Low-rank persistent probability representation for higher-order role discovery

Role discovery is an emerging research area in the analysis of social networks, biological networks, and neural networks. The fundamental idea of role discovery is partitioning the vertices based on their structural features in a graph. However, most existing studies construct the role representations by learning the pairwise connectivity structures, thus weak to represent the relations among multiple vertices, namely higher-order structures. To this effect, this paper proposes a novel approach termed Low-Rank Persistent Probability representation (LRPP) for higher-order role discovery. In detail, we define local Dowker filtration to compute the persistent homology of the neighbor subgraph, which models the connection patterns among vertices and their neighbors. Then, we propose the Persistent Probability (PP) based on Dirichlet equivalence to vectorize the obtained features. Since the persistent probability matrix is sparse, we finally construct the low-rank representation to cluster the higher-order roles by embedding PP vectors into a series of linear subspaces. We conduct experiments of role discovery on both labeled and unlabeled datasets. The experimental results show that the proposed LRPP outperforms the baseline methods in role discovery and demonstrate that LRPP can effectively represent the higher-order role in graphs.