  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.
