Nucleus Decomposition in Probabilistic Graphs: Hardness and Algorithms

Abstract

Finding dense components in graphs is of great importance in analysing the structure of networks. Popular and computationally feasible frameworks for discovering dense subgraphs are core and truss decompositions. Recently, SarÄśyÃijce et al. introduced nucleus decomposition, a generalization which uses higher-order structures and can reveal interesting subgraphs that can be missed by core and truss decompositions.In this paper, we present nucleus decomposition in probabilistic graphs. We study the most interesting case of nucleus decomposition, k-(3, 4)-nucleus, which asks for maximal subgraphs where each triangle is contained in k 4-cliques. The major questions we address are: How to define meaningfully nucleus decomposition in probabilistic graphs? How hard is computing nucleus decomposition in probabilistic graphs? Can we devise efficient algorithms for exact or approximate nucleus decomposition in large graphs?We present three natural definitions of nucleus decomposition in probabilistic graphs: local, global, and weakly-global. We show that the local version is in PTIME, whereas global and weakly-global are #P-hard and NP-hard, respectively. We present an efficient and exact dynamic programming approach for the local case and furthermore, present statistical approximations that can scale to large datasets without much loss of accuracy. For global and weaklyglobal decompositions, we complement our intractability results by proposing efficient algorithms that give approximate solutions based on search space pruning and Monte-Carlo sampling. Our extensive experimental results show the scalability and efficiency of our algorithms. Compared to probabilistic core and truss decompositions, nucleus decomposition significantly outperforms in terms of density and clustering metrics.1 Or (k − 2) triangles depending on the definition.

Theoretically and Practically Efficient Parallel Nucleus Decomposition

“…Sariyüce et al later proposed parallel algorithms for nucleus decomposition based on local computation [55]. Recent work has studied nucleus decomposition in probabilistic graphs [23].…”