Elsevier

Physics Reports

Volume 424, Issues 4–5, February 2006, Pages 175-308
Physics Reports

Complex networks: Structure and dynamics

https://doi.org/10.1016/j.physrep.2005.10.009Get rights and content

Abstract

Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.

Introduction

Networks are all around us, and we are ourselves, as individuals, the units of a network of social relationships of different kinds and, as biological systems, the delicate result of a network of biochemical reactions. Networks can be tangible objects in the Euclidean space, such as electric power grids, the Internet, highways or subway systems, and neural networks. Or they can be entities defined in an abstract space, such as networks of acquaintances or collaborations between individuals.

Historically, the study of networks has been mainly the domain of a branch of discrete mathematics known as graph theory. Since its birth in 1736, when the Swiss mathematician Leonhard Euler published the solution to the Königsberg bridge problem (consisting in finding a round trip that traversed each of the bridges of the prussian city of Königsberg exactly once), graph theory has witnessed many exciting developments and has provided answers to a series of practical questions such as: what is the maximum flow per unit time from source to sink in a network of pipes, how to color the regions of a map using the minimum number of colors so that neighboring regions receive different colors, or how to fill n jobs by n people with maximum total utility. In addition to the developments in mathematical graph theory, the study of networks has seen important achievements in some specialized contexts, as for instance in the social sciences. Social networks analysis started to develop in the early 1920s and focuses on relationships among social entities, as communication between members of a group, trades among nations, or economic transactions between corporations.

The last decade has witnessed the birth of a new movement of interest and research in the study of complex networks, i.e. networks whose structure is irregular, complex and dynamically evolving in time, with the main focus moving from the analysis of small networks to that of systems with thousands or millions of nodes, and with a renewed attention to the properties of networks of dynamical units. This flurry of activity, triggered by two seminal papers, that by Watts and Strogatz on small-world networks, appeared in Nature in 1998, and that by Barabási and Albert on scale-free networks appeared one year later in Science, has seen the physics’ community among the principal actors, and has been certainly induced by the increased computing powers and by the possibility to study the properties of a plenty of large databases of real networks. These include transportation networks, phone call networks, the Internet and the World Wide Web, the actors’ collaboration network in movie databases, scientific coauthorship and citation networks from the Science Citation Index, but also systems of interest in biology and medicine, as neural networks or genetic, metabolic and protein networks.

The massive and comparative analysis of networks from different fields has produced a series of unexpected and dramatic results. The first issue that has been faced is certainly structural. The research on complex networks begun with the effort of defining new concepts and measures to characterize the topology of real networks. The main result has been the identification of a series of unifying principles and statistical properties common to most of the real networks considered. A relevant property regards the degree of a node, that is the number of its direct connections to other nodes. In real networks, the degree distribution P(k), defined as the probability that a node chosen uniformly at random has degree k or, equivalently, as the fraction of nodes in the graph having degree k, significantly deviates from the Poisson distribution expected for a random graph and, in many cases, exhibits a power law (scale-free) tail with an exponent γ taking a value between 2 and 3. Moreover, real networks are characterized by correlations in the node degrees, by having relatively short paths between any two nodes (small-world property), and by the presence of a large number of short cycles or specific motifs.

These empirical findings have initiated a revival of network modelling, since the models proposed in mathematical graph theory turned out to be very far from the real needs. Scientists had to do with the development of new models to mimic the growth of a network and to reproduce the structural properties observed in real topologies. The structure of a real network is the result of the continuous evolution of the forces that formed it, and certainly affects the function of the system. So that this stage of the research was motivated by the expectancy that understanding and modelling the structure of a complex network would lead to a better knowledge of its evolutionary mechanisms, and to a better cottoning on its dynamical and functional behavior.

And, indeed, it was shown that the coupling architecture has important consequences on the network functional robustness and response to external perturbations, as random failures, or targeted attacks. At the same time, it outcropped for the first time the possibility of studying the dynamical behavior of large assemblies of dynamical systems interacting via complex topologies, as the ones observed empirically. This led to a series of evidences pointing to the crucial role played by the network topology in determining the emergence of collective dynamical behavior, such as synchronization, or in governing the main features of relevant processes that take place in complex networks, such as the spreading of epidemics, information and rumors.

A number of review articles [1], [2], [3], [4] and books [5], [6], [7], [8] on complex networks, which the reader may find useful to consult, have already appeared in the literature. Watts’ pioneering book on the subject deals with the structure and the dynamics of small-world networks [5], while Strogatz’ review article in the Nature's special issue on complex systems contains a discussion on networks of dynamical units [1]. Albert and Barabási [2], and Dorogovtsev and Mendes [3], [7] have mainly focused their reviews on models of growing graphs, from the point of view of statistical mechanics. The review by Newman is a critical account on the field [4], containing an accurate list of references, an exhaustive overview on structural properties, measures and models, and also a final chapter devoted to processes taking place on networks. Four other references are worthwhile to mention at the beginning of this Report. They are the collection of contributed papers edited by Bornholdt and Schuster [6], that edited by Pastor-Satorras et al. [9], that edited by Ben-Naim et al. [10], and the book by Pastor-Satorras and Vespignani on the analysis and modelling of the Internet [8]. There is also a series of popular books on complex networks available on the market for the lay audience [11], [12], [13]. See for instance Buchanan's Nexus, for having the point of view of a science journalist on the field [11]. Furthermore, a variety of related books dealing with networks in specific fields of research has been published. In the context of graph theory, the books by Bollobás [14], [15], West [16] and Harary [17] deserve to be quoted. The textbooks by Wasserman and Faust [18] and by Scott [19] are widely known among people working in social networks analysis. Refs. [20], [21], [22] are, instead, useful sources for the description of the standard graph algorithms.

Is this subject deserving another report? At least three reasons have motivated our work.

The first is that new research lines have emerged, covering novel topics and problems in network structure. An example is the fresh and increasingly challenging care to study weighted networks, i.e. networks in which a real number is associated to each link. This is motivated by the fact that in most of the real cases a complex topology is often associated with a large heterogeneity in the capacity and intensity of the connections. Paradigmatic cases are the existence of strong and weak ties between individuals in social systems, different capabilities of transmitting electric signals in neural networks, unequal traffic on the Internet. Ignoring such a diversity in the interactions would mean leaving away a lot of information on complex networks which is, instead, available and very useful for their characterization. A further novel topic concerns spatial networks. While most of the early works on complex networks have focused on the characterization of the topological properties, the spatial aspect has received less attention, when not neglected at all. However, it is not surprising that the topology could be constrained by the geographical embedding. For instance, the long range connections in a spatial network are constrained by the Euclidean distance, this having important consequences on the network's statistical properties. Also the degree is constrained because the number of edges that can be connected to a single node is limited by the physical space to connect them. This is particularly evident in planar networks (e.g. networks forming vertices whenever two edges cross), as urban streets, where only a small number of streets can cross in an intersection. And even in non-planar spatial networks, such as airline networks the number of connections is limited by the space available at the airport. These facts contribute to make spatial networks different from other complex networks. Along with a full account on structural properties, the present review includes all this novel material, and its many applications to relevant concrete situations.

The second reason is that most of the interest in the subject has lately switched to investigate the dynamical behavior of networks, with a special emphasis on how the network structure affects the properties of a networked dynamical system. An example is the concerned attention to study the emergence of collective synchronized dynamics in complex networks, from the point of view of relating the propensity for synchronization of a network to the interplay between topology and local properties of the coupled dynamical systems. This phenomenon, indeed, represents a crucial feature in many relevant circumstances. For instance, evidence exists that some brain diseases are the result of an abnormal and, some times, abrupt synchronization of a large number of neural populations, so that the investigation on the network mechanisms involved in the generation, maintenance and propagation of the epileptic disorders is an issue nowadays at the forefront of neuroscience. Synchronization phenomena are very relevant also in sociology to gather a better understanding of the mechanisms underlying the formation of social collective behaviors, as the sudden emergence of new habits, fashions or leading opinions. A large portion of the second part of this Report is devoted to summarize the main achievements that have been obtained so far in dealing with collective behaviors in complex networks, reviewing the major ideas and concepts that have been developed, and assessing the rigorous results that are nowadays available.

Finally, we present a survey of a series of topics that are currently attracting much attention in the scientific community. These include the problem of building manageable algorithms to find community structures, the issue of searching within a complex network, and the modelling of adaptive networks.

Community structures are an important property of complex networks. For example, tightly connected groups of nodes in a social network represent individuals belonging to social communities, tightly connected groups of nodes in the World Wide Web often correspond to pages on common topics, while communities in cellular and genetic networks are somehow related to functional modules. Consequently, finding the communities within a network is a powerful tool for understanding the functioning of the network, as well as for identifying a hierarchy of connections within a complex architecture.

Another relevant problem is how to reach a node of the network from another one, by navigating the network often in the absence of information on the global structure, or how to optimize a searching procedure based only on some local information on the network topology.

Adaptive and dynamical wirings are a peculiarity of those networks that are themselves dynamical entities. This means that the topology is not fixed, or grown, once forever. Instead it is allowed to evolve and adapt in time, driven by some external action, or by the action of the internal elements, or following specific predetermined evolving rules. This step forward has been motivated by the need of suitably modelling some specific cases, such as genetic regulatory networks, ecosystems, financial markets, as well as to properly describe a series of technologically relevant problems emerging, e.g., in mobile and wireless connected units. In some cases, the research work has just begun, and, even though the results are not so firmly established, we believe that the state of the art calls for future relevant achievements.

The Report is organized as follows.

Chapter 2 is about network structure. We describe some of the common properties observed in the topology of real networks, and how they are measured. We then briefly review the main models that have been proposed over the years, focusing on random graphs, small-world models and scale-free networks. Finally, we give a special emphasis to the study and modelling of weighted networks, as well as networks with a spatial structure.

In Chapter 3 we discuss the network robustness against external perturbations consisting in the malfunctioning, or the deliberate damage, of some of its components. We review both static and dynamical approaches. Specifically, we describe percolation processes on uncorrelated and correlated networks, cascading failures, and congestion in transportation and communication networks.

In Chapter 4 we consider cellular automata on complex topologies and we analyze a series of models for the spreading of epidemics and rumors.

Chapter 5 is concerned with the emergence of collective synchronized dynamics in complex networks. In this context, we review the most significant advancement represented by the Master Stability Function approach, giving conditions in the wiring topology that maximize the propensity for synchronization of a network. We furthermore consider networks whose dynamical units evolve nonlinearly, and we review the main results obtained with networks of chaotic maps, networks of chaotic systems, and with networks of periodic oscillators.

Chapter 6 summarizes some applications to real networks such as the Internet and the World Wide Web, social networks, networks describing the interaction between cell components, and neural networks. The chapter contains issues concerning both the structure and the dynamics of such networks.

Finally, in Chapter 7 we consider three topics that have recently attracted a large interest in the scientific community. We first discuss algorithms for partitioning large networks into community structures. We then review the recent advancements in finding reliable and fast ways for navigation and searching in a complex network. The chapter ends with a discussion of adaptive and dynamical wirings.

Section snippets

The structure of complex networks

The material in this chapter is intended to serve as a brief account of the recent developments in the characterization and modelling of the structural properties of a network. We shall first introduce definitions and notations, and discuss the basic quantities used to describe the topology of a network. Then, we shall move to the analysis of the properties observed in real networks, and provide the reader with a brief review of the models motivated by the empirical observations. The chapter

Static and dynamic robustness

Robustness refers to the ability of a network to avoid malfunctioning when a fraction of its constituents is damaged. This is a topic of obvious practical reasons, as it affects directly the efficiency of any process running on top of the network, and it was one of the first issues to be explored in the literature on complex networks. In this chapter, we review the main results concerning the resilience of networks to both random failures and intentional attacks. The problem can be encountered

Spreading processes

Cellular automata on complex topologies are systems in which each node of the network represents an agent that can be in only one of a finite number of states. Time is discrete, and at each time step, the next state of each agent is computed as a function of its state and of the states of its neighbors on the network. The formalism for cellular automata was introduced by von Neumann in the 1940s as a framework to study the process of reproduction [341]. Currently, cellular automata are

Synchronization and collective dynamics

The emergence of collective and synchronized dynamics in large networks of coupled units has been investigated since the beginning of the nineties in different contexts and in a variety of fields, ranging from biology and ecology [371], [383], [384], to semiconductor lasers [385], [386], [387], [388], to electronic circuits [389], [390].

This chapter aims at reviewing some different techniques that have been proposed for assessing the propensity for synchronization of a given networked system.

Applications

In this chapter we discuss a series of applications to real networks. These include both issues concerning the structure of the networks and their dynamics. We shall review some structural aspects of social networks and consider two kind of dynamics involving social networks: opinion formation and game models. We shall then discuss the statistical properties of the Internet and of the World Wide Web. Finally we shall focus of complex networks of interest to biology and medicine, such as

Other topics

In this final chapter we consider three topics that have recently attracted a large interest in the scientific community. We first discuss the problem of partitioning large graphs into their community structures, i.e. tightly connected subgraphs. This is basically a structural problem. We then review the recent advancements in finding reliable and fast ways for navigation and searching in a complex network. The chapter ends with a discussion of adaptive and dynamical wirings.

Acknowledgements

At the end of a such work, we would like to acknowledge all scientists with whom we had (and have) interactions on this topic. Their sharing of very stimulating and fruitful discussions (or unpublished results on the subjects treated in this report) is largely responsible for our effort to provide a new account of this rapidly growing field of research.

In this spirit, the authors would like to thank R. Albert, A. Amann, V. Anishchenko, A. Arenas, A. Barabási, M. Baranger, E. Barreto, B.

Note added in proof

After the completion of the Report we learned about recent works that are important contributions to the subjects reviewed here. In the following, these additional References are listed, together with the number of the Section they refer to.

Section1.1:
U. Brandes, T. Erlebach (Eds.), Network Analysis: Methodological Foundations, Springer, Berlin, 2005.
Section 2.1:
K.I. Goh, G. Salvi, B. Kahng, D. Kim, preprint cond-mat/0508332.
Section 2.3.1:
I. Derényi, G. Palla, T. Vicsek, Phys. Rev. Lett. 94

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