Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Donniell E. Fishkind, Felix Parker, Hamilton Sawczuk, Lingyao Meng, Eric Bridgeford, Avanti Athreya, Carey E. Priebe, Vince Lyzinski

TL;DR

The Phantom Alignment Strength Conjecture introduced here provides a principled and practical means to approach the issue of phantom alignment strength and provides empirical evidence for the conjecture, and explores its consequences.

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.

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

TL;DR

The Phantom Alignment Strength Conjecture introduced here provides a principled and practical means to approach the issue of phantom alignment strength and provides empirical evidence for the conjecture, and explores its consequences.

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.

Paper Structure

This paper contains 12 sections, 3 theorems, 14 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Seeded graph matching, alignment strength
  3. The correlated Bernoulli random graph model
  4. Phantom Alignment Strength Conjecture, consequences
  5. Empirical evidence in favor of the Phantom Alignment Strength Conjecture
  6. Of hockey sticks and phantom alignment strength
  7. Of hockey sticks and broken hockey sticks
  8. Phantom alignment strength vs theoretical matchability threshold
  9. Block settings
  10. Real data; matching graphs to noisy renditions
  11. Real data; matching same objects under different modalities
  12. Notable mentions and future directions, plus caveats

Key Result

Theorem 1

Suppose $\mu$ is bounded away from $0$ and $1$, over all $\mathfrak{n}$. Then it holds that $\mathfrak{str}' (\varphi^*) - \varrho_T \stackrel{a.s.}{\rightarrow} 0$.

Figures (9)

  • Figure 1: For each $\varrho_e$ from $0$ to $1$ in increments of $.025$, alignment strength of $\hat{\varphi}$ for $100$ independent realizations when all Bernoulli probabilities were $0.5$ (in particular, $\varrho_T=\varrho_e$), with $n=15$ nonseeds, $s=15$ seeds, a green asterisk if $\hat{\varphi}=\varphi^*$, else a red asterisk.
  • Figure 2: Alignment strength $\mathfrak{str} (\hat{\varphi}_{\textup{SGM}})$ plotted against total correlation $\varrho_T$ for the synthetic data experiments in Section \ref{['sec:broken']}, separated according to the number of seeds $s$. The number of nonseeds was $n=1000$, and only the case of $\mu'=0.5$ is shown here. Match ratio of each experiment is color coded green, blue, or red according to the legend above. Subfigures (g) and (h) are zooms into subfigures (c) and (d), to increase the granularity so that the thresholding is better seen.
  • Figure 3: Phantom alignment strength as a function of $n$, fitted to $f_p(n):=d_p+ c_p \sqrt{\frac{\log n}{n}}$.
  • Figure 4: Experiment A in Section \ref{['sec:block']}; here $F_{1,1}$, $F_{1,2}$, $F_{2,2}$ are resp. point mass at $0.3$, $0.4$, $0.5$.
  • Figure 7: Section \ref{['sec:noisy']} experiments; LHS is noisy connectome, RHS is corresponding synthetic.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3