Asymptotically perfect seeded graph matching without edge correlation (and applications to inference)

TL;DR

The paper introduces OmniMatch, an algorithm that achieves asymptotically perfect seeded graph matching in multi-graph settings without relying on edge correlation, with applications in inference tasks like hypothesis testing.

Contribution

The paper presents OmniMatch, a novel algorithm that guarantees asymptotically perfect matching in multi-graph scenarios without edge correlation, under mild assumptions.

Findings

OmniMatch asymptotically perfectly aligns unseeded vertices.

The algorithm improves hypothesis testing power by correcting vertex misalignments.

Effective in connectomics and machine translation data examples.

Abstract

We present the OmniMatch algorithm for seeded multiple graph matching. In the setting of $d$-dimensional Random Dot Product Graphs (RDPG), we prove that under mild assumptions, OmniMatch with $s$ seeds asymptotically and efficiently perfectly aligns $O(s^{\alpha})$ unseeded vertices -- for $\alpha<2\wedge d/4$ -- across multiple networks even in the presence of no edge correlation. We demonstrate the effectiveness of our algorithm across numerous simulations and in the context of shuffled graph hypothesis testing. In the shuffled testing setting, testing power is lost due to the misalignment/shuffling of vertices across graphs, and we demonstrate the capacity of OmniMatch to correct for misaligned vertices prior to testing and hence recover the lost testing power. We further demonstrate the algorithm on a pair of data examples from connectomics and machine translation.