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Statistical Mechanics of Complex Networks REVIEWS OF MODERN PHYSICS, VOLUME 74, JANUARY 2002 Statistical mechanics of complex networks Re´ka Albert* and Albert-La´szlo´ Baraba´si Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556 (Published 30 January 2002) Complex networks describe a wide range of systems in nature and society. Frequently cited examples include the cell, a network of chemicals linked by chemical reactions, and the Internet, a network of routers and computers connected by physical links. While traditionally these systems have been modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks are governed by robust organizing principles. This article reviews the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, the authors discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, the emerging theory of evolving networks, and the interplay between topology and the network’s robustness against failures and attacks. CONTENTS C. Random graphs with power-law degree distribution 66 D. Bipartite graphs and the clustering coefficient 66 I. Introduction 48 VI. Small-World Networks 67 II. The Topology of Real Networks: Empirical Results 49 A. The Watts-Strogatz model 67 A. World Wide Web 49 B. Properties of small-world networks 68 B. Internet 50 1. Average path length 68 C. Movie actor collaboration network 52 2. Clustering coefficient 69 D. Science collaboration graph 52 3. Degree distribution 70 E. The web of human sexual contacts 52 4. Spectral properties 70 F. Cellular networks 52 VII. Scale-Free Networks 71 G. Ecological networks 53 A. The Baraba´si-Albert model 71 H. Phone call network 53 B. Theoretical approaches 71 I. Citation networks 53 C. Limiting cases of the Baraba´si-Albert model 73 J. Networks in linguistics 53 D. Properties of the Baraba´si-Albert model 74 K. Power and neural networks 54 1. Average path length 74 L. Protein folding 54 2. Node degree correlations 75 III. Random-Graph Theory 54 3. Clustering coefficient 75 A. The Erdo˝ s-Re´nyi model 54 4. Spectral properties 75 B. Subgraphs 55 VIII. The Theory of Evolving Networks 76 A. Preferential attachment ⌸(k)76 C. Graph evolution 56 1. Measuring ⌸(k) for real networks 76 D. Degree distribution 57 2. Nonlinear preferential attachment 77 E. Connectedness and diameter 58 3. Initial attractiveness 77 F. Clustering coefficient 58 B. Growth 78 G. Graph spectra 59 1. Empirical results 78 IV. Percolation Theory 59 2. Analytical results 78 A. Quantities of interest in percolation theory 60 C. Local events 79 B. General results 60 Ͻ 1. Internal edges and rewiring 79 1. The subcritical phase (p pc)60 Ͼ 2. Internal edges and edge removal 79 2. The supercritical phase (p pc)61 D. Growth constraints 80 C. Exact solutions: Percolation on a Cayley tree 61 1. Aging and cost 80 D. Scaling in the critical region 62 2. Gradual aging 81 E. Cluster structure 62 E. Competition in evolving networks 81 F. Infinite-dimensional percolation 62 1. Fitness model 81 G. Parallels between random-graph theory and 2. Edge inheritance 82 percolation 63 F. Alternative mechanisms for preferential V. Generalized Random Graphs 63 attachment 82 A. Thresholds in a scale-free random graph 64 1. Copying mechanism 82 B. Generating function formalism 64 2. Edge redirection 82 1. Component sizes and phase transitions 65 3. Walking on a network 83 2. Average path length 65 4. Attaching to edges 83 G. Connection to other problems in statistical mechanics 83 *Present address: School of Mathematics, University of Min- 1. The Simon model 83 nesota, Minneapolis, Minnesota 55455. 2. Bose-Einstein condensation 85 0034-6861/2002/74(1)/47(51)/$35.00 47 ©2002 The American Physical Society 48 R. Albert and A.-L. Baraba´si: Statistical mechanics of complex networks IX. Error and Attack Tolerance 86 condensed-matter physics, ranging from percolation to A. Numerical results 86 Bose-Einstein condensation. 1. Random network, random node removal 87 Traditionally the study of complex networks has been 2. Scale-free network, random node removal 87 3. Preferential node removal 87 the territory of graph theory. While graph theory ini- B. Error tolerance: analytical results 88 tially focused on regular graphs, since the 1950s large- C. Attack tolerance: Analytical results 89 scale networks with no apparent design principles have D. The robustness of real networks 90 been described as random graphs, proposed as the sim- 1. Communication networks 90 plest and most straightforward realization of a complex 2. Cellular networks 91 network. Random graphs were first studied by the Hun- 3. Ecological networks 91 garian mathematicians Paul Erdo˝ s and Alfre´dRe´nyi. X. Outlook 91 According to the Erdo˝ s-Re´nyi model, we start with N A. Dynamical processes on networks 91 B. Directed networks 92 nodes and connect every pair of nodes with probability Ϫ C. Weighted networks, optimization, allometric p, creating a graph with approximately pN(N 1)/2 scaling 92 edges distributed randomly. This model has guided our D. Internet and World Wide Web 93 thinking about complex networks for decades since its E. General questions 93 introduction. But the growing interest in complex sys- F. Conclusions 94 tems has prompted many scientists to reconsider this Acknowledgments 94 modeling paradigm and ask a simple question: are the References 94 real networks behind such diverse complex systems as the cell or the Internet fundamentally random? Our in- tuition clearly indicates that complex systems must dis- I. INTRODUCTION play some organizing principles, which should be at some level encoded in their topology. But if the topology Complex weblike structures describe a wide variety of of these networks indeed deviates from a random graph, systems of high technological and intellectual impor- we need to develop tools and measurements to capture tance. For example, the cell is best described as a com- in quantitative terms the underlying organizing prin- plex network of chemicals connected by chemical reac- ciples. tions; the Internet is a complex network of routers and In the past few years we have witnessed dramatic ad- computers linked by various physical or wireless links; vances in this direction, prompted by several parallel de- fads and ideas spread on the social network, whose velopments. First, the computerization of data acquisi- nodes are human beings and whose edges represent tion in all fields led to the emergence of large databases various social relationships; the World Wide Web is an on the topology of various real networks. Second, the enormous virtual network of Web pages connected by increased computing power allowed us to investigate hyperlinks. These systems represent just a few of the networks containing millions of nodes, exploring ques- many examples that have recently prompted the scien- tions that could not be addressed before. Third, the slow tific community to investigate the mechanisms that de- but noticeable breakdown of boundaries between disci- termine the topology of complex networks. The desire plines offered researchers access to diverse databases, to understand such interwoven systems has encountered allowing them to uncover the generic properties of com- significant challenges as well. Physics, a major benefi- plex networks. Finally, there is an increasingly voiced ciary of reductionism, has developed an arsenal of suc- need to move beyond reductionist approaches and try to cessful tools for predicting the behavior of a system as a understand the behavior of the system as a whole. Along whole from the properties of its constituents. We now this route, understanding the topology of the interac- understand how magnetism emerges from the collective tions between the components, i.e., networks, is un- behavior of millions of spins, or how quantum particles avoidable. lead to such spectacular phenomena as Bose-Einstein Motivated by these converging developments and cir- condensation or superfluidity. The success of these mod- cumstances, many new concepts and measures have eling efforts is based on the simplicity of the interactions been proposed and investigated in depth in the past few between the elements: there is no ambiguity as to what years. However, three concepts occupy a prominent interacts with what, and the interaction strength is place in contemporary thinking about complex net- uniquely determined by the physical distance. We are at works. Here we define and briefly discuss them, a discus- a loss, however, to describe systems for which physical sion to be expanded in the coming sections. distance is irrelevant or for which there is ambiguity as Small worlds: The small-world concept in simple to whether two components interact. While for many terms describes the fact that despite their often large complex systems with nontrivial network topology such size, in most networks there is a relatively short path ambiguity is naturally present, in the past few years we between any two nodes. The distance between two have increasingly recognized that the tools of statistical nodes is defined as the number of edges along the short- mechanics offer an ideal framework for describing these est path connecting them. The most popular manifesta- interwoven systems as well. These developments have tion of small worlds is the ‘‘six degrees of separation’’ introduced new and challenging problems for statistical concept, uncovered by the social psychologist Stanley physics and unexpected links to major topics in Milgram (1967), who concluded that there was a path of Rev. Mod. Phys., Vol. 74, No. 1, January 2002 R. Albert and A.-L. Baraba´si: Statistical mechanics of complex networks 49 acquaintances with a typical length of about six between P͑k͒ϳkϪ␥. (2) most pairs of people in the United States (Kochen, Such networks are called scale free (Baraba´si and Al- 1989).
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