On the efficacy of higher-order spectral clustering under weighted stochastic block models

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\textendash a fundamental problem in network analysis, at the level of motifs. In particular, higher-order spectral clustering has been developed, where the notion of \emph{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 {\it 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.