Hypothesis Testing for Equality of Latent Positions in Random Graphs

Abstract

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 hypotheses test include testing whether two vertices in a stochastic block model or degree-corrected stochastic block model graph have the same block membership vectors. 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 problem of choosing between the standard stochastic block model and its degree-corrected variant. The effectiveness of our proposed tests are illustrated via both simulation studies and real data applications.

Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence

“…For example, our results do not imply consistent estimation for the row-wise covariance matrices Σi, though we do show if one has covariances that are only distinct between clusters, then one can estimate the limiting covariance matrix Si for each row of U . Our techniques could therefore be appropriately modified to develop two-sample asymptotically valid confidence regions or test statistics, such as in deriving a Hotelling T 2 analogue for the singular vectors as in ; Du and Tang (2021). Furthermore, one possibility for further inference would be to consider drawing several matrices M + E independently from the same distribution, assuming the rows are matched together between samples, in which case one could leverage existing statistical methodology to conduct two-sample tests of hypothesis.…”

Lost in the Shuffle: Testing Power in the Presence of Errorful Network Vertex Labels

“…As in [16], we will choose to control the sparsity of our graphs via ν and not through the latent position matrix X or block probability matrix Λ. As such, we will implicitly make the following assumption throughout the remainder for all RDPGs and positive semidefinite SBMs (when viewed as RDPGs):…”

“…These results (and analogues for unscaled variants of the ASE) have laid the groundwork for myriad subsequent inference results, including clustering [51,39,33,49], classification [54], time-series analysis [10,45], and vertex nomination [19,60], among others. In the shuffled testing analysis that we consider herein, we will use the following ASE consistency result from [48,16].…”

“…Numerous methods exist in the literature for (semi)parametric testing across network samples, where one of the parameters being leveraged is the correspondence of labels across networks; see for example [52,16,3,34]. There are also nonparametric methods that estimate and compare network distributions, and ignore the information contained in the vertex labels; see, for example, [53,1].…”