Correlated random networks

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= summation operator exp([-betaH(c)]. The simplest cases are uncorrelated random networks such as the well-known Erdös-Rényi graphs. Here we study more general interactions H(c) which lead to correlations, for example, between the connectivities of adjacent vertices. In particular, such correlations occur in optimized networks described by partition functions in the limit beta--> infinity. They are argued to be a crucial signature of evolutionary design in biological networks.