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

2026-07-10

AI Summary

This work addresses the problem of graph matching for sparse and weakly correlated Erdős–Rényi graphs by proposing an efficient algorithm based on local tree correlation tests. Departing from conventional thresholding rules, the method employs a ranking-based strategy to achieve matching. Under the conditions that the correlation parameter $ s \in (\sqrt{C_{\text{Otter}}}, 1] $ and the signal strength $ \lambda = (\log n)^{\alpha + o(1)} $ with $ 0 < \alpha < 1 $, the algorithm achieves almost exact recovery with high probability in nearly quadratic time $ n^{2+o(1)} $. This represents the first such result in this regime. In contrast to existing approaches whose computational complexity sharply increases as $ s \to \sqrt{C_{\text{Otter}}} $, the proposed method significantly improves both efficiency and applicability.

Abstract

This paper studies graph matching under the correlated $\text{Erdős-Rényi}$ (ER) graph pair model. This model first samples an $\mathrm{ER}(n,\fracλ{ns})$ base graph, whose edges are then independently subsampled twice with probability $s$ to produce two correlated $\mathrm{ER}(n,\fracλ{n})$ 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 $λ=(\log n)^{α+o(1)}$ for some $α\in(0,1)$ 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 $λ$, 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.

Problem

• graph matching
• correlated Erdős–Rényi graphs
• almost exact recovery
• quadratic time complexity
• local tree correlation

Innovation

• graph matching
• local tree correlation
• rank-based algorithm
• almost exact recovery
• correlated Erdős–Rényi graphs