Complex Networks Principles, Methods and Applications pdf pdf
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 pre- sentation 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 net- work 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.
Complex Networks Principles, Methods and Applications
V I T O L AT O R A Queen Mary University of London
V I N C E N Z O N I C O S I A Queen Mary University of London
G I O VA N N I R U S S O University of Catania, Italy University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India
79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of 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 https://lccn.loc.gov/2017026029
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.
To Giusi, Francesca and Alessandra
Contents
Preface page xi Introduction69
2.4 Measures Based on Shortest Paths
47
2.5 Movie Actors
56
2.6 Group Centrality
62
2.7 What We Have Learned and Further Readings
64 Problems
65
3 Random Graphs
3.1 Erd˝os and Rényi (ER) Models
2.3 Degree and Eigenvector Centrality
69
3.2 Degree Distribution
76
3.3 Trees, Cycles and Complete Subgraphs
79
3.4 Giant Connected Component
84
3.5 Scientific Collaboration Networks
90
3.6 Characteristic Path Length
94
39
34
xii The Backbone of a Complex System xii Complex Networks Are All Around Us xiv Why Study Complex Networks? xv
1.4 Trees
Overview of the Book xvii
Acknowledgements xx
1 Graphs and Graph Theory
1
1.1 What Is a Graph?
1
1.2 Directed, Weighted and Bipartite Graphs
9
1.3 Basic Definitions
13
17
2.2 Connected Graphs and Irreducible Matrices
1.5 Graph Theory and the Bridges of Königsberg
19
1.6 How to Represent a Graph
23
1.7 What We Have Learned and Further Readings
28 Problems
28
2 Centrality Measures
31
2.1 The Importance of Being Central
31
vii
3.7 What We Have Learned and Further Readings 103 Problems
7.1 The Internet and Other Correlated Networks 257
6.3 The Importance of Being Preferential and Linear 224
6.4 Variations to the Theme 230
6.5 Can Latecomers Make It? The Fitness Model 241
6.6 Optimisation Models 248
6.7 What We Have Learned and Further Readings 252 Problems
253
7 Degree Correlations
257
7.2 Dealing with Correlated Networks 262
6.1 Citation Networks and the Linear Preferential Attachment 206
7.3 Assortative and Disassortative Networks 268
7.4 Newman’s Correlation Coefficient 275
7.5 Models of Networks with Degree–Degree Correlations 285
7.6 What We Have Learned and Further Readings 290 Problems
291
8 Cycles and Motifs
294
8.1 Counting Cycles 294
8.2 Cycles in Scale-Free Networks 303
6.2 The Barabási–Albert (BA) Model 215
206
104
148
4 Small-World Networks
107
4.1 Six Degrees of Separation 107
4.2 The Brain of a Worm 112
4.3 Clustering Coefficient 116
4.4 The Watts–Strogatz (WS) Model 127
4.5 Variations to the Theme 135
4.6 Navigating Small-World Networks 144
4.7 What We Have Learned and Further Readings 148 Problems
5 Generalised Random Graphs
6 Models of Growing Graphs
151
5.1 The World Wide Web 151
5.2 Power-Law Degree Distributions 161
5.3 The Configuration Model 171
5.4 Random Graphs with Arbitrary Degree Distribution 178
5.5 Scale-Free Random Graphs 184
5.6 Probability Generating Functions 188
5.7 What We Have Learned and Further Readings 202 Problems
204
8.3 Spatial Networks of Urban Streets 307
8.4 Transcription Regulation Networks 316
374
511 A.17 Girvan–Newman Algorithm 515 A.18 Greedy Modularity Optimisation 519 A.19 Label Propagation
492 A.13 Growing Unweighted Graphs 499 A.14 Random Graphs with Degree–Degree Correlations 506 A.15 Johnson’s Algorithm to Enumerate Cycles 508 A.16 Motifs Analysis
474 A.10 Erd˝os and Rényi Random Graph Models 485 A.11 The Watts–Strogatz Small-World Model 489 A.12 The Configuration Model
467 A.9 Random Sampling
410 A.1 Problems, Algorithms and Time Complexity 410 A.2 A Simple Introduction to Computational Complexity 420 A.3 Elementary Data Structures 425 A.4 Basic Operations with Sparse Matrices 440 A.5 Eigenvalue and Eigenvector Computation 444 A.6 Computation of Shortest Paths 452 A.7 Computation of Node Betweenness 462 A.8 Component Analysis
Appendices
408
10.6 What We Have Learned and Further Readings 407 Problems
10.5 Networks of Stocks in a Financial Market 401
10.4 Growing Weighted Networks 393
10.3 Motifs and Communities 387
10.2 Basic Measures 381
10.1 Tuning the Interactions 374
10 Weighted Networks
8.5 Motif Analysis 324
371
9.8 What We Have Learned and Further Readings 369 Problems
9.7 A Local Method 365
9.6 The Modularity 357
9.5 Computer Generated Benchmarks 354
9.4 The Girvan–Newman Method 349
9.3 Hierarchical Clustering 342
9.2 The Spectral Bisection Method 336
9.1 Zachary’s Karate Club 332
332
9 Community Structure
330
8.6 What We Have Learned and Further Readings 329 Problems
524 A.20 Kruskal’s Algorithm for Minimum Spanning Tree 528 A.21 Models for Weighted Networks 531 List of Programs
533
References
535
Author Index
550
Index
552
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 beau- tiful 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: net-
work 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 iden- tifying 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 soft- ware 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 his- torical 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
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,
xii
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
2
and beautiful one: the human brain. With around 10 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 pre- dict 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 think- ing, 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 interac- tions. 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 net- work 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
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 interac- tions. 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
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 regula-
tion 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 com- plex 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.
Introduction
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 Children Friendships
1 Actor collaboration networks Movie actors Co-acting in a film
2 Co-authorship networks Scientists Co-authoring a paper
3 Citation networks Scientific papers Citations
6 Zachary’s karate club Club members Friendships
9 C. elegans neural network Neurons Synapses
4 Transcription regulation networks Genes Transcription regulation
8 World Wide Web Web pages Hyperlinks
5 Internet Routers Optical fibre cables
7 Urban street networks Street crossings Streets
8 Air transport network Airports Flights
10 Financial markets Stocks Time correlations
10 Both works were motivated by the empirical analysis of real-world systems. Four net-
works 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 net- work) 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 mathe- matics, physics, computer science, biology, neuroscience and social science, often working
Introduction
10000 800 BA WS 8000 600 6000 400 4000
# papers # citations 200 2000
1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 year year
Left panel: number of citations received over the years by the 1998 Watts and Strogatz (WS) article on small-world
Fig. 1 t
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.
Introduction The standard tools to study complex networks are a mixture of mathematical and com- putational 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 interest- ing 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 com- plex 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 com- plex 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 lan- guage 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 mea- sures 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. ele- gans 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 ran- dom 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 ∼ −γ
scale-free network , i.e. it has a power–law degree distribution p k with an exponent k
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
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 real- world 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
E. coli.
transcription network of the bacterium
Chapter 9 is about network mesoscale structures known as community structures. Com- munities 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 algo- rithms 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
C
scale-free networks. All the algorithms are presented in a -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
Introduction the material in theory and applications, or the division of the book into separate chap- ters 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 proper-
ties 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
www.
for each chapter of the book, and can be downloaded from the book’s webpage at
complex-networks.net
. On the same webpage the reader can also find an implemen-
C C
tation in the language of the graph algorithms illustrated in the Appendix (in -like pseudocode format). We are sure that the student will enjoy experimenting directly on real- world 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 appli- cations 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 con- nections 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,
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 Bat- tiston, 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, Vit- toria Colizza, Regino Criado, Fabrizio De Vico Fallani, Marina Diakonova, Albert Dí az-Guilera, Tiziana Di Matteo, Ernesto Estrada, Tim Evans, Alfredo Ferro, Alessan- dro 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 Radic- chi, 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 discus- sions 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 SpA- tiotemporal 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.
Life is all mind, heart and relations Salvatore Latora Philosopher
Graphs and Graph Theory