The Phantom Alignment Strength Conjecture: Practical use of graph matching alignment strength to indicate a meaningful graph match

TL;DR

This paper introduces the Phantom Alignment Strength Conjecture, a practical method to distinguish meaningful graph matches from misleading noise-induced alignments, supported by empirical evidence.

Contribution

The paper proposes the Phantom Alignment Strength Conjecture, offering a new practical approach to assess the significance of graph matching results.

Findings

Empirical evidence supports the conjecture.

Guidelines for identifying true graph matches.

Insights into noise effects on alignment strength.

Abstract

The alignment strength of a graph matching is a quantity that gives the practitioner a measure of the correlation of the two graphs, and it can also give the practitioner a sense for whether the graph matching algorithm found the true matching. Unfortunately, when a graph matching algorithm fails to find the truth because of weak signal, there may be "phantom alignment strength" from meaningless matchings that, by random noise, have fewer disagreements than average (sometimes substantially fewer); this alignment strength may give the misleading appearance of significance. A practitioner needs to know what level of alignment strength may be phantom alignment strength and what level indicates that the graph matching algorithm obtained the true matching and is a meaningful measure of the graph correlation. The {\it Phantom Alignment Strength Conjecture} introduced here provides a principled and practical means to approach this issue. We provide empirical evidence for the conjecture, and explore its consequences.