Testing for high-dimensional geometry in random graphs

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...

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Vydáno v:Random structures & algorithms Ročník 49; číslo 3; s. 503 - 532
Hlavní autoři: Bubeck, Sébastien, Ding, Jian, Eldan, Ronen, Rácz, Miklós Z.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Hoboken Blackwell Publishing Ltd 01.10.2016
Wiley Subscription Services, Inc
Témata:
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
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Shrnutí: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
Bibliografie:ark:/67375/WNG-B3WNKCDC-S
NSF DMS 1313596
istex:D33BB8AA7D7292B21ECD2C0232F76C85298AE6D0
Supported by NSF (to J.D.) (DMS 1313596); NSF (to M.Z.R.) (DMS 1106999).
ArticleID:RSA20633
NSF DMS 1106999
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SourceType-Scholarly Journals-1
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20633