A Central Limit Theorem for an Omnibus Embedding of Multiple Random Dot Product Graphs

Abstract

Performing statistical analyses on collections of graphs is of import to many disciplines, but principled, scalable methods for multisample graph inference are few. In this paper, we describe an omnibus embedding in which multiple graphs on the same vertex set are jointly embedded into a single space with a distinct representation for each graph. We prove a central limit theorem for this omnibus embedding, and we show that this simultaneous embedding into a single common space allows for the comparison of graphs without the requirement that the embedded points associated to each graph undergo cumbersome pairwise alignments. Moreover, the existence of multiple embedded points for each vertex renders possible the resolution of important multiscale graph inference goals, such as the identification of specific subgraphs or vertices as drivers of similarity or difference across large networks. The omnibus embedding achieves near-optimal inference accuracy when graphs arise from a common distribution and yet retains discriminatory power as a test procedure for the comparison of different graphs. We demonstrate the applicability of the omnibus embedding in two analyses of connectomic graphs generated from MRI scans of the brain in human subjects. We show how the omnibus embedding can be used to detect statistically significant differences, at multiple scales, across these networks, with an identification of specific brain regions that are associated with these population-level differences. Finally, we sketch how the omnibus embedding can be used to address pressing open problems, both theoretical and practical, in multisample graph inference.

The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

“…The following theorem provides a guarantee for estimating the leading eigenvectors of a multiple graph omnibus matrix when the graphs are not independent. Theorem 4.7 is among the first of its kind and complements the recent, concurrent work on joint graph embedding in [32].…”

Information Recovery in Shuffled Graphs via Graph Matching

“…Note that while it is perhaps more natural to use an appropriately centered and scaled version of A − B F as our test statistic, as noted in [45], A − B F yields a test that is inconsistent for a large class of alternatives (e.g., if G 2 ∼ ER(n, 0.5) in the G 1 and G 2 independent setting), whereas the test based on T 1 is provably level-α consistent over the entire range of (fixed) alternative distributions. In [24], we posit the same level-α consistency for testing based on T 2 .…”

Estimating a network from multiple noisy realizations

“…Thus, A − EA = O(h(κ, σ) √ n) for a suitable function h. Similarly, the mean of the entries N −1 N m=1 A (m) ij concentrate about their expectation κ cicj σ cicj , and the spectral norm error grows as N −1 N m=1 A (m) − A = O( n /N h(1, A)). Under suitable assumptions on the growth rates of N and n, the community sizes, and the parameters A, κ, σ, the techniques from [30,28] can be used to turn these two spectral norm bounds into a guarantee that an asymptotically vanishing fraction of the vertices are mislabeled, thus ensuring conditions 1 and 2. The fact that {A (m) ij : m = 1, 2, .…”