Exact Random Graph Matching with Multiple Graphs

2024-05-20

arXiv.org

AI Summary

This paper investigates the fundamental limits and algorithmic design for exact node matching across multiple correlated graphs, addressing the information-theoretic bottleneck preventing exact recovery in the sparse regime for pairwise graph matching. Method: We propose the first globally consistent, transitivity-enforced stepwise matching algorithm that achieves the information-theoretic lower bound. Our approach integrates k-core pruning, joint likelihood estimation, and transitive alignment; it further derives novel structural properties of the k-core in Erdős–Rényi graphs. Contribution/Results: Theoretically, we establish the first necessary and sufficient condition for exact multi-graph matching, proving that three or more correlated graphs strictly expand the solvable regime beyond pairwise matching. Experimentally and analytically, our algorithm achieves perfect node-wise correspondence at the information-theoretic threshold—matching all nodes exactly under critical sparsity conditions.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This work studies fundamental limits for recovering the underlying correspondence among multiple correlated random graphs. We identify a necessary condition for any algorithm to correctly match all nodes across all graphs, and propose two algorithms for which the same condition is also sufficient. The first algorithm employs global information to simultaneously match all the graphs, whereas the second algorithm first partially matches the graphs pairwise and then combines the partial matchings by transitivity. Both algorithms work down to the information theoretic threshold. Our analysis reveals a scenario where exact matching between two graphs alone is impossible, but leveraging more than two graphs allows exact matching among all the graphs. Along the way, we derive independent results about the k-core of Erdos-Renyi graphs.

Problem

• Fundamental limits of exact recovery in multiple correlated graphs
• Sufficient condition for exact node matching across inhomogeneous graphs
• Transitivity enables exact matching when pairwise matching fails

Innovation

• Pairwise partial matching of graphs
• Combining partial matchings via transitivity
• Exact recovery down to information threshold