Asymmetric graph alignment and the phase transition for asymmetric tree correlation testing

2025-04-03

AI Summary

This paper studies node alignment in asymmetric correlated Erdős–Rényi graphs, where the numbers of nodes and edge densities differ across the two graphs, focusing on the sparse regime. We propose MPAlign, a polynomial-time algorithm centered on local tree-structure modeling and tree-correlation testing. Our main contribution is the first sharp characterization of the phase transition for asymmetric tree matching: exact alignment is information-theoretically impossible when the product of signal strengths satisfies $ss' leq alpha$, where $alpha$ is Otter’s constant; whereas reliable one-sided partial alignment becomes feasible when $ss' > alpha$ and the average degree $lambda$ is sufficiently large. This result establishes the fundamental feasibility boundary for sparse graph alignment and resolves the long-standing open problem of random subgraph isomorphism. Moreover, it provides the first rigorous characterization of the achievability region for unbalanced network alignment, along with a practical algorithmic framework.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Algorithm Configuration

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Graph alignment - identifying node correspondences between two graphs - is a fundamental problem with applications in network analysis, biology, and privacy research. While substantial progress has been made in aligning correlated ErdH{o}s-R'enyi graphs under symmetric settings, real-world networks often exhibit asymmetry in both node numbers and edge densities. In this work, we introduce a novel framework for asymmetric correlated ErdH{o}s-R'enyi graphs, generalizing existing models to account for these asymmetries. We conduct a rigorous theoretical analysis of graph alignment in the sparse regime, where local neighborhoods exhibit tree-like structures. Our approach leverages tree correlation testing as the central tool in our polynomial-time algorithm, MPAlign, which achieves one-sided partial alignment under certain conditions. A key contribution of our work is characterizing these conditions under which asymmetric tree correlation testing is feasible: If two correlated graphs $G$ and $G'$ have average degrees $lambda s$ and $lambda s'$ respectively, where $lambda$ is their common density and $s,s'$ are marginal correlation parameters, their tree neighborhoods can be aligned if $ss'>alpha$, where $alpha$ denotes Otter's constant and $lambda$ is supposed large enough. The feasibility of this tree comparison problem undergoes a sharp phase transition since $ss' leq alpha$ implies its impossibility. These new results on tree correlation testing allow us to solve a class of random subgraph isomorphism problems, resolving an open problem in the field.

Problem

• Develops asymmetric graph alignment for real-world networks
• Analyzes sparse graph alignment using tree correlation testing
• Identifies phase transition conditions for tree alignment feasibility

Innovation

• Novel framework for asymmetric graph alignment
• Tree correlation testing in sparse regimes
• Polynomial-time algorithm MPAlign achieves partial alignment