Core Decomposition on Uncertain Graphs Revisited

Qiangqiang Dai, Rong-Hua Li, Guoren Wang, Rui Mao, Zhiwei Zhang, Ye Yuan

Abstract: Core decomposition on uncertain graphs is a fundamental problem in graph analysis. Given an uncertain graph <inline-formula><tex-math notation="LaTeX">$\mathcal {G}$</tex-math></inline-formula> , the core decomposition problem is to determine all <inline-formula><tex-math notation="LaTeX">$(k,\eta)\text{-cores}$</tex-math></inline-formula> in <inline-formula><tex-math notation="LaTeX">$\mathcal {G}$</tex-math></inline-formula> , where a <inline-formula><tex-math notation="LaTeX">$(k,\eta)\text{-core}$</tex-math></inline-formula> is a maximal subgraph of <inline-formula><tex-math notation="LaTeX">$\mathcal {G}$</tex-math></inline-formula> such that each node has an <inline-formula><tex-math notation="LaTeX">$\eta \text{-}$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">${\mathsf {degree}}$</tex-math></inline-formula> no less than <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> within the subgraph. The <inline-formula><tex-math notation="LaTeX">$\eta \text{-}$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">${\mathsf {degree}}$</tex-math></inline-formula> of a node <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> is defined as the maximum integer <inline-formula><tex-math notation="LaTeX">$r$</tex-math></inline-formula> such that the probability that <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> has a degree no less than <inline-formula><tex-math notation="LaTeX">$r$</tex-math></inline-formula> is larger than or equal to the threshold <inline-formula><tex-math notation="LaTeX">$\eta \in [0,1]$</tex-math></inline-formula> . The state-of-the-art algorithm for solving this problem is based on a peeling technique which iteratively removes the nodes with the smallest <inline-formula><tex-math notation="LaTeX">$\eta \text{-}$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">${\mathsf {degrees}}$</tex-math></inline-formula> and also dynamically updates their neighbors’ <inline-formula><tex-math notation="LaTeX">$\eta \text{-}$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">${\mathsf {degrees}}$</tex-math></inline-formula> . Unfortunately, we find that such a peeling algorithm with the dynamical <inline-formula><tex-math notation="LaTeX">$\eta \text{-}$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">${\mathsf {degree}}$</tex-math></inline-formula> updating technique is incorrect due to the inaccuracy of the recursive floating-point number division operations involved in the dynamical updating procedure. To correctly compute the <inline-formula><tex-math notation="LaTeX">$(k,\eta)\text{-cores}$</tex-math></inline-formula> , we first propose a bottom-up algorithm based on an on-demand <inline-formula><tex-math notation="LaTeX">$\eta \text{-}$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">${\mathsf {degree}}$</tex-math></inline-formula> computational strategy. To further improve the efficiency, we also develop a more efficient top-down algorithm with several nontrivial optimization techniques. Both of our algorithms do not involve any floating-point number division operations, thus the correctness can be guaranteed. In addition, we also develop the parallel variants of all the proposed algorithms. Finally, we conduct extensive experiments to evaluate the proposed algorithms using five large real-life datasets. The results show that our algorithms are at least three orders of magnitude faster than the existing exact algorithms on large uncertain graphs. The results also demonstrate the high scalability and parallel performance of the proposed algorithms.