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. Several works have studied the community detection problem–a fundamental problem in network analysis at the level of motifs. In particular, the higher-order spectral clustering has been developed, where the notion of motif adjacency matrix is introduced as the algorithm's input. However, how the higher-order spectral clustering works and when it performs better than its edge-based counterpart remain largely unknown. To elucidate these problems, the higher-order spectral clustering is investigated from a statistical perspective. The clustering performance of the higher-order spectral clustering is theoretically studied under a weighted stochastic block model, and the resulting bounds are compared with the corresponding results of the edge-based spectral clustering. The upper bounds and simulations show that when the network is dense and the edge weights have a weak signal, higher-order spectral clustering can lead to a performance gain in clustering. Real data experiments also corroborate the merits of higher-order spectral clustering.