Statistical Mechanics of Complex Networks R´eka Albert1;2 and Albert-L´aszl´o Barab´asi2 1School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455 2Department of Physics, 225 Nieuwland Science Hall, University of Notre Dame, Notre Dame, Indiana 46556 Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks is governed by robust organizing principles. Here we review 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, we discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, as well as the interplay between topology and the network's robustness against failures and attacks. CONTENTS 1. Average path length 24 2. Clustering coefficient 25 I. INTRODUCTION 2 3. Degree distribution 26 II. THE TOPOLOGY OF REAL NETWORKS: 4. Spectral properties 26 EMPIRICAL RESULTS 3 VII. THE SCALE-FREE MODEL 27 1. World-Wide Web 3 A. Definition of the scale-free (SF) model 27 2. Internet 5 B. Theoretical approaches 28 3. Movie actor collaboration network 5 C. Limiting cases of the SF model 29 4. Science collaboration graph 5 D. Properties of the SF model 30 5. The web of human sexual contacts 6 1. Average path length 30 6. Cellular networks 6 2. Node degree correlations 31 7. Ecological networks 6 3. Clustering coefficient 31 8. Phone-call network 6 4. Spectral properties 32 9. Citation networks 7 VIII. EVOLVING NETWORKS 32 10. Networks in linguistics 7 A. Preferential attachment Π(k) 32 11. Power and neural networks 7 1. Measuring Π(k) for real networks 32 12. Protein folding 7 2. Nonlinear preferential attachment 33 III. RANDOM GRAPH THEORY 9 3. Initial attractiveness 34 A. The Erd}os-R´enyi model 9 B. Growth 34 B. Subgraphs 10 1. Empirical results 34 C. Graph Evolution 11 2. Analytical results 34 D. Degree Distribution 12 C. Local events 35 E. Connectedness and Diameter 13 1. Internal edges and rewiring 35 F. Clustering coefficient 13 2. Internal edges and edge removal 36 G. Graph spectra 14 D. Growth constraints 36 IV. PERCOLATION THEORY 14 1. Aging and cost 36 A. Quantities of interest in percolation theory 15 2. Gradual aging 37 arXiv:cond-mat/0106096 v1 6 Jun 2001 B. General results 16 E. Competition in evolving networks 37 C. Exact solutions: percolation on a Cayley tree 16 1. Fitness model 37 D. Scaling in the critical region 17 2. Edge inheritance 38 E. Cluster structure 17 F. Alternative mechanisms for preferential F. Infinite dimensional percolation 18 attachment 38 G. Parallels between random graph theory and G. Connection to other problems in statistical percolation 18 mechanics 40 V. GENERALIZED RANDOM GRAPHS 19 1. The Simon model 40 A. Thresholds in a scale-free random graph 19 2. Bose-Einstein condensation 41 B. Generating function formalism 20 IX. ERROR AND ATTACK TOLERANCE 42 1. Component sizes and phase transitions 20 A. Numerical results 42 2. Average path length 21 1. Random network, random node removal 43 C. Random graphs with power-law degree 2. Scale-free network, random node removal 44 distribution 21 3. Preferential node removal 44 D. Bipartite graphs and the clustering coefficient 22 B. Error tolerance: analytical results 44 VI. SMALL-WORLD NETWORKS 23 C. Attack tolerance: analytical results 45 A. The Watts-Strogatz (WS) model 23 D. The robustness of real networks 46 B. Properties of small-world networks 24 X. OUTLOOK 48 1 1. Dynamics on networks 48 and most straightforward realization of a complex net- 2. Directed networks 49 work. Random graphs were first studied by the Hungar- 3. Weighted networks, optimization, allometric ian mathematicians Paul Erd}os and Alfr´ed R´enyi. Ac- scaling 49 cording to the Erd}os-R´enyi (ER) model, we start with 4. Internet and World-Wide Web 50 N nodes and connect every pair of nodes with probabil- 5. General questions 50 ity p, creating a graph with approximately pN(N 1)=2 6. Conclusions 51 edges distributed randomly. This model has guided− our 7. Acknowledgments 51 References 51 thinking about complex networks for decades after its in- troduction. But the growing interest in complex systems prompted many scientists to reconsider this modeling I. INTRODUCTION paradigm and ask a simple question: are real networks behind such diverse complex systems as the cell or the Internet, fundamentally random? Our intuition clearly Complex weblike structures describe a wide variety indicates that complex systems must display some orga- of systems of high technological and intellectual impor- nizing principles which should be at some level encoded tance. For example, the cell is best described as a com- in their topology as well. But if the topology of these plex network of chemicals connected by chemical reac- networks indeed deviates from a random graph, we need tions; the Internet is a complex network of routers and to develop tools and measures to capture in quantitative computers linked by various physical or wireless links; terms the underlying organizing principles. fads and ideas spread on the social network whose nodes are human beings and edges represent various social re- In the past few years we witnessed dramatic advances lationships; the Wold-Wide Web is an enormous virtual in this direction, prompted by several parallel develop- network of webpages connected by hyperlinks. These ments. First, the computerization of data acquisition in systems represent just a few of the many examples that all fields led to the emergence of large databases on the have recently prompted the scientific community to in- topology of various real networks. Second, the increased vestigate the mechanisms that determine the topology of computing power allows us to investigate networks con- complex networks. The desire to understand such inter- taining millions of nodes, exploring questions that could woven systems has brought along significant challenges not be addressed before. Third, the slow but noticeable as well. Physics, a major beneficiary of reductionism, breakdown of boundaries between disciplines offered re- has developed an arsenal of successful tools to predict searchers access to diverse databases, allowing them to the behavior of a system as a whole from the properties uncover the generic properties of complex networks. Fi- of its constituents. We now understand how magnetism nally, there is an increasingly voiced need to move be- emerges from the collective behavior of millions of spins, yond reductionist approaches and try to understand the or how do quantum particles lead to such spectacular behavior of the system as a whole. Along this route, un- phenomena as Bose-Einstein condensation or superfluid- derstanding the topology of the interactions between the ity. The success of these modeling efforts is based on components, i.e. networks, is unavoidable. the simplicity of the interactions between the elements: Motivated by these converging developments and cir- there is no ambiguity as to what interacts with what, cumstances, many quantities and measures have been and the interaction strength is uniquely determined by proposed and investigated in depth in the past few years. the physical distance. We are at a loss, however, in de- However, three concepts occupy a prominent place in scribing systems for which physical distance is irrelevant, contemporary thinking about complex networks. Next or there is ambiguity whether two components interact. we define and briefly discuss them, a discussion to be While for many complex systems with nontrivial network expanded in the coming chapters. topology such ambiguity is naturally present, in the past Small worlds: The small world concept in simple few years we increasingly recognize that the tools of sta- terms describes the fact that despite their often large tistical mechanics offer an ideal framework to describe size, in most networks there is a relatively short path be- these interwoven systems as well. These developments tween any two nodes. The distance between two nodes have brought along new and challenging problems for sta- is defined as the number of edges along the shortest path tistical physics and unexpected links to major topics in connecting them. The most popular manifestation of condensed matter physics, ranging from percolation to "small worlds" is the "six degrees of separation" con- Bose-Einstein condensation. cept, uncovered by the social psychologist Stanley Mil- Traditionally the study of complex networks has been gram (1967), who concluded that there was a path of ac- the territory of graph theory. While graph theory ini- quaintances with typical length about six between most tially focused on regular graphs, since the 1950's large- pairs of people in the United States (Kochen 1989). The scale networks with no apparent design principles were small world property appears to characterize most com- described as random graphs, proposed as the simplest plex networks: the actors in Hollywood are on average 2 within three costars from each other, or the chemicals These three concepts, small path length, clustering and in a cell are separated typically by three reactions. The scale-free degree distribution have initiated a revival of small world concept, while intriguing, is not an indication network modeling in the past few years, resulting in the of a particular organizing principle. Indeed, as Erd}os and introduction and study of three main classes of model- R´enyi have demonstrated, the typical distance between ing paradigms.