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We study the convergence rate of average consensus algorithms in networks with stochastic communication failures. We show how the system dynamics can be modeled by a discrete-time linear system with multiplicative random coefficients. This formulation captures many types of random networks including networks with links failures, node failures, and network partitions. With this formulation, we use first-order spectral perturbation analysis to analyze the mean-square convergence rate under various network conditions. Our analysis reveals that in large networks, the effect of communication failures on the convergence rate is similar to the effect of changing the weight assigned to the communication links. We also show that in large networks, when the probability of communication failure is small, correlation in communication failures plays a negligible role in the convergence rate of the algorithm.