Matching and mixing: Matchability of graphs under Markovian error

TL;DR

This paper studies graph matching under a Markovian noise model, establishing thresholds for anonymization and mixing times, and demonstrates how structured graphs can be anonymized faster than the noise model's mixing time.

Contribution

It introduces a novel Markovian noise model for graph evolution, derives new anonymization thresholds, and compares these with the mixing properties for different graph structures.

Findings

For Erd ext{"o}s-Rényi graphs, thresholds are of order Θ(n^2 log n).

Structured models like the Stochastic Block Model can be anonymized faster, in O(n^α log n) for α<2.

Simulations confirm theoretical bounds on real-world networks.

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.