Fast Maximal Clique Enumeration on Uncertain Graphs: A Pivot-based Approach

Maximal clique enumeration on uncertain graphs is a fundamental problem in uncertain graph analysis. In this paper, we study a problem of enumerating all maximal (𝑘, 𝜂)-cliques on an uncertain graph G, where a vertex set 𝐻 of G is a maximal (𝑘, 𝜂)-clique if (1) 𝐻 (|𝐻 | ≥ 𝑘) is a clique with probability no less than 𝜂, and (2) 𝐻 is a maximal vertex set satisfying (1). The state-of-the-art algorithms for enumerating all maximal (𝑘, 𝜂)-cliques are based on a set enumeration technique which are often very costly. This is because the set enumeration based techniques may explore all subsets of a maximal (𝑘, 𝜂)-clique, thus resulting in many unnecessary computations. To overcome this issue, we propose several novel and efficient pivot-based algorithms to enumerate all maximal (𝑘, 𝜂)cliques based on a newly-developed pivot-based pruning principle. Our pivot-based pruning principle is very general which can be applied to speed up the enumeration of any maximal subgraph that satisfies a hereditary property. Here the hereditary property means that if a maximal subgraph 𝐻 satisfies a property P, any subgraph of 𝐻 also meets P. To the best of our knowledge, our work is the first to systematically explore the idea of pivot for maximal clique enumeration on uncertain graphs. In addition, we also develop a nontrivial size-constraint based pruning technique and a new graph reduction technique to further improve the efficiency. Extensive experiments on nine real-world graphs demonstrate the efficiency, effectiveness, and scalability of the proposed algorithms.