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