Tame sparse exponential random graphs

TL;DR

This paper provides a precise second-order estimate for the probability of many vertices forming triangles in sparse random graphs, enabling the design of a tunable exponential random graph model with proven large deviation properties.

Contribution

It introduces a novel exponential random graph model based on triangle counts, with a second-order probability estimate and a consistent parameter estimator.

Findings

Second-order accuracy in probability estimates for triangle-rich sparse graphs.

Ability to tune the model to achieve any desired fraction of vertices in triangles.

Derivation of the large deviation principle for the number of edges.

Abstract

In this paper, we obtain a precise estimate of the probability that the sparse binomial random graph contains a large number of vertices in a triangle. The estimate of log of this probability is correct up to second order, and enables us to propose an exponential random graph model based on the number of vertices in a triangle. Specifically, by tuning a single parameter, we can with high probability induce any given fraction of vertices in a triangle. Moreover, in the proposed exponential random graph model we derive the large deviation principle for the number of edges. As a byproduct, we propose a consistent estimator of the tuning parameter.