DETECTION THRESHOLDS FOR THE β-MODEL ON SPARSE GRAPHS

• 成果类型: Article
• 署名作者: Mukherjee, Rajarshi; Mukherjee, Sumit; Sen, Subhabrata
• 署名单位: University of California System; University of California Berkeley; Columbia University; Microsoft; Microsoft
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1585
• 发表日期: 2018
• 页码: 1288-1317
• 关键词: p-asterisk models; HIGHER CRITICISM; random networks; degree sequence; number; distributions; regression

摘要

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.