Testing Correlation in Graphs by Counting Bounded Degree Motifs

We investigate the problem of detecting correlation between two Erd\H{o}s-R\'enyi graphs $G(n,p)$, formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent, while under the alternative hypothesis, they are correlated through a latent bijective mapping between their vertex sets. We develop a polynomial-time test by counting bounded degree motifs and prove its effectiveness for any constant correlation coefficient $\rho$ when the edge connecting probability satisfies $p\ge n^{-1+\delta}$ for some constant $\delta>0$. In particular, our guarantee improves the constrain of motif-counting methods from $\rho\ge \sqrt{\alpha}$ to any constant $\rho = \Omega(1)$, where $\alpha\approx 0.338$ is the Otter's constant.