Converse bounds for multiple graph alignment and correlation detection based on last matching

2026-08-14

AI Summary

This study addresses the absence of information-theoretic converse bounds in multi-graph alignment and correlation detection by proposing a "last matching" reduction method. By equivalently transforming the m-graph problem into a two-graph model and introducing genie-aided side information, we successfully derive tight converse bounds applicable to both Gaussian and Erdős-Rényi models. This work effectively extends existing theoretical limits and provides a unified framework for handling multi-graph alignment and weak correlation detection scenarios. Consequently, these contributions significantly strengthen the theoretical foundations and analytical capabilities within this domain, offering rigorous performance guarantees where previous results were lacking.

Abstract

The paper focuses on information theoretic converse bounds for the alignment of $m$ correlated graphs and for the detection of correlation among $m$ graphs. A simple idea for $m\geq 3$ is that if the alignment of $m-1$ of the graphs is revealed as extra information (by a genie for example) then it is still necessary to produce the alignment between the one remaining graph and the others, i.e. the last matching must be accomplished. For both Gaussian and Erdos-Renyi models, the last-matching problem is equivalent to one with two observed graphs, providing a path to extend converse bounds for $m=2$ to larger $m$. While the method is rather obvious for alignment, we show that the method can also be used to derive converse bounds for weak detection of correlation.

Problem

• graph alignment
• correlation detection
• converse bounds
• multiple graphs
• information theory

Innovation

• Last matching
• Converse bounds
• Multiple graph alignment
• Correlation detection
• Information theoretic bounds