Testing for high-dimensional geometry in random graphs

• Published in: Random structures & algorithms Vol. 49; no. 3; pp. 503 - 532
• Main Authors: Bubeck, Sébastien, Ding, Jian, Eldan, Ronen, Rácz, Miklós Z.
• Format: Journal Article
• Language: English
• Published: Hoboken Blackwell Publishing Ltd 01.10.2016; Wiley Subscription Services, Inc
• Subjects: Boundaries; Computational efficiency; Constants; Graphs; high-dimensional geometric structure; hypothesis testing; Mathematical analysis; Null hypothesis; Optimization; random geometric graphs; random graphs; signed triangles; Vectors (mathematics)
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20633

We study the problem of detecting the presence of an underlying high‐dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erdős‐Rényi random graph G(n, p). Under the alternative, the graph is generated from the G(n,p,d) model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere Sd−1, and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., p is a constant), we propose a near‐optimal and computationally efficient testing procedure based on a new quantity which we call signed triangles. The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix. In the sparse regime, we make a conjecture for the optimal detection boundary. We conclude the paper with some preliminary steps on the problem of estimating the dimension in G(n,p,d). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 503–532, 2016