On the Minimax Misclassification Ratio of Hypergraph Community Detection

Abstract

In this paper, 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 (SBM) from graphs to d-uniform hypergraphs, the asymptotic minimax mismatch ratio (the risk function in the community detection problem) is characterized. For proving the achievability, we propose a two-step polynomial time algorithm that provably 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. Compared to the existing works in SBM, the main technical challenge lies in the complex structure of error events since community relations become much more complicated. We tackle the challenge by a series of non-trivial generalizations. To ensure the optimality of the refinement algorithm, all possible misclassification patterns among a wealth of community relations should be considered and treated with care when concentration inequalities are applied. Moreover, in proving the performance guarantee of the spectral clustering algorithm, we have to deal with random matrices whose entries are no longer independent as opposed to its graph counterpart. 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 Rényi divergence of order 1/2 between two Bernoulli distributions. The Bernoulli distributions 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. I (Eli) Chien was with the

Exact Recovery in the General Hypergraph Stochastic Block Model

“…1) Nodes are partitioned into k ≥ 2 non-overlapping communities, in which each node is assigned to one of these k communities with probabilities {p i } k i=1 . This generalizes the setting in [30], [34] in which k = 2 and p 1 = p 2 , and the setting in [31], [40]- [42] wherein the k communities are of equal or approximately equal sizes (i.e., p i ≈ p j for all…”

“…In particular, Ghoshdastidar and Dukkipati [26] first proposed a random hypergraph model called the d-uniform hypergraph stochastic block model (d-HSBM), in which each subset of nodes with cardinality d is generated independently as an order-d hyperedge with a certain probability that depends on the communities that the d nodes belong to. Subsequent researchers further investigated the recovery limits of d-HSBMs by developing various hypergraph clustering algorithms (such as spectral clustering methods [27]- [33], semidefinite programming-based methods [34]- [36], tensor decomposition-based methods [37], approximate-message passing algorithms [38], [39], etc) with theoretical guarantees, and characterizing the minimax misclassification ratio [40]- [42] as well as the exact recovery criterion for the special case of two symmetric communities [34].…”

“…For example, in [30], [31], [33]- [35] the hyperedge probability can only take two values depending on whether all the d nodes belong to the same community. Although other works such as [40]- [42] relax the restrictions in [30], [31], [33]- [35], they are nevertheless particularizations of our general HSBM model. We are aware that the assumption on hyperedge probabilities made in [26]- [29] is similar to ours, but their focus is to characterize the performance of a hypergraph spectral clustering method contained therein in terms of the fraction of misclassified nodes, whereas we aim to quantify necessary and sufficient conditions for exact recovery (see Definition 1 for details).…”

You are AllSet: A Multiset Function Framework for Hypergraph Neural Networks

“…Learning on hypergraphs has attracted significant attention due to its ability to capture higher-order structural information [58,59]. In the statistical learning and information theory literature, researchers have studied community detection problem for hypergraph stochastic block models [60,61,62]. In the machine learning community, methods that mitigate drawbacks of CE have been reported in [6,14,12,63].…”

Community detection in the sparse hypergraph stochastic block model

“…Subsequently, they extended their results to sparse, nonuniform hypergraphs [19–21]. For exact recovery, it was shown that the phase transition occurs in the regime of logarithmic average degrees in [11, 12, 29] and the exact threshold was given in [27], by a generalization of the techniques in [2]. Almost exact recovery for HSBMs was studied in [11, 12, 21].…”