Hypothesis Testing for Equality of Latent Positions in Random Graphs

We consider the hypothesis testing problem that two vertices i and j of a generalized random dot product graph have the same latent positions, possibly up to scaling. Special cases of this hypothesis test include testing whether two vertices in a stochastic block model or degree-corrected stochastic block model graph have the same block membership vectors, or testing whether two vertices in a popularity adjusted block model have the same community assignment. We propose several test statistics based on the empirical Mahalanobis distances between the ith and jth rows of either the adjacency or the normalized Laplacian spectral embedding of the graph. We show that, under mild conditions, these test statistics have limiting chi-square distributions under both the null and local alternative hypothesis, and we derived explicit expressions for the non-centrality parameters under the local alternative. Using these limit results, we address the model selection problems including choosing between the standard stochastic block model and its degree-corrected variant, and choosing between the ER model and stochastic block model. The effectiveness of our proposed tests are illustrated via both simulation studies and real data applications.