Persistent Homology of Graph Embeddings

Abstract

Popular network models such as the mixed membership and standard stochastic block model are known to exhibit distinct geometric structure when embedded into R d using spectral methods. The resulting point cloud concentrates around a simplex in the first model, whereas it separates into clusters in the second. By adopting the formalism of generalised random dotproduct graphs in [26], we demonstrate that both of these models, and different mixing regimes in the case of mixed membership, may be distinguished by the persistent homology of the underlying point distribution in the case of adjacency spectral embedding. Moreover, despite non-identifiability issues, we show that the persistent homology of the support of the distribution and its super-level sets can be consistently estimated. As an application of our consistency results, we provide a topological hypothesis test for distinguishing the standard and mixed membership stochastic block models.

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

“…Theorem 3 (Theorem 7 of Solanki et al (2019)). Suppose F is a (p, q)-admissible distribution; that is for all x, y ∈ supp(F ), x I p,q y ∈ [0, 1].…”

“…Before proving Proposition 1, we include some important related results that we will require. First, Theorem 7 in Solanki et al (2019) says that when we have a (p, q)-admissible distribution, the support is bounded.…”

Spectral embedding and the latent geometry of multipartite networks

“…Additionally, the matrices Q n and Q −1 n have bounded spectral norm almost surely [24,26]. For the remainder of the proof, we suppress the dependence of Q n and G (k) n on n. We now construct a matrix X ∈ R n×D such that the multipartite random dot product graph (A, Y, z) is a generalized random dot product graph (A, X), and show that it satisfies the conditions of Lemmas 15 and 16.…”

Spectral clustering under degree heterogeneity: a case for the random walk Laplacian

“…, X n ) = T −1 X (see Section 4). Since t i = Θ(nρ n ) and X , the support of F , is a bounded set [44], we have that…”