A nonparametric two-sample hypothesis testing problem for random graphs. (English) Zbl 1450.62040

Tang, Minh; Athreya, Avanti; Sussman, Daniel L.; Lyzinski, Vince; Priebe, Carey E.

Bernoulli 23, No. 3, 1599-1630 (2017).

Summary: We consider the problem of testing whether two independent finite-dimensional random dot product graphs have generating latent positions that are drawn from the same distribution, or distributions that are related via scaling or projection. We propose a test statistic that is a kernel-based function of the estimated latent positions obtained from the adjacency spectral embedding for each graph. We show that our test statistic using the estimated latent positions converges to the test statistic obtained using the true but unknown latent positions and hence that our proposed test procedure is consistent across a broad range of alternatives. Our proof of consistency hinges upon a novel concentration inequality for the suprema of an empirical process in the estimated latent positions setting.

MSC:

62G10	Nonparametric hypothesis testing
05C80	Random graphs (graph-theoretic aspects)
60F05	Central limit and other weak theorems
60F17	Functional limit theorems; invariance principles
62G20	Asymptotic properties of nonparametric inference

Keywords:

empirical process; nonparametric graph inference; random dot product graph