Affiliations: [a] Faculty of Computer Science and Engineering, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi, Pakistan
| [b] Department of Computer Engineering, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Pakistan
Correspondence:
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Corresponding author. Zahid Halim, Faculty of Computer Science and Engineering, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi, Pakistan. E-mail: [email protected].
Abstract: A graph is a network that can represent the communication between a variety of data elements. The data can have uncertainty, primarily due to the heterogeneity of data sources. Moreover, it is sometimes difficult to assure the existence of a link between data elements; compelling to consider the data as a probabilistic entity. Extracting densely connected regions from a graph is a key task of the intelligent systems. The enumeration of dense substructures in a graph can help to identify important patterns. This can have many applications in medical image processing, accident analysis, and surveillance, to name a few. One such dense substructure is a clique, where all nodes are directly connected to each other. An α-maximal clique in an uncertain graph is a clique with a minimum probability α, such that it is not a subset of any other clique of the same weight. Extracting all α-maximal cliques is an NP-Complete problem. This work focuses on reducing the time consumed to enumerate all α-maximal cliques in a graph. Another focus of this work is to reduce the CPU (Central Processing Unit) cycles for efficient enumeration of all α-maximal cliques. An algorithm is proposed that computes all weighted maximal cliques in an uncertain graph. The worst-case asymptotic time complexity of the algorithm is O(n2n). The proposed algorithm utilizes the h-index concept to form cliques with vertex degree greater than h. The algorithm builds cliques at two levels of enumeration. The first level finds the α-maximal cliques with a descending order in sizes. On each successive α-maximal clique iteration of the first level, the second level tracks and deletes all subsets of the clique. The second level is to ensure the fact that all subsets of an α-maximal clique are cliques. The proposed algorithm is compared with two recent maximal clique enumeration algorithms, namely: MULE (Maximal Uncertain Clique Enumeration) and LMC (Listing all maximal cliques in large sparse real-world graphs). Real-world benchmark uncertain graphs are utilized for the experimental evaluation. The results suggest better performance of the proposed approach in terms of the time consumption.
Keywords: Graphs, clique, enumeration, uncertain data
