Matching recovery threshold for correlated random graphs

Abstract

For two correlated graphs which are independently sub-sampled from a common Erdős–Rényi graph $\mathbf{G}(\mathit{n},\mathit{p})$, we wish to recover their latent vertex matching from the observation of these two graphs without labels. When $\mathit{p}={\mathit{n}}^{-\mathit{\alpha }\mathbf{+}\mathit{o}(1)}$ for $\mathit{\alpha }\in (0,1]$, we establish a sharp information-theoretic threshold for whether it is possible to correctly match a positive fraction of vertices. Our result sharpens a constant factor in a recent work by Wu, Xu and Yu.