Solution of the 2-star model of a network

TL;DR

This paper provides an analytical solution to the 2-star exponential random graph model, revealing phase transition behavior and symmetry breaking phenomena in network structures.

Contribution

It offers the first exact analytical expressions for the 2-star model, clarifying its phase structure and transition properties.

Findings

Identification of a symmetry-broken phase in the 2-star model

Analytic expressions for key network quantities

Demonstration of a continuous phase transition

Abstract

The p-star model or exponential random graph is among the oldest and best-known of network models. Here we give an analytic solution for the particular case of the 2-star model, which is one of the most fundamental of exponential random graphs. We derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition.