Correlated random networks

TL;DR

This paper develops a statistical framework for analyzing correlated random networks, extending beyond uncorrelated Erdős-Rényi graphs to include interactions that induce correlations, which are significant in biological network evolution.

Contribution

It introduces a general statistical theory for networks with correlations, including optimized networks, highlighting their importance in biological evolution.

Findings

Developed a partition function-based statistical model for correlated networks

Identified correlations as signatures of evolutionary design

Analyzed optimized networks in the limit of zero temperature

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 $Z=\sum_{\c} \exp {[}-\beta \H(\c) {]}$. The simplest cases are uncorrelated random networks such as the well-known Erd\"os-R\'eny graphs. Here we study more general interactions $\H(\c)$ 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.