Alignment Strength and Correlation for Graphs

TL;DR

This paper investigates the relationship between alignment strength and correlation in graphs with Bernoulli distributions, introducing a new correlation measure $ ho_T$ that influences graph matching performance.

Contribution

It establishes a theoretical link between alignment strength and a novel correlation measure $ ho_T$, and demonstrates how $ ho_T$ affects graph matching complexity and success.

Findings

Alignment strength converges to $ ho_T$ in correlated Bernoulli graphs.

Graph matchability depends on $ ho_T$, showing threshold behavior.

Matching runtime is influenced by the correlation measure $ ho_T$.

Abstract

When two graphs have a correlated Bernoulli distribution, we prove that the alignment strength of their natural bijection strongly converges to a novel measure of graph correlation $\rho_T$ that neatly combines intergraph with intragraph distribution parameters. Within broad families of the random graph parameter settings, we illustrate that exact graph matching runtime and also matchability are both functions of $\rho_T$, with thresholding behavior starkly illustrated in matchability.