Counting Cohesive Subgraphs with Hereditary Properties

The classic clique model has properties of hereditaries and cohesiveness. Here hereditaries means a subgraph of a clique is still a clique. Counting small cliques in a graph is a fundamental operation of numerous applications. However, the clique model is often too restrictive for practical use, leading to the focus on other relaxed-cliques with properties of hereditaries and cohesiveness. To address this issue, we investigate a new problem of counting general hereditary cohesive subgraphs (HCS). All subgraphs with properties of hereditaries and cohesiveness can be called a kind of HCS. To count HCS, we propose a general framework called HCSPivot, which can be applied to count all kinds of HCS. HCSPivot can count most HCS combinatorially without explicitly listing them. Two additional noteworthy features of HCSPivot are its ability to (1) simultaneously count HCS of any size and (2) simultaneously count HCS for each node or each edge. Based on our HCSPivot framework, we propose two novel algorithms with several carefully designed pruning techniques to count s-defective cliques and s-plexes, which are two specific types of HCS. We conduct extensive experiments on 8 large real-world graphs, and the results demonstrate the high efficiency and effectiveness of our solutions.