Nucleus Decomposition in Probabilistic Graphs: Hardness and Algorithms

TL;DR

This paper extends nucleus decomposition to probabilistic graphs, analyzing computational hardness and proposing efficient algorithms for local, global, and weakly-global cases, with experimental validation showing improved density and clustering over existing methods.

Contribution

It introduces probabilistic nucleus decomposition, establishes complexity results, and develops scalable algorithms for different cases, including approximations for large graphs.

Findings

Local nucleus decomposition is in PTIME.

Global and weakly-global are #P-hard and NP-hard.

Proposed algorithms outperform existing methods in density and clustering metrics.

Abstract

Finding dense components in graphs is of great importance in analyzing the structure of networks. Popular and computationally feasible frameworks for discovering dense subgraphs are core and truss decompositions. Recently, Sariyuce 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 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. 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.