Approximating the cumulant generating function of triangles in the Erd\"os-R\'enyi random graph

TL;DR

This paper investigates the cumulant generating function of triangles in Erdős-Rényi graphs, identifying phase transitions and structural properties of large graphs through graphon analysis.

Contribution

It introduces a detailed analysis of the edge-triangle model, revealing a replica symmetry breaking transition and characterizing the structure of typical graphs in different phases.

Findings

Identifies a phase transition in the model.

Characterizes the structure of graphs in the broken symmetry phase.

Provides a variational approach to analyze the infinite volume limit.

Abstract

We study the pressure of the "edge-triangle model", which is equivalent to the cumulant generating function of triangles in the Erd\"os-R\'enyi random graph. By analyzing finite graphs of increasing volume, as well as the graphon variational problem in the infinite volume limit, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.