We study the mean time for a random walk to traverse between two arbitrary sites of the Erdos-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful

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... tity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-monotonically with p near the percolation transition. Much of this behavior can be understood by simple heuristic arguments.