Information-Theoretic Limits and Strong Consistency on Binary Non-uniform Hypergraph Stochastic Block Models

TL;DR

This paper establishes the fundamental limits and proposes an optimal algorithm for node classification in non-uniform hypergraph stochastic block models, achieving strong consistency across different regimes.

Contribution

It identifies the information-theoretic threshold for strong consistency and introduces a novel refinement algorithm that is provably optimal in all regimes.

Findings

Threshold for strong consistency identified via Generalized Hellinger distance.

Proposed refinement algorithm achieves optimality and strong consistency.

Aggregating all uniform layers improves clustering accuracy in diverging degree regimes.

Abstract

We investigate the unsupervised node classification problem on random hypergraphs under the non-uniform Hypergraph Stochastic Block Model (HSBM) with two equal-sized communities. In this model, edges appear independently with probabilities depending only on the labels of their vertices. We identify the threshold for strong consistency, expressed in terms of the Generalized Hellinger distance. Below this threshold, strong consistency is impossible, and we derive the Information-Theoretic (IT) lower bound on the expected mismatch ratio. Above the threshold, the parameter space is typically divided into two disjoint regions. When only the aggregated adjacency matrices are accessible, while one-stage algorithms accomplish strong consistency with high probability in the region far from the threshold, they fail in the region closer to the threshold. We propose a new refinement algorithm which, in conjunction with the initial estimation, provably achieves strong consistency throughout the entire region above the threshold, and attains the IT lower bound when below the threshold, proving its optimality. This novel refinement algorithm applies the power iteration method to a weighted adjacency matrix, where the weights are determined by hyperedge sizes and the initial label estimate. Unlike the constant degree regime where a subset selection of uniform layers is necessary to enhance clustering accuracy, in the scenario with diverging degrees, each uniform layer contributes non-negatively to clustering accuracy. Therefore, aggregating information across all uniform layers yields better performance than using any single layer alone.