Discovering Hierarchical Subgraphs of K-Core-Truss

Discovering dense subgraphs in a graph is a fundamental graph mining task, which has a wide range of applications in social networks, biology and visualization to name a few. Even the problem of computing most cohesive subgraphs is NP-hard (like clique, quasi-clique, k -densest subgraph), there exists a polynomial time algorithm for computing the k -core and k -truss. In this paper, we propose a novel dense subgraph model, 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 , which leverages on a new type of important edges based on the basis of k -core and k -truss. We investigate the structural properties of the 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 model. Compared to k -core and k -truss, 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 can significantly discover the interesting and important structural information out the scope of k -core and k -truss. We study two useful problems of 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 decomposition and 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 search. In particular, we develop a k -core-truss decomposition algorithm to find all 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 in a graph G by iteratively removing edges with the smallest 𝖽𝖾𝗀𝗋𝖾𝖾 - 𝗌𝗎𝗉𝗉𝗈𝗋𝗍 . In addition, we offer a 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 search algorithm to identifying a particular 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 containing a given query node such that the core number k is the largest. Extensive experiments on several web-scale real-world datasets show the effectiveness and efficiency of 𝗄 - 𝖼𝗈𝗋𝖾 - 𝗍𝗋𝗎𝗌𝗌 model and proposed algorithms.