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Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
The inability of contemporary science to describe systems composed of nonidentical elements that have diverse and nonlocal interactions currently limits advances in many disciplines, ranging from molecular biology to computer science (1). The difficulty of describing these systems lies partly in their topology: Many of them form rather complex networks whose vertices are the elements of the system and whose edges represent the interactions between them. For example, living systems form a huge genetic network whose vertices are proteins and genes, the chemical interactions between them representing edges (2). At a different organizational level, a large network is formed by the nervous system, whose vertices are the nerve cells, connected by axons (3). But equally complex networks occur in social science, where vertices are individuals or organizations and the edges are the social interactions between them (4), or in the World Wide Web (WWW), whose vertices are HTML documents connected by links pointing from one page to another (5, 6). Because of their large size and the complexity of their interactions, the topology of these networks is largely unknown.
Traditionally, networks of complex topology have been described with the random graph theory of Erdos and Renyi (ER) (7), but in the absence of data on large networks, the predictions of the ER theory were rarely tested in the real world. However, driven by the computerization of data acquisition, such topological information is increasingly available, raising the possibility of understanding the dynamical and topological stability of large networks.
Here we report on the existence of a high degree of self-organization characterizing the large-scale properties of complex networks. Exploring...