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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| Published in | Random structures & algorithms Vol. 49; no. 3; pp. 503 - 532 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Hoboken
Blackwell Publishing Ltd
01.10.2016
Wiley Subscription Services, Inc |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1042-9832 1098-2418 |
| DOI | 10.1002/rsa.20633 |
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| Abstract | 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 |
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| AbstractList | 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 Erds-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 S 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 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 model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere , 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 . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 503–532, 2016 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 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 Erds-Renyi random graph G(n, p). Under the alternative, the graph is generated from the [Formulaomitted] model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere [Formulaomitted], 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 [Formulaomitted]. Random Struct. Alg., 49, 503-532, 2016 |
| Author | Bubeck, Sébastien Ding, Jian Eldan, Ronen Rácz, Miklós Z. |
| Author_xml | – sequence: 1 givenname: Sébastien surname: Bubeck fullname: Bubeck, Sébastien email: sebubeck@microsoft.com organization: Theory Group, Microsoft Research, and Department of Operations Research and Financial Engineering, Princeton University, Berkeley – sequence: 2 givenname: Jian surname: Ding fullname: Ding, Jian email: jianding@galton.uchicago.edu organization: Department of Statistics, University of Chicago, Berkeley – sequence: 3 givenname: Ronen surname: Eldan fullname: Eldan, Ronen email: roneneldan@gmail.com organization: Department of Computer Science & Engineering, University of Washington, Berkeley – sequence: 4 givenname: Miklós Z. surname: Rácz fullname: Rácz, Miklós Z. email: racz@stat.berkeley.edu organization: Department of Statistics, University of California, Berkeley |
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| Cites_doi | 10.1016/S0925-7721(97)00014-X 10.1137/1.9781611973105.132 10.3150/13-BEJ565 10.1145/2554797.2554840 10.1214/EJP.v16-967 10.1093/biomet/20A.1-2.32 10.1214/aos/1009210544 10.1093/acprof:oso/9780198506263.001.0001 10.1109/TNSE.2015.2397592 10.1007/s11856-015-1169-5 10.1007/s00440-014-0576-6 10.1038/30918 10.1214/009117906000000917 10.1214/11-AOS904 10.1007/s10959-013-0519-7 10.1214/14-AOS1208 10.1198/016214502388618906 |
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| References_xml | – reference: M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, NY, USA, 1964. – reference: R. Eldan, An efficiency upper bound for inverse covariance estimation, Israel J Math 207 (2015), 1-9. – reference: S. Bubeck, E. Mossel, and M. Z. Rácz, On the influence of the seed graph in the preferential attachment model, IEEE Trans Network Sci Eng 2 (2015), 30-39. – reference: P. J. Bickel, A. Chen, and E. Levina, The method of moments and degree distributions for network models, Ann Stat 39 (2011), 2280-2301. – reference: P. D. Hoff, A. E. Raftery, and M. S. Handcock, Latent space approaches to social network analysis, J Am Stat Assoc 97 (2002), 1090-1098. – reference: D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature 393 (1998), 440-442. – reference: M. Penrose, Random geometric graphs, Volume 5 of Oxford studies in probability, Oxford University Press, Oxford, UK, 2003. – reference: T. Jiang and D. 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| Snippet | 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... 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... |
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| SubjectTerms | 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) |
| Title | Testing for high-dimensional geometry in random graphs |
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