  Higher-order structures of networks, namely, small subgraphs of networks
(also called network motifs), are widely known to be crucial and essential to
the organization of networks. There has been a few work studying the community
detection problem -- a fundamental problem in network analysis, at the level of
motifs. In particular, higher-order spectral clustering has been developed,
where the notion of motif adjacency matrix is introduced as the input of the
algorithm. However, it remains largely unknown that how higher-order spectral
clustering works and when it performs better than its edge-based counterpart.
To elucidate these problems, we investigate higher-order spectral clustering
from a statistical perspective. In particular, we theoretically study the
clustering performance of higher-order spectral clustering under a weighted
stochastic block model and compare the resulting bounds with the corresponding
results of edge-based spectral clustering. It turns out that when the network
is dense with weak signal of weights, higher-order spectral clustering can
really lead to the performance gain in clustering. We also use simulations and
real data experiments to support the findings.
