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Complex Networks: Structure and Dynamics S Physics Reports 424 (2006) 175–308 www.elsevier.com/locate/physrep Complex networks: Structure and dynamics S. Boccalettia,∗, V. Latorab,c, Y. Morenod,e, M. Chavezf , D.-U. Hwanga aCNR-Istituto dei Sistemi Complessi, Largo E. Fermi, 6, 50125 Florence, Italy bDipartimento di Fisica e Astronomia, Universitá di Catania, Via S. Sofia, 64, 95123 Catania, Italy cIstituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S. Sofia, 64, 95123 Catania, Italy dInstituto de Biocomputación y Física de Sistemas Complejos, Universidad de Zaragoza, Zaragoza 50009, Spain eDepartamento de Fisica Teórica, Universidad de Zaragoza, Zaragoza 50009, Spain f Laboratoire de Neurosciences Cognitives et Imagerie Cérébrale (LENA) CNRS UPR-640, Hôpital de la Salpêtrière. 47 Bd. de l’Hôpital, 75651 Paris CEDEX 13, France Accepted 27 October 2005 Available online 10 January 2006 editor: I. Procaccia 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. © 2005 Elsevier B.V. All rights reserved. PACS: 05.45.−a Contents 1. Introduction ...........................................................................................................177 1.1. The network approach to nature .....................................................................................177 1.2. Outline of the report ...............................................................................................179 2. The structure of complex networks........................................................................................180 2.1. Definitions and notations ...........................................................................................180 2.1.1. Node degree, degree distributions and correlations ...............................................................181 2.1.2. Shortest path lengths, diameter and betweenness .................................................................182 2.1.3. Clustering .................................................................................................183 2.1.4. Motifs ....................................................................................................184 ∗ Corresponding author. E-mail address: [email protected] (S. Boccaletti). 0370-1573/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2005.10.009 176 S. Boccaletti et al. / Physics Reports 424 (2006) 175–308 2.1.5. Community structures .......................................................................................184 2.1.6. Graph spectra ..............................................................................................185 2.2. Topology of real networks ..........................................................................................186 2.2.1. The small-world property ....................................................................................187 2.2.2. Scale-free degree distributions ................................................................................187 2.2.3. Some examples .............................................................................................188 2.3. Networks models .................................................................................................191 2.3.1. Random graphs .............................................................................................191 2.3.2. Generalized random graphs ...................................................................................192 2.3.3. Small-world networks .......................................................................................193 2.3.4. Static scale-free networks ....................................................................................194 2.3.5. Evolving scale-free networks .................................................................................195 2.4. Weighted networks ................................................................................................198 2.4.1. Measures for weighted networks ..............................................................................198 2.4.2. Real weighted networks......................................................................................201 2.4.3. Modelling weighted networks .................................................................................203 2.5. Spatial networks ..................................................................................................205 2.5.1. The small-world behavior in Euclidean space ....................................................................207 2.5.2. Scale-free networks in Euclidean space .........................................................................208 2.5.3. Modelling real geographical networks ..........................................................................209 3. Static and dynamic robustness ...........................................................................................213 3.1. Static tolerance to errors and attacks ..................................................................................213 3.1.1. Numerical results ...........................................................................................213 3.1.2. Resilience in uncorrelated networks ............................................................................214 3.1.3. Resilience in correlated networks ..............................................................................216 3.1.4. Tolerance to attacks .........................................................................................219 3.2. Dynamical effects .................................................................................................220 3.2.1. Modelling cascading failures .................................................................................220 3.2.2. Congestion in communication networks ........................................................................225 4. Spreading processes ....................................................................................................228 4.1. Epidemic spreading ...............................................................................................228 4.1.1. The homogeneous mixing hypothesis ..........................................................................229 4.1.2. Uncorrelated networks .......................................................................................230 4.1.3. Correlated networks .........................................................................................232 4.1.4. Fluctuating diseases .........................................................................................233 4.2. Rumor spreading ..................................................................................................234 5. Synchronization and collective dynamics ..................................................................................237 5.1. Introduction to synchronization ......................................................................................237 5.2. The Master stability function approach ...............................................................................238 5.3. Network propensity for synchronization ..............................................................................241 5.3.1. Synchronization in weighted networks: coupling matrices with real spectra ...........................................241 5.3.2. Synchronization in weighted networks: coupling matrices with complex spectra.......................................243 5.4. Synchronization of coupled oscillators ................................................................................246 5.5. Synchronization of chaotic dynamics .................................................................................248 5.6. Other collective behaviors in networks of ordinary differential equations ...................................................250 6. Applications ..........................................................................................................251 6.1. Social networks ...................................................................................................251 6.1.1. Structure ..................................................................................................251 6.1.2. Dynamics I: opinion formation ................................................................................253 6.1.3.
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