Efficient (k, α)-Maximal-Cliques Enumeration Over Uncertain Graphs

Abstract

A maximal clique ( ${MC}$ ) is a complete subgraph satisfying that no other cliques can take it as their proper subgraph. Given an uncertain graph, the $top\text{-}K\,\,{MC}\text{s}$ enumeration problem studies how to return $k {MC}\text{s}$ with the highest rank value. Existing algorithms rank ${MC}\text{s}$ according to their probabilities, thus usually return ${MC}\text{s}$ with higher probabilities but less number of vertices, and fail to return large ${MC}\text{s}$ that convey more useful information. Considering this problem, this paper studies the problem of enumerating $top\text{-}K\,\,{MC}\text{s}$ . Our approach returns $k {MC}\text{s}$ with the most number of vertices satisfying that their probabilities $\geq \alpha $ , where each ${MC}$ is called an $\alpha $ - ${MC}$ , and computing $k$ largest $\alpha $ - ${MC}\text{s}$ is called as $(k,\alpha)$ - ${MC}\text{s}$ . We propose an efficient $(k,\alpha)$ - ${MC}\text{s}$ enumeration algorithm, $Top\text{-}KMC$ , which works in three steps, including partition, enumeration and verification. Here, partition means that we compute the set $\mathcal {M}$ of all ${MC}\text{s}$ without considering the probability information, as if the graph is partitioned into a set of subgraphs. Enumeration means that we compute $\alpha $ - ${MC}\text{s}$ from each ${MC}$ of $\mathcal {M}$ . As each such subgraph is an ${MC}$ , the cost of computing common neighbors for finding $\alpha $ - ${MC}\text{s}$ can be reduced. Verification means that we need to verify whether an $\alpha $ - ${MC}$ is a subgraph of another $\alpha $ - ${MC}$ . If not, it is an $\alpha $ - ${MC}$ ; otherwise, it is a useless $\alpha $ - ${MC}$ and should be removed. We further propose an optimized algorithm $Top\text{-}KMC+$ to reduce both time and space by merging the above three steps into a whole step. The experimental results on real datasets show that both $Top\text{-}KMC$ and $Top\text{-}KMC+$ can return $k$ largest $\alpha $ - ${MC}\text{s}$ efficiently.