Counting Cohesive Subgraphs with Hereditary Properties

Counting small cohesive subgraphs in a graph is a fundamental operation with numerous applications in graph analysis. Previous studies on cohesive subgraph counting are mainly based on the clique model, which aim to count the number of $k$-cliques in a graph with a small $k$. However, the clique model often proves too restrictive for practical use. To address this issue, we investigate a new problem of counting cohesive subgraphs that adhere to the hereditary property. Here the hereditary property means that if a graph $G$ has a property $\mathcal{P}$, then any induced subgraph of $G$ also has a property $\mathcal{P}$. To count these hereditary cohesive subgraphs (\hcss), we propose a new listing-based framework called \hcslist, which employs a backtracking enumeration procedure to count all \hcss. A notable limitation of \hcslist is that it requires enumerating all \hcss, making it intractable for large and dense graphs due to the exponential growth in the number of \hcss with respect to graph size. To overcome this limitation, we propose a novel pivot-based framework called \hcspivot, which can count most \hcss in a combinatorial manner without explicitly listing them. Two additional noteworthy features of \hcspivot is its ability to (1) simultaneously count \hcss of any size and (2) simultaneously count \hcss for each vertex or each edge, while \hcslist is only capable of counting a specific size of \hcs and obtaining a total count of \hcss in a graph. We focus specifically on two \hcs: $s$-defective clique and $s$-plex, with several non-trivial pruning techniques to enhance the efficiency. We conduct extensive experiments on 8 large real-world graphs, and the results demonstrate the high efficiency and effectiveness of our solutions.