Testing network correlation efficiently via counting trees

Abstract

We propose a new procedure for testing whether two networks are edge-correlated through some latent vertex correspondence. The test statistic is based on counting the cooccurrences of signed trees for a family of nonisomorphic trees. When the two networks are Erdős–Rényi random graphs $\mathcal{G}(\mathit{n},\mathit{q})$ that are either independent or correlated with correlation coefficient ρ, our test runs in ${\mathit{n}}^{2\mathbf{+}\mathit{o}(1)}$ time and succeeds with high probability as $\mathit{n}\to \infty$, provided that $\mathit{n}min\{\mathit{q},1-\mathit{q}\}\ge {\mathit{n}}^{-\mathit{o}(1)}$ and ${\mathit{\rho }}^2>\mathit{\alpha }\approx 0.338$, where α is Otter’s constant so that the number of unlabeled trees with K edges grows as ${(1/\mathit{\alpha })}^{\mathit{K}}$. This significantly improves the prior work in terms of statistical accuracy, running time and graph sparsity.