🤖 AI Summary
Whether “counting” (estimating the number and sizes of communities) is computationally easier than “locating” (exactly recovering community assignments) in community detection.
Method: We establish the first computational lower bound for the planted–planted hypothesis testing problem, leveraging the low-degree polynomial framework, statistical-to-computational phase transition analysis, and classical hypothesis testing theory.
Results: We rigorously prove that community counting and exact community recovery are computationally equivalent—refuting the conjecture that counting is inherently easier than locating. This constitutes the first rigorous evidence of computational intractability for distinguishing between multiple planted distribution models. The result reveals the intrinsic hardness of counting in community structure inference and provides foundational insights into computational limits in statistical learning and graphical models.
📝 Abstract
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.
