Testing Correlation in Graphs by Counting Bounded Degree Motifs

Correlation analysis is a fundamental step for extracting meaningful insights from complex datasets. In this paper, 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. 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 $p\ge n^{-2/3}$. Our results overcome the limitation requiring $\rho \ge \sqrt{\alpha}$, where $\alpha\approx 0.338$ is the Otter's constant, extending it to any constant $\rho$. Methodologically, bounded degree motifs -- ubiquitous in real networks -- make the proposed statistic both natural and scalable. We also validate our method on synthetic and real co-citation networks, further confirming that this simple motif family effectively captures correlation signals and exhibits strong empirical performance.