Characterizing convergence speed is one of the most important research challenges in the design of distributed consensus algorithms for networked multi-agent systems. In this paper, we consider a group of agents that communicate via a dynamically switching random information network. Each link in the network, which represents the directed/undirected information flow between any ordered/unordered pair of agents, could be subject to failure with a certain probability. Hence we model the information flow using dynamically switching random graphs. We characterize the convergence speed for the distributed discrete-time consensus algorithm over a variety of random networks with arbitrary weights. In particular, we propose the asymptotic and per-step (mean square) convergence factors as measures of the convergence speed and derive the exact value for the per-step (mean square) convergence factor. Numerical examples are also given to illustrate our theoretical results. (C) 2009 Elsevier Ltd. All rights reserved.