Enumeration of Maximal Cliques from an Uncertain Graph

Abstract

We consider the enumeration of dense substructures (maximal cliques) from an uncertain graph. For parameter 0 ;a ;1, we define the notion of an a-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of a-maximal cliques possible within a (uncertain) graph. We present an algorithm to enumerate a-maximal cliques whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm. KeywordsRuntime, Proteins, Algorithm design and analysis, Social network services, Uncertainty, Bioinformatics Disciplines Electrical and Computer EngineeringComments This is a manuscript of an article published as Mukherjee, Arko Provo, Pan Xu, and Srikanta Tirthapura. Abstract-We consider the enumeration of dense substructures (maximal cliques) from an uncertain graph. For parameter 0 < a < 1, we define the notion of an a-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of a-maximal cliques possible within a (uncertain) graph. We present an algorithm to enumerate a-maximal cliques whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm.

Efficient Probabilistic K-Core Computation on Uncertain Graphs

“…Upperbound. In [12], the uncertain clique, namely clique defined based on possible worlds semantics is a global model where the probability of an clique is a subgraph where is clique in the possible instance graphs with probability at least . In the deterministic case, we may safely claim that if a node is not a k-core member, it cannot contribute to any ( )-clique because ( )-clique is a k-core.…”

“…In Section II-D and II-E, we further discuss the difference of the models, and show that both models are useful in reallife applications. In particular, we remark that the proposed ( , )-core model is useful in the k-core membership related applications, especially the scenarios in which the possible world semantics is employed in a global way such as influence spread on independent cascade (IC) model [11] and uncertain clique computation [12]. Challenges and Contributions.…”

Enumerating Top-k Quasi-Cliques

“…Much attention has been paid to the problem of enumerating cliques, which are complete dense structures in a graph, e.g. [1], [2], [3], [4], [5], [6]. Usually, however, dense subgraphs are not cliques.…”

Incremental maintenance of maximal cliques in a dynamic graph

“…The current best bound on the time complexity of output-sensitive maximal clique enumeration on a dense graph G = (V, E) is due to [27] which runs with O(M (n)) time delay (the interval between outputting two maximal cliques), where M (n) is the time complexity for multiplying two n×n matrix, which is O(n 2.376 ). Further work in this direction includes [21] and [19], who consider the enumeration of maximal independent sets in lexicographic order, [6], who consider the external memory model, and [31], who consider uncertain graphs. Extensions to parallel frameworks such as MapReduce or MPI are presented in [30,39].…”