On the asymptotics of constrained exponential random graphs

TL;DR

This paper investigates the behavior of exponential random graph models under constraints, such as fixed edge counts, exploring phase transitions and typical graph structures, with specific results for the edge-triangle model.

Contribution

It provides general theoretical results for constrained exponential random graphs and applies these to analyze the edge-triangle model with fixed edge density.

Findings

Identification of phase transition phenomena under constraints

Characterization of typical graph structures in constrained models

Application to the edge-triangle model with fixed density

Abstract

The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to get concrete answers in the edge-triangle model with fixed density of edges.