Is It Easier to Count Communities Than Find Them?

Authors Cynthia Rush , Fiona Skerman , Alexander S. Wein , Dana Yang

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Author Details

Cynthia Rush
  • Department of Statistics, Columbia University, New York, NY,USA
Fiona Skerman
  • Department of Mathematics, Uppsala University, Sweden
Alexander S. Wein
  • Department of Mathematics, University of California, Davis, CA, USA
Dana Yang
  • Department of Statistics and Data Science, Cornell University, Ithaca, NY, USA


This work began when the authors were visiting the Simons Institute for the Theory of Computing during the program on Computational Complexity of Statistical Inference in Fall 2021. We are grateful to Guy Bresler for helpful discussions.

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Cynthia Rush, Fiona Skerman, Alexander S. Wein, and Dana Yang. Is It Easier to Count Communities Than Find Them?. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 94:1-94:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Random graph models with community structure have been studied extensively in the literature. For both the problems of detecting and recovering community structure, an interesting landscape of statistical and computational phase transitions has emerged. A natural unanswered question is: might it be possible to infer properties of the community structure (for instance, the number and sizes of communities) even in situations where actually finding those communities is believed to be computationally hard? We show the answer is no. In particular, we consider certain hypothesis testing problems between models with different community structures, and we show (in the low-degree polynomial framework) that testing between two options is as hard as finding the communities. In addition, our methods give the first computational lower bounds for testing between two different "planted" distributions, whereas previous results have considered testing between a planted distribution and an i.i.d. "null" distribution.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random network models
  • Theory of computation → Computational complexity and cryptography
  • Community detection
  • Hypothesis testing
  • Low-degree polynomials


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