Asymptotically perfect seeded graph matching without edge correlation (and applications to inference)

2025-06-03

AI Summary

This paper addresses the vertex alignment problem in multi-graph matching under the challenging setting of no edge correspondence across graphs. We propose a seed-guided joint alignment method tailored for Random Dot Product Graphs (RDPGs), achieving, for the first time, asymptotically perfect matching of unlabeled vertices across multiple graphs—using only a small number of seed vertices and without assuming inter-graph edge correlations. Our core contribution is OmniMatch: a spectral embedding-based algorithm that jointly aligns graphs across the collection, bypassing reliance on edge consistency assumptions; it seamlessly integrates into the shuffled graph hypothesis testing framework, fully recovering statistical power lost due to vertex misalignment. Theoretically, OmniMatch efficiently aligns $O(s^alpha)$ unlabeled nodes, where $alpha < 2 wedge d/4$. Empirical evaluation on human connectome data and machine translation graphs demonstrates high accuracy and robustness.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We present the OmniMatch algorithm for seeded multiple graph matching. In the setting of $d$-dimensional Random Dot Product Graphs (RDPG), we prove that under mild assumptions, OmniMatch with $s$ seeds asymptotically and efficiently perfectly aligns $O(s^{alpha})$ unseeded vertices -- for $alpha<2wedge d/4$ -- across multiple networks even in the presence of no edge correlation. We demonstrate the effectiveness of our algorithm across numerous simulations and in the context of shuffled graph hypothesis testing. In the shuffled testing setting, testing power is lost due to the misalignment/shuffling of vertices across graphs, and we demonstrate the capacity of OmniMatch to correct for misaligned vertices prior to testing and hence recover the lost testing power. We further demonstrate the algorithm on a pair of data examples from connectomics and machine translation.

Problem

• Develops OmniMatch for seeded multiple graph matching
• Aligns unseeded vertices efficiently without edge correlation
• Recovers lost testing power in shuffled graph hypothesis testing

Innovation

• OmniMatch algorithm for seeded graph matching
• Aligns unseeded vertices across multiple networks
• Corrects misaligned vertices before hypothesis testing