The prevalence of graph data has brought a lot of attention to cohesive and dense subgraph mining. In contrast with the large number of indexes proposed to help mine dense subgraphs in general graphs, only very few indexes are proposed for the same in bipartite graphs. In this work, we present the index called $\alpha \mathsf{}\mathrm{(}\beta \mathrm{)}$-core number on vertices, which reflects the maximal cohesive and dense subgraph a vertex can be in, to help enumerate the $\mathrm{(}\alpha \mathrm{,}\beta \mathrm{)}$-cores, a commonly used dense structure in bipartite graphs. To address the problem of extremely high time and space cost for enumerating the $\mathrm{(}\alpha \mathrm{,}\beta \mathrm{)}$-cores, we first present a linear time and space algorithm for computing the $\alpha \mathsf{}\mathrm{(}\beta \mathrm{)}$-core numbers of vertices. We further propose core maintenance algorithms, to update the core numbers of vertices when a graph changes by avoiding recalculations. Experimental results on different real-world and synthetic datasets demonstrate the effectiveness and efficiency of our algorithms.

A public-private-graph (pp-graph) is developed to model social networks with hidden relationships, and it consists of one public graph in which edges are visible to all users, and multiple private graphs in which edges are only visible to its endpoint users. In contrast with conventional graphs where the edges can be visible to all users, it lacks accurate indexes to evaluate the importance of a vertex in a pp-graph. In this paper, we first propose a novel concept, public-private-core (pp-core) number based on the k-core number, which integrally considers both the public graph and private graphs of vertices, to measure how critical a user is. We then give an efficient algorithm for the pp-core number computation, which takes only linear time and space. Considering that the graphs can be always evolving over time, we also present effective algorithms for pp-core maintenance after the graph changes, avoiding redundant re-computation of pp-core number. Extension experiments conducted on real-world social networks show that our algorithms achieve good efficiency and stability. Compared to recalculating the pp-core numbers of all vertices, our maintenance algorithms can reduce the computation time by about 6–8 orders of magnitude.