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We propose a nonparametric two-sample test statistic for low-rank,conditionally independent edge random graphs whose edge probability matriceshave negative eigenvalues and arbitrarily close eigenvalues. Our proposed teststatistic involves using the maximum mean discrepancy applied to suitablyrotated rows of a graph embedding, where the rotation is estimated usingoptimal transport. We show that our test statistic, appropriately scaled, isconsistent for sufficiently dense graphs, and we study its convergence underdifferent sparsity regimes. In addition, we provide empirical evidencesuggesting that our novel alignment procedure can perform better than thena\"ive alignment in practice, where the na\"ive alignment assumes an eigengap.

Nonparametric Two-Sample Hypothesis Testing for Random Graphs with Negative and Repeated Eigenvalues