A detailed investigation into near degenerate exponential random graphs

TL;DR

This paper investigates the phase transition behavior of exponential random graph models, revealing how they shift abruptly from sparse to dense networks without intermediate states, and provides explicit structural characterizations.

Contribution

It offers a detailed analysis of near degenerate behavior in exponential random graphs, including explicit asymptotic structure characterizations as parameters vary.

Findings

Models transition sharply from sparse to dense graphs

Explicit characterization of asymptotic graph structures

Insights into phase transition mechanisms in network models

Abstract

The exponential family of random graphs has been a topic of continued research interest. Despite the relative simplicity, these models capture a variety of interesting features displayed by large-scale networks and allow us to better understand how phases transition between one another as tuning parameters vary. As the parameters cross certain lines, the model asymptotically transitions from a very sparse graph to a very dense graph, completely skipping all intermediate structures. We delve deeper into this near degenerate tendency and give an explicit characterization of the asymptotic graph structure as a function of the parameters.