Phase transition in the two star Exponential Random Graph Model

TL;DR

This paper introduces a simulation method for the two star exponential random graph model, explores its phase transition behavior, and establishes parallels with the Curie-Weiss model, providing insights into its structural properties.

Contribution

It presents a novel simulation scheme for the two star ERGM and characterizes its phase transition boundaries, linking it to well-known physical models.

Findings

Phase transition boundaries similar to Curie-Weiss model

Validated phase transition predictions with concentration results

Provided a new simulation approach for the two star ERGM

Abstract

This paper gives a way to simulate from the two star probability distribution on the space of simple graphs via auxiliary variables. Using this simulation scheme, the model is explored for various domains of the parameter values, and the phase transition boundaries are identified, and shown to be similar as that of the Curie-Weiss model of statistical physics. Concentration results are obtained for all the degrees, which further validate the phase transition predictions.