Exact matching of random graphs with constant correlation

This paper deals with the problem of graph matching or network alignment for Erd?s–Rényi graphs, which can be viewed as a noisy average-case version isomorphism problem. Let G and $$G'$$ G(n, p) graphs marginally, identified their adjacency matrices. Assume that are correlated such $${\mathbb {E}}[G_{ij} G'_{ij}] = p(1-\alpha )$$ . For permutation $$\pi $$ representing latent between vertices , denote by $$G^\pi obtained from permuting Observing we aim to recover In this work, show every $$\varepsilon \in (0,1]$$ there is $$n_0>0$$ depending on absolute constants $$\alpha _0, R > 0$$ following property. $$n \ge n_0$$ $$(1+\varepsilon ) \log n \le np n^{\frac{1}{R n}}$$ $$0< \alpha < \min (\alpha _0,\varepsilon /4)$$ There polynomial-time algorithm F {P}}\{F(G^\pi ,G')=\pi \}=1-o(1)$$ first recovers exact constant correlation high probability. The based comparison partition trees associated vertices.