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This paper investigates the average-consensus problem of first-order discrete-time multi-agent networks in uncertain communication environments. Each agent can only use its own and neighbors' information to design its control input. To attenuate the communication noises, a distributed stochastic approximation type protocol is used. By using probability limit theory and algebraic graph theory, consensus conditions for this kind of protocols are obtained: (A) For the case of fixed topologies, a necessary and sufficient condition for mean square average-consensus is given, which is also sufficient for almost sure consensus. (B) For the case of time-varying topologies, sufficient conditions for mean square average-consensus and almost sure consensus are given, respectively. Especially, if the network switches between jointly-containing-spanning-tree, instantaneously balanced graphs, then the designed protocol can guarantee that each individual state converges, both almost surely and in mean square, to a common random variable, whose expectation is right the average of the initial states of the whole system, and whose variance describes the static maximum mean square error between each individual state and the average of the initial states of the whole system.