Achieving Almost Exact Recovery in Almost Quadratic Time: Rank-Based Graph Matching via Local Tree Correlation Tests

This paper studies graph matching under the correlated $\text{Erd\H{o}s-R\'{e}nyi}$ (ER) graph pair model. This model first samples an $\mathrm{ER}(n,\frac{\lambda}{ns})$ base graph, whose edges are then independently subsampled twice with probability $s$ to produce two correlated $\mathrm{ER}(n,\frac{\lambda}{n})$ graphs. We propose a graph matching algorithm that has $n^{2+o(1)}$ time complexity and achieves almost exact recovery with high probability under the assumptions $\lambda=(\log n)^{\alpha+o(1)}$ for some $\alpha\in(0,1)$ and $s\in(\sqrt{C_{\mathrm{Otter}}},1]$, where $C_{\mathrm{Otter}}\approx 0.338$ is Otter's tree-counting constant. This is the first algorithm with almost quadratic time complexity in this regime of $\lambda$, while the best known result in this regime is the chandelier-counting algorithm with time complexity $O(n^{c(s)})$, where $c(s)\rightarrow \infty$ as $s$ approaches $\sqrt{C_\mathrm{Otter}}$ from above. The proposed algorithm is based on local tree correlation tests. It uses a rank-based algorithm to match the vertex pairs instead of threshold-based rules in the literature. This avoids the need of computing an explicit threshold, which is computationally difficult to obtain. To prove the almost exact recovery result, we establish a new analysis of tree correlation tests in the diverging-degree regime, where both the mean degree and the tree depth grow with $n$. Based on this new result, we establish the existence of a threshold for a threshold-based graph matching algorithm via local tree correlation tests. Finally, we couple the performance of the rank-based algorithm with the threshold-based algorithm to show almost exact recovery.