Scientific Understanding of Consciousness
Consciousness as an Emergent Property of Thalamocortical Activity

Complex Systems, Growth and Preferential Attachment, Self-Organization



Science 15 October 1999: Vol. 286 no. 5439 pp. 509-512

Emergence of Scaling in Random Networks

Albert-László Barabási and Réka Albert

Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA.


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. 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. At a different organizational level, a large network is formed by the nervous system, whose vertices are the nerve cells, connected by axons. But equally complex networks occur in social science, where vertices are individuals or organizations and the edges are the social interactions between them, or in the World Wide Web (WWW), whose vertices are HTML documents connected by links pointing from one page to another. Because of their large size and the complexity of their interactions, the topology of these networks is largely unknown.

Growth and preferential attachment are mechanisms common to a number of complex systems, including business networks, social networks (describing individuals or organizations), transportation networks, and so on. Consequently, we expect that the scale-invariant state observed in all systems for which detailed data has been available to us is a generic property of many complex networks, with applicability reaching far beyond the quoted examples. A better description of these systems would help in understanding other complex systems as well, for which less topological information is currently available, including such important examples as genetic or signaling networks in biological systems. We often do not think of biological systems as open or growing, because their features are genetically coded. However, possible scale-free features of genetic and signaling networks could reflect the networks' evolutionary history, dominated by growth and aggregation of different constituents, leading from simple molecules to complex organisms. With the fast advances being made in mapping out genetic networks, answers to these questions might not be too far away. Similar mechanisms could explain the origin of the social and economic disparities governing competitive systems, because the scale-free inhomogeneities are the inevitable consequence of self-organization due to the local decisions made by the individual vertices, based on information that is biased toward the more visible (richer) vertices, irrespective of the nature and origin of this visibility.

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