Power-law decay of the degree-sequence probabilities of multiple random graphs with application to graph isomorphism

TL;DR

This paper develops a framework to compare degree-sequence probabilities in multiple Erdős-Rényi graphs and applies it to bound the likelihood of graph isomorphism events, revealing power-law decay behaviors.

Contribution

It extends an approximation framework to analyze degree-sequence probabilities in multiple graphs and applies it to bound isomorphism-related event probabilities.

Findings

Probability bounds decay as a power law in the approximation model

The same bounds hold in the original model for small probability events

Framework can be applied to analyze graph isomorphism probabilities

Abstract

We consider events over the probability space generated by the degree sequences of multiple independent Erd\H{o}s-R\'enyi random graphs, and consider an approximation probability space where such degree sequences are deemed to be sequences of i.i.d. random variables. We show that, for any sequence of events with probabilities asymptotically smaller than some power law in the approximation model, the same upper bound also holds in the original model. We accomplish this by extending an approximation framework proposed in a seminal paper by McKay and Wormald. Finally, as an example, we apply the developed framework to bound the probability of isomorphism-related events over multiple independent random graphs.