Higher-Order Spectral Clustering under Superimposed Stochastic Block Model

TL;DR

This paper introduces the Superimposed Stochastic Block Model (SupSBM) to incorporate higher-order structures in network community detection, and analyzes the performance of spectral clustering methods within this framework.

Contribution

The paper develops the SupSBM model capturing realistic network features and provides rigorous bounds on spectral clustering error, extending analysis to hypergraph models and mixed edge-hyperedge observations.

Findings

Non-asymptotic bounds on misclustering error for SupSBM.

Bounds on spectral clustering errors for standard and hypergraph SBMs.

A criterion for choosing between edge-based and hyperedge-based spectral clustering.

Abstract

Higher-order motif structures and multi-vertex interactions are becoming increasingly important in studies that aim to improve our understanding of functionalities and evolution patterns of networks. To elucidate the role of higher-order structures in community detection problems over complex networks, we introduce the notion of a Superimposed Stochastic Block Model (SupSBM). The model is based on a random graph framework in which certain higher-order structures or subgraphs are generated through an independent hyperedge generation process, and are then replaced with graphs that are superimposed with directed or undirected edges generated by an inhomogeneous random graph model. Consequently, the model introduces controlled dependencies between edges which allow for capturing more realistic network phenomena, namely strong local clustering in a sparse network, short average path length, and community structure. We proceed to rigorously analyze the performance of a number of recently proposed higher-order spectral clustering methods on the SupSBM. In particular, we prove non-asymptotic upper bounds on the misclustering error of spectral community detection for a SupSBM setting in which triangles or 3-uniform hyperedges are superimposed with undirected edges. As part of our analysis, we also derive new bounds on the misclustering error of higher-order spectral clustering methods for the standard SBM and the 3-uniform hypergraph SBM. Furthermore, for a non-uniform hypergraph SBM model in which one directly observes both edges and 3-uniform hyperedges, we obtain a criterion that describes when to perform spectral clustering based on edges and when on hyperedges, based on a function of hyperedge density and observation quality.