Information-Theoretic Limits on Exact Subgraph Alignment Problem

2026-01-08

arXiv.org

AI Summary

This work addresses the subgraph alignment problem, which arises in domains such as computer vision, social network analysis, and bioinformatics, and entails locating an embedded small graph pattern within a larger graph and recovering the corresponding vertex mapping. The paper presents the first formal definition of this problem, establishes a theoretical framework based on the Erdős–Rényi subgraph-pair model, and derives near-tight information-theoretic limits for exact recovery using tools from information theory. These results offer a statistical perspective on the recoverability of the NP-hard subgraph isomorphism problem and characterize the fundamental information-theoretic limits of subgraph alignment, thereby providing a rigorous theoretical foundation and performance benchmark for future algorithmic development.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

The graph alignment problem aims to identify the vertex correspondence between two correlated graphs. Most existing studies focus on the scenario in which the two graphs share the same vertex set. However, in many real-world applications, such as computer vision, social network analysis, and bioinformatics, the task often involves locating a small graph pattern within a larger graph. Existing graph alignment algorithms and analysis cannot directly address these scenarios because they are not designed to identify the specific subset of vertices where the small graph pattern resides within the larger graph. Motivated by this limitation, we introduce the subgraph alignment problem, which seeks to recover both the vertex set and/or the vertex correspondence of a small graph pattern embedded in a larger graph. In the special case where the small graph pattern is an induced subgraph of the larger graph and both the vertex set and correspondence are to be recovered, the problem reduces to the subgraph isomorphism problem, which is NP-complete in the worst case. In this paper, we formally formulate the subgraph alignment problem by proposing the Erdos-Renyi subgraph pair model together with some appropriate recovery criterion. We then establish almost-tight information-theoretic results for the subgraph alignment problem and present some novel approaches for the analysis.

Problem

• subgraph alignment
• graph matching
• subgraph isomorphism
• vertex correspondence
• information-theoretic limits

Innovation

• subgraph alignment
• information-theoretic limits
• graph correspondence
• Erdos-Renyi subgraph model
• subgraph isomorphism