maxi San Miguel, Corrado Santoro, M. Ángeles Serrano, Simone
Severini, Emanuele Strano, Michael Szell, Bosiljka Tadi´c, Constantino Tsallis, Stefan
Thurner, Hugo Touchette, Petra Vértes, Lucio Vinicius for the many stimulating discussions and for their useful comments. We thank in particular Olle Persson, Luciano Da
Fontoura Costa, Vittoria Colizza, and Rosario Mantegna for having provided us with their
network data sets.
We acknowledge the European Commission project LASAGNE (multi-LAyer SpAtiotemporal Generalized NEtworks), Grant 318132 (STREP), the EPSRC project GALE,
Grant EP/K020633/1, and INFN FB11/TO61, which have supported and made possible
our work at the various stages of this project.
Finally, we thank our families for their never-ending support and encouragement.

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Life is all mind, heart and relations
Salvatore Latora
Philosopher

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1

Graphs and Graph Theory

Graphs are the mathematical objects used to represent networks, and graph theory is the
branch of mathematics that deals with the study of graphs. Graph theory has a long history. The notion of the graph was introduced for the first time in 1763 by Euler, to settle
a famous unsolved problem of his time: the so-called Königsberg bridge problem. It is no
coincidence that the first paper on graph theory arose from the need to solve a problem from
the real world. Also subsequent work in graph theory by Kirchhoff and Cayley had its root
in the physical world. For instance, Kirchhoff’s investigations into electric circuits led to
his development of a set of basic concepts and theorems concerning trees in graphs. Nowadays, graph theory is a well-established discipline which is commonly used in areas as
diverse as computer science, sociology and biology. To give some examples, graph theory
helps us to schedule airplane routing and has solved problems such as finding the maximum flow per unit time from a source to a sink in a network of pipes, or colouring the

regions of a map using the minimum number of different colours so that no neighbouring
regions are coloured the same way. In this chapter we introduce the basic definitions, setting up the language we will need in the rest of the book. We also present the first data set
of a real network in this book, namely Elisa’s kindergarten network. The two final sections
are devoted to, respectively, the proof of the Euler theorem and the description of a graph
as an array of numbers.

1.1 What Is a Graph?
The natural framework for the exact mathematical treatment of a complex network is a
branch of discrete mathematics known as graph theory [48, 47, 313, 150, 272, 144]. Discrete mathematics, also called finite mathematics, is the study of mathematical structures
that are fundamentally discrete, i.e. made up of distinct parts, not supporting or requiring
the notion of continuity. Most of the objects studied in discrete mathematics are countable sets, such as integers and finite graphs. Discrete mathematics has become popular in
recent decades because of its applications to computer science. In fact, concepts and notations from discrete mathematics are often useful to study or describe objects or problems
in computer algorithms and programming languages. The concept of the graph is better
introduced by the two following examples.
1

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