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Testing for high‐dimensional geometry in random graphs
Author(s) -
Bubeck Sébastien,
Ding Jian,
Eldan Ronen,
Rácz Miklós Z.
Publication year - 2016
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20633
Subject(s) - mathematics , random graph , combinatorics , conjecture , random matrix , vertex (graph theory) , graph , discrete mathematics , eigenvalues and eigenvectors , physics , quantum mechanics
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 sphereS d − 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