Complex networks: Structure and dynamics

S BOCCALETTI, V LATORA, Y MORENO, M CHAVEZ, D HWANG
2006 Physics reports  
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
more » ... cterizing 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. The network approach to nature 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
doi:10.1016/j.physrep.2005.10.009 fatcat:us3vssqn6ngcxdi4ise2znlcdi