# On the Minimax Misclassification Ratio of Hypergraph Community Detection

arXiv: https://arxiv.org/abs/1802.00926 · HTML: https://mlsanity.com/paper/1802.00926/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Feb 2018.

## Built on by (most cited first, 11 of 11)

- [You are AllSet: A Multiset Function Framework for Hypergraph Neural Networks](https://mlsanity.com/paper/2106.13264/) (2021)
- [Exact Recovery in the General Hypergraph Stochastic Block Model](https://mlsanity.com/paper/2105.04770/) (2021)
- [Community Detection in General Hypergraph via Graph Embedding](https://mlsanity.com/paper/2103.15035/) (2021)
- [Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model](https://mlsanity.com/paper/2112.11671/) (2021)
- [Sparse random hypergraphs: Non-backtracking spectra and community detection](https://mlsanity.com/paper/2203.07346/) (2022)
- [Multiway Spherical Clustering via Degree-Corrected Tensor Block Models](https://mlsanity.com/paper/2201.07401/) (2022)
- [Community Detection in the Hypergraph SBM: Exact Recovery Given the Similarity Matrix](https://mlsanity.com/paper/2208.12227/) (2022)
- [Multilayer hypergraph clustering using the aggregate similarity matrix](https://mlsanity.com/paper/2301.11657/) (2023)
- [Information-Theoretic Limits and Strong Consistency on Binary Non-uniform Hypergraph Stochastic Block Models](https://mlsanity.com/paper/2306.06845/) (2023)
- [Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph boldsymbolβ-Model](https://mlsanity.com/paper/2307.02818/) (2023)
- [Community Detection in the Multi-View Stochastic Block Model](https://mlsanity.com/paper/2401.09510/) (2024)

## Abstract (arXiv)

Community detection in hypergraphs is explored. Under a generative hypergraph model called "d-wise hypergraph stochastic block model" (d-hSBM) which naturally extends the Stochastic Block Model from graphs to d-uniform hypergraphs, the asymptotic minimax mismatch ratio is characterized. For proving the achievability, we propose a two-step polynomial time algorithm that achieves the fundamental limit. The first step of the algorithm is a hypergraph spectral clustering method which achieves partial recovery to a certain precision level. The second step is a local refinement method which leverages the underlying probabilistic model along with parameter estimation from the outcome of the first step. To characterize the asymptotic performance of the proposed algorithm, we first derive a sufficient condition for attaining weak consistency in the hypergraph spectral clustering step. Then, under the guarantee of weak consistency in the first step, we upper bound the worst-case risk attained in the local refinement step by an exponentially decaying function of the size of the hypergraph and characterize the decaying rate. For proving the converse, the lower bound of the minimax mismatch ratio is set by finding a smaller parameter space which contains the most dominant error events, inspired by the analysis in the achievability part. It turns out that the minimax mismatch ratio decays exponentially fast to zero as the number of nodes tends to infinity, and the rate function is a weighted combination of several divergence terms, each of which is the Renyi divergence of order 1/2 between two Bernoulli's. The Bernoulli's involved in the characterization of the rate function are those governing the random instantiation of hyperedges in d-hSBM. Experimental results on synthetic data validate our theoretical finding that the refinement step is critical in achieving the optimal statistical limit.