Nucleus Decomposition in Probabilistic Graphs: Hardness and Algorithms

Fatemeh Esfahani, Venkatesh Srinivasan, Alex Thomo, Kui Wu

Abstract: Finding dense components in graphs is of great importance in analysing the structure of networks. Popular frameworks for discovering dense subgraphs are core and truss decompositions. Recently, Sarıyüce et al. introduced nucleus decomposition, which uses <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex> -cliques contained in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$s$</tex> -eliques, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$s &gt; r$</tex> , as the basis for defining dense subgraphs. Nucleus decomposition can reveal interesting subgraphs that can be missed by core and truss decompositions. In this paper, we present nucleus decomposition in probabilistic graphs. 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. Further, we present statistical approximations that can scale to bigger datasets without much loss of accuracy. For global and weakly-global decompositions we complement our intractability results by proposing efficient algorithms that give approximate solutions based on search space pruning and Monte-Carlo sampling. Extensive experiments 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.