The importance sampling technique for understanding rare events in Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper develops an importance sampling method based on large deviation principles to efficiently estimate the probability of rare large subgraph counts, like triangles, in dense Erdős-Rényi graphs, highlighting the limitations of naive exponential tilts.

Contribution

It introduces an asymptotically optimal importance sampling scheme using exponential random graphs tailored to rare event conditioning, improving estimation accuracy.

Findings

Exponential tilt from large deviations is not always optimal.

Optimal importance sampling involves a specific exponential random graph parameter choice.

Scheme is effective in both replica symmetric and parts of replica breaking phases.

Abstract

In dense Erd\H{o}s-R\'enyi random graphs, we are interested in the events where large numbers of a given subgraph occur. The mean behavior of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, can one develop efficient numerical schemes to estimate the probability of an Erd\H{o}s-R\'enyi graph containing an excessively large number of a fixed given subgraph? Using the large deviation principle we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring within Erd\H{o}s-R\'enyi graphs. We show that the exponential tilt suggested directly by the large deviation principle does not always yield an optimal scheme. The exponential tilt used in the importance sampling scheme comes from a generalized class of exponential random graphs. Asymptotic optimality, a measure of the efficiency of the importance sampling scheme, is achieved by a special choice of the parameters in the exponential random graph that makes it indistinguishable from an Erd\H{o}s-R\'enyi graph conditioned to have many triangles in the large network limit. We show how this choice can be made for the conditioned Erd\H{o}s-R\'enyi graphs both in the replica symmetric phase as well as in parts of the replica breaking phase to yield asymptotically optimal numerical schemes to estimate this rare event probability.