Asymptotic Null Distributions of Moran's $I$ and Assortativity in Large Networks

2026-10-06

AI Summary

This study addresses the problem that the asymptotic distributions of Moran's I and Newman's assortativity under the null hypothesis remain unknown in large networks. Assuming Gaussian node attributes independent of network structure, we systematically analyze the convergence behavior and topological effects of both dependence measures through asymptotic theory and probabilistic derivations, with validation via simulation experiments. The results reveal that network heterogeneity plays a decisive role in the validity of normal approximations, demonstrating that dominant nodes can invalidate both statistics. Furthermore, this work establishes the precise conditions under which convergence rates and limiting distributions hold. These findings provide essential theoretical corrections for independence testing in complex networks.

Abstract

This study investigates the asymptotic behavior of two dependence measures defined on networks, Moran's $I$ statistic and Newman's assortativity, under the null hypothesis that a Gaussian node attribute $Y$ is independent of the network structure. We demonstrate that the structure of the network directly affects the convergence rate to normality of these measures as the size of the network increases. We further establish that, in some instances, the mean values of these dependence measures under the null hypothesis remain non-negligible asymptotically and must therefore be explicitly accounted for when calculating the test statistics. Applications to a variety of simulated and real networks also reveal that the normal approximation performs well only when the network is not strongly heterogeneous. Network topology determines both the convergence rate to normality and whether the limiting distribution is Gaussian. In dense networks whose degree heterogeneity does not vanish, we further show that assortativity can fail to be a valid test statistic even though Moran's $I$ remains well behaved, whereas a dominating node invalidates both.

Problem

• Moran's I
• Assortativity
• Asymptotic null distribution
• Network topology
• Large networks

Innovation

• Moran's I
• Assortativity
• Asymptotic null distribution
• Network topology
• Degree heterogeneity