The feasibility of multi-graph alignment: a Bayesian approach

TL;DR

This paper investigates the thresholds for successful multi-graph alignment using a Bayesian framework, revealing an all-or-nothing phase transition in Gaussian models and thresholds in sparse Erdős-Rényi models.

Contribution

It introduces a Bayesian estimation approach over metric spaces to analyze multi-graph alignment thresholds in different probabilistic models.

Findings

Exact alignment is achievable above a critical threshold in Gaussian models.

Below the threshold, partial or any alignment is statistically impossible.

Identifies thresholds in sparse Erdős-Rényi models and conjectures partial alignment feasibility above them.

Abstract

We establish thresholds for the feasibility of random multi-graph alignment in two models. In the Gaussian model, we demonstrate an "all-or-nothing" phenomenon: above a critical threshold, exact alignment is achievable with high probability, while below it, even partial alignment is statistically impossible. In the sparse Erd\H{o}s-R\'enyi model, we rigorously identify a threshold below which no meaningful partial alignment is possible and conjecture that above this threshold, partial alignment can be achieved. To prove these results, we develop a general Bayesian estimation framework over metric spaces, which provides insight into a broader class of high-dimensional statistical problems.