Existence of a symmetric bipodal phase in the edge-triangle model

TL;DR

This paper investigates the structure of entropy-maximizing graphons in the edge-triangle model, identifying conditions under which symmetric bipodal structures are optimal and when they are not.

Contribution

It proves the existence of a symmetric bipodal phase near certain densities and shows its absence below a specific edge density threshold.

Findings

Symmetric bipodal graphons are optimal near certain densities.

Non-bipodal graphons are optimal below a specific edge density.

The paper delineates the phase transition in the edge-triangle model.

Abstract

In the edge-triangle model with edge density close to 1/2 and triangle density below 1/8 we prove that the unique entropy-maximizing graphon is symmetric bipodal. We also prove that,for any edge density $e$ less than $e_0 = (3-\sqrt{3})/6 \approx 0.2113$ and triangle density slightly less than $e^3$, the entropy-maximizing graphon is not symmetric bipodal.