Principles of statistical mechanics of random networks

TL;DR

This paper develops a statistical mechanics framework for uncorrelated random networks, deriving their properties and revealing conditions for fat-tailed degree distributions to exist.

Contribution

It introduces a method to construct equilibrium uncorrelated networks with arbitrary degree distributions and analyzes the conditions for fat-tailed distributions to emerge.

Findings

Equilibrium uncorrelated networks can have arbitrary degree distributions.

Fat-tailed degree distributions require a critical average degree.

A phase with a condensate of edges emerges at the critical point.

Abstract

We develop a statistical mechanics approach for random networks with uncorrelated vertices. We construct equilibrium statistical ensembles of such networks and obtain their partition functions and main characteristics. We find simple dynamical construction procedures that produce equilibrium uncorrelated random graphs with an arbitrary degree distribution. In particular, we show that in equilibrium uncorrelated networks, fat-tailed degree distributions may exist only starting from some critical average number of connections of a vertex, in a phase with a condensate of edges.