Correlated random networks

Abstract

We develop a statistical theory of networks. A network is a set of vertices and links given by its adjacency matrix $\c$, and the relevant statistical ensembles are defined in terms of a partition function $Z=\sum_{\c} \exp {[}-β\H(\c) {]}$. The simplest cases are uncorrelated random networks such as the well-known Erdös-Rény graphs. Here we study more general interactions $\stackrel{˝}{(}\stackrel{¸}{)}$ which lead to {\em correlations}, for example, between the connectivities of adjacent vertices. In particular, such correlations occur in {\em optimized} networks described by partition functions in the limit $\beta \to \infty$. They are argued to be a crucial signature of evolutionary design in biological networks.

4 pages Revex

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• Statistical mechanics of complex networks
• Evolution of networks
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