Matching and mixing: Matchability of graphs under Markovian error

2026-01-27

AI Summary

This study investigates the matching and anonymization thresholds of graph sequences under edge-dependent Markovian noise, specifically modeled via edgelighter random walks. Focusing on graphs generated by the Erdős–Rényi (ER) model and the stochastic block model (SBM), the work establishes the first theoretical framework for anonymization thresholds in the presence of edge-dependent noise by integrating graph matching algorithms with mixing time analysis of Markov chains. The main contributions include proving that for ER graphs, both the anonymization threshold and the mixing time scale as Θ(n² log n), whereas structured SBM graphs can achieve faster anonymization—prior to global mixing—with thresholds as low as O(n^α log n) for α < 2. Simulations and empirical validation on real-world networks, including Facebook and email graphs, confirm the practical relevance of the theoretical findings.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We consider the problem of graph matching for a sequence of graphs generated under a time-dependent Markov chain noise model. Our edgelighter error model, a variant of the classical lamplighter random walk, iteratively corrupts the graph $G_0$ with edge-dependent noise, creating a sequence of noisy graph copies $(G_t)$. Much of the graph matching literature is focused on anonymization thresholds in edge-independent noise settings, and we establish novel anonymization thresholds in this edge-dependent noise setting when matching $G_0$ and $G_t$. Moreover, we also compare this anonymization threshold with the mixing properties of the Markov chain noise model. We show that when $G_0$ is drawn from an Erd\H{o}s-R\'enyi model, the graph matching anonymization threshold and the mixing time of the edgelighter walk are both of order $\Theta(n^2\log n)$. We further demonstrate that for more structured model for $G_0$ (e.g., the Stochastic Block Model), graph matching anonymization can occur in $O(n^\alpha\log n)$ time for some $\alpha<2$, indicating that anonymization can occur before the Markov chain noise model globally mixes. Through extensive simulations, we verify our theoretical bounds in the settings of Erd\H{o}s-R\'enyi random graphs and stochastic block model random graphs, and explore our findings on real-world datasets derived from a Facebook friendship network and a European research institution email communication network.

Problem

• graph matching
• Markovian noise
• anonymization threshold
• edge-dependent noise
• mixing time

Innovation

• graph matching
• Markovian noise
• anonymization threshold
• mixing time
• edgelighter model