In this paper, we study the robust consensus problem for a set of discrete-time linear agents to coordinate over an uncertain communication network, which is to achieve consensus against the transmission errors and noises resulted from the information exchange between the agents. We model the network by means of communication links subject to multiplicative stochastic uncertainties, which are susceptible to describing packet dropout, random delay, and fading phenomena. Different communication topologies, such as undirected graphs and leader-follower graphs, are considered. We derive conditions for robust consensus in the mean square sense. The results show that under a fundamental condition relating the network synchronizability, the unstable agent dynamics, and the channel uncertainty variances, the consensus can be ensured and consensus protocols can be designed based on the state information transmitted over the uncertain channels, by solving a modified algebraic Riccati equation. Furthermore, this condition is guaranteed to hold whenever the agents contain no eigenvalues strictly outside the unit circle and is thus necessary and sufficient under the circumstance.