Solution for the properties of a clustered network

TL;DR

This paper provides an exact mean-field analytic solution for Strauss's network model with clustering, revealing a phase transition associated with network degeneracy and dense subgraph formation.

Contribution

It introduces an exact mean-field solution for a clustered network model and characterizes the phase transition leading to degeneracy.

Findings

Degenerate region corresponds to a symmetry-broken phase.

Degeneracy onset is a first-order phase transition.

Analytic solution matches previous simulation observations.

Abstract

We study Strauss's model of a network with clustering and present an analytic mean-field solution which is exact in the limit of large network size. Previous computer simulations have revealed a degenerate region in the model's parameter space in which triangles of adjacent edges clump together to form unrealistically dense subgraphs, and perturbation calculations have been found to break down in this region at all orders. Our analytic solution shows that this region corresponds to a classic symmetry-broken phase and that the onset of the degeneracy corresponds to a first-order phase transition in the density of the network.