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The minimum connected dominating set (MCDS) of a given graph G is the smallest sub-graph of G such that every vertex in G belongs either to the sub-graph or is adjacent to a vertex of the sub-graph. Finding the MCDS in an arbitrary graph is a NP-Hard problem, and several approximation algorithms have been proposed for solving this problem in deterministic graphs, but to the best of our knowledge no work has been done on finding the MCDS in stochastic graphs. In this paper, the MCDS problem in the stochastic graphs is first introduced, and then a learning automata-based approximation algorithm called SCDS is proposed for solving this problem when the probability distribution function of the vertex weight is unknown. It is shown that by a proper choice of the parameters of the proposed algorithm, the probability with which the proposed algorithm find the MCDS is close enough to unity. The simulation results show the efficiency of the proposed algorithm in terms of the number of samplings.