Finding Minimum Vertex Covering in Stochastic Graphs: A Learning Automata Approach

Structural and behavioral parameters of many real networks such as social networks are unpredictable, uncertain and time varying parameters and for these reasons deterministic graphs for modeling such networks are too restrictive to solve most of the real network problems. It seems that stochastic graphs, in which weights associated to the vertices are random variable, may be a better graph model for real world networks. Once we use stochastic graph as the model for a network, every feature of the graph such as path, spanning tree, clique, dominating set and cover set should be treated as a stochastic feature. For example, choosing stochastic graph as a graph model of an online social network and defining community structure in terms of clique, the concept of stochastic clique may be used to study community structures properties or defining influence spreading according to the coverage of influential users, the concept of stochastic vertex covering may be used to study spread of influence. In this paper, minimum vertex covering in stochastic graphs is first defined and then four learning automata-based algorithms for solving minimum vertex covering problem in stochastic graphs where the probability distribution functions of the weights associated with the vertices of the graph are unknown are proposed. It is shown that by a proper choice of the parameters of the proposed algorithms, one can make the probability of finding minimum vertex cover in stochastic graph as close to unity as possible. Experimental results on synthetic stochastic graphs reveal that at a certain confidence level the proposed algorithms significantly outperform standard sampling method in terms of the number of samples needed to be taken from the vertices of the stochastic graph.