evolution networks

Evolution of networks
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S.N. Dorogovtsev1,2,∗ and J.F.F. Mendes1,†
Departamento de F´ ısica and Centro de F´ ısica do Porto, Faculdade de Ciˆ encias, Universidade do Porto Rua do Campo Alegre 687, 4169-007 Porto, Portugal 2 A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
Hale Waihona Puke arXiv:cond-mat/0106144 v2 7 Sep 2001
CONTENTS I. Introduction 2 II. Historical background 2 III. Structural characteristics of evolving networks 2 III A. Degree 3 III B. Shortest path 3 III C. Clustering coefficient 3 III D. Size of the giant component 4 III E. Other many-vertex characteristics 4 IV. Notions of equilibrium and non-equilibrium networks 5 V. Evolving networks in Nature 6 V A. Networks of citations of scientific papers 6 V B. Networks of collaborations 7 V C. Communications networks, the WWW and Internet 8 V C 1. Structure of the Internet 8 V C 2. Structure of the WWW 10 V D. Biological networks 12 V D 1. Structure of neural networks 12 V D 2. Networks of metabolic reactions 12 V D 3. Protein networks 13 V D 4. Ecological and food webs 13 V D 5. Word Web of human language 14 V E. Electronic circuits 14 V F. Other networks 15 VI. Classical random graphs, the Erd¨ os-R´ enyi model 15 VII. Small-world networks 15 VII A. The Watts-Strogatz model and its variations 16 VII B. The smallest-world network 18 VII C. Other possibilities to obtain large clustering coefficient 20 VIII. Growing exponential networks 20 IX. Scale-free networks 21
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I. INTRODUCTION
II. HISTORICAL BACKGROUND
The Internet and World Wide Web are perhaps the most impressive creatures of our civilization (Baran 1964). [1]. Their influence on us is incredible. They are part of our life, of our world. Our present and our future are impossible without them. Nevertheless, we know much less about them than one may expect. We know surprisingly little of their structure and hierarchical organization, their global topology, their local properties, and various processes occurring within them. This knowledge is needed for the most effective functioning of the Internet and WWW, for ensuring their safety, and for utilizing all of their possibilities. Certainly, the understanding of such problems is a topic not of computer science and applied mathematics, but rather of non-equilibrium statistical physics. In fact, these wonderful communications nets [2–10] are only particular examples of a great class of evolving networks. Numerous networks, e.g., collaboration networks [11–15], public relations nets [16–20], citations of scientific papers [21–27], some industrial networks [11,12,28], transportation networks [29,30], nets of relations between enterprises and agents in financial markets [31], telephone call graphs [32], many biological networks [33–45], food and ecological webs [46–52], etc., belong to it. The finiteness of these networks sets serious restrictions on extracting useful experimental data because of strong size effects and, often, insufficient statistics. The large size of the Internet and WWW and their extensive and easily accessible documentation allow reliable and informative experimental investigation of their structure and properties. Unfortunately, the statistical theory of neural networks [53,54] seems to be rather useless for the understanding of problems of the evolution of networks, since this advanced theory does not seriously touch on the main question arising for real networks – how networks becomes specifically structured during their growth. Quite recently, general features of structural organization of such networks were discovered [2,3,5,6,27,41,55,56,58]. It has become clear that their complex scale-free structure is a natural consequence of the principles of their growth. Some simple basic ideas have been proposed. Self-organization of growing networks and processes occurring within them have been related [59–63] to corresponding phenomena (growth phenomena [64], self-organization [65–67] and self-organized criticality [68–70], percolation [71–73], localization, etc.) being studied by physicists for a long time. The goal of our paper is to review the recent rapid progress in understanding the evolution of networks using ideas and methods of statistical physics. The problems that we discuss relate to computer science, mathematics, physics, engineering, biology, economy, and social sciences. Here, we present the point of view of physicists. To restrict ourselves, we do not dwell on Boolean and neural networks. 2