The phantom alignment strength conjecture: practical use of graph matching alignment strength to indicate a meaningful graph match

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 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.

Clustered Graph Matching for Label Recovery and Graph Classification

“…al. in [23]. In particular, our result in Erdős-Rényi model simulation part showed matching objective functions similar to the "hockey stick" matchability plots in their paper.…”

“…In particular, our result in Erdős-Rényi model simulation part showed matching objective functions similar to the "hockey stick" matchability plots in their paper. Both our work and that in [23] deal with edgewise correlations, and we are working to unify our results and use some of our results and computations to support the backbones of the phantom alignment strength conjecture, and find some explanation or causation of the "hockey sticks" matchability plots. In turn, we will be able to propose more precise conditions on when our three fore-mentioned matching algorithms will behave similarly and when they will differ significantly.…”

“…While subtle, we do see that this transition point occurs at a larger value of q when n and m generally increase as expected. The nature of the jump, and the relatively flat objective function value post-jump, across all the figures when SGM fails is indicative of the presence of phantom alignment strength after this critical threshold; see [23] for further detail.…”

Lost in the Shuffle: Testing Power in the Presence of Errorful Network Vertex Labels

“…It is a natural next step to explore more the interplay between matching and subsequent inference (here testing). Namely, we could consider the following questions (among others): how the signal in an imperfectly recovered matching affects power loss as opposed to a random misalignment; how a probabilistic alignment (where the unknown in vertex labels is encoded into a stochastic matrix giving probabilities of alignment) can be incorporated into the testing framework; and how to use matching metrics (e.g., alignment strength [20,21]) to estimate the size and membership of U k,n when this is unknown a priori.…”

Bisected graph matching improves automated pairing of bilaterally homologous neurons from connectomes

“…In other words, we can expect the performance of any graph matching algorithm to be poor (with respect to the true permutation) if the network correlations are weak. Given this phenomenon, [64] discussed how to assess these so-called “phantom alignments,” proposing methods for assessing whether a matching is likely to have arisen by pure chance under some model of uncorrelated networks. While these kinds of methods could also be applied to the bisected graph matching setting, they still do not provide a method for inferring a better matching for these weakly correlated networks, so we did not explore them further here.…”