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June 2018 Detection thresholds for the $\beta$-model on sparse graphs
Rajarshi Mukherjee, Sumit Mukherjee, Subhabrata Sen
Ann. Statist. 46(3): 1288-1317 (June 2018). DOI: 10.1214/17-AOS1585


In this paper, we study sharp thresholds for detecting sparse signals in $\beta$-models for potentially sparse random graphs. The results demonstrate interesting interplay between graph sparsity, signal sparsity and signal strength. In regimes of moderately dense signals, irrespective of graph sparsity, the detection thresholds mirror corresponding results in independent Gaussian sequence problems. For sparser signals, extreme graph sparsity implies that all tests are asymptotically powerless, irrespective of the signal strength. On the other hand, sharp detection thresholds are obtained, up to matching constants, on denser graphs. The phase transitions mentioned above are sharp. As a crucial ingredient, we study a version of the higher criticism test which is provably sharp up to optimal constants in the regime of sparse signals. The theoretical results are further verified by numerical simulations.


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Rajarshi Mukherjee. Sumit Mukherjee. Subhabrata Sen. "Detection thresholds for the $\beta$-model on sparse graphs." Ann. Statist. 46 (3) 1288 - 1317, June 2018.


Received: 1 August 2016; Revised: 1 May 2017; Published: June 2018
First available in Project Euclid: 3 May 2018

zbMATH: 1392.62131
MathSciNet: MR3798004
Digital Object Identifier: 10.1214/17-AOS1585

Primary: 62C20 , 62G10 , 62G20

Keywords: beta model , Detection boundary , higher criticism , sparse random graphs , sparse signals

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 3 • June 2018
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