Tripodal structure in undersaturated random graphs

TL;DR

This paper explores the phase structure of typical graphs in the Strauss model, revealing a tripodal phase with a discontinuous transition to bipodal graphs and internal phase changes.

Contribution

It identifies a tripodal phase in the Strauss model's parameter space and characterizes the phase transitions, including a boundary discontinuity and internal phase change.

Findings

Typical graphs are tripodal below a certain edge density

Discontinuous phase transition at the boundary to bipodal phase

Internal phase transition within the tripodal region

Abstract

We numerically investigate typical graphs in a region of the Strauss model of random graphs with constraints on the densities of edges and triangles. This region, where typical graphs had been expected to be bipodal but turned out to be tripodal, involves edge densities $e$ below $e_0 = (3-\sqrt{3})/6 \approx 0.2113$ and triangle densities $t$ slightly below $e^3$. We determine the extent of this region in $(e,t)$ space and show that there is a discontinuous phase transition at the boundary between this region and a bipodal phase. We further show that there is at least one phase transition within this region, where the parameters describing typical graphs change discontinuously.