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 Erdős-Rényi model, the graph matching anonymization threshold and the mixing time of the edgelighter walk are both of order $Θ(n^2\log n)$. We further demonstrate that for more structured model for $G_0$ (e.g., the Stochastic Block Model), graph matching anonymization can occur in $O(n^α\log n)$ time for some $α<2$, 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 Erdős-Rényi 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.

翻译：我们研究了在时间依赖的马尔可夫链噪声模型下生成的一系列图的图匹配问题。我们的边照明器误差模型是经典照明器随机游走的一种变体，它通过边依赖的噪声迭代地破坏图$G_0$，从而产生一系列噪声图副本$(G_t)$。图匹配领域的许多文献主要关注边独立噪声设置下的匿名化阈值，而我们在匹配$G_0$和$G_t$时，针对这种边依赖噪声设置建立了新的匿名化阈值。此外，我们还比较了该匿名化阈值与马尔可夫链噪声模型的混合特性。我们证明，当$G_0$从Erdős-Rényi模型中抽取时，图匹配匿名化阈值与边照明器游走的混合时间均为$Θ(n^2\log n)$阶。我们进一步证明，对于结构更复杂的$G_0$模型（例如随机分块模型），图匹配匿名化可能在$O(n^α\log n)$时间内发生，其中$α<2$，这表明匿名化可能在马尔可夫链噪声模型全局混合之前发生。通过大量模拟，我们在Erdős-Rényi随机图和随机分块模型随机图的设置下验证了我们的理论界限，并在源自Facebook友谊网络和一个欧洲研究机构电子邮件通信网络的真实数据集上探索了我们的发现。