Abstract
We study the problem of testing the existence of a dense subhypergraph. The null hypothesis corresponds to an Erdős-Rényi uniform random hypergraph and the alternative hypothesis corresponds to a uniform random hypergraph that contains a dense subhypergraph. We establish sharp detection boundaries in both scenarios: (1) the edge probabilities are known; (2) the edge probabilities are unknown. In both scenarios, sharp detection boundaries are characterized by the appropriate model parameters. Asymptotically powerful tests are provided when the model parameters fall in the detectable regions. Our results indicate that the detectable regions for general hypergraph models are dramatically different from their graph counterparts.
Funding Statement
Shang’s research was supported in part by NSF DMS 1764280 and 1821157.
Acknowledgements
We would like to express our sincere thanks to the Editor Markus Reiß and the anonymous AE and reviewers for constructive and helpful suggestions.
Citation
Mingao Yuan. Zuofeng Shang. "Sharp detection boundaries on testing dense subhypergraph." Bernoulli 28 (4) 2459 - 2491, November 2022. https://doi.org/10.3150/21-BEJ1425
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