Dimension reduction in vertex-weighted exponential random graphs

TL;DR

This paper analyzes vertex-weighted exponential random graphs, revealing their approximation as mixtures of graphs near fixed points, and explores how solutions relate to community structure and weight regimes.

Contribution

It introduces a novel analysis of vertex-weighted exponential random graphs, including fixed point approximations and behavior under different weight conditions.

Findings

Graphs approximate mixtures near fixed points.

Solutions close to block vectors indicate community structure.

Behavior varies with positive, small, and negative weights.

Abstract

We investigate the behavior of vertex-weighted exponential random graphs. We show that vertex-weighted exponential random graphs with edge weights induced by products of independent vertex weights are approximate mixtures of graphs whose vertex weight vector is a near fixed point of a certain vector equation. For graphs with Hamiltonians counting cliques, it is demonstrated that, under appropriate conditions, every solution to this equation is close to a block vector with a small number of communities. We prove that for the cases of positive weights and small weights in the Hamiltonian in particular, the vector equation has a unique solution. Lastly, the behavior of vertex-weighted exponential random graphs counting triangles is studied in detail and the solution to the vector equation is shown to approach the zero vector as the weight diverges to negative infinity for sufficiently large networks.