We consider the effect of network throughput on the convergence of a specific class of distributed averaging algorithms, called consensus algorithms. These algorithms rely on iterative computation of the desired average by message passing among the nodes. It is thus assumed that the rate of convergence should benefit from greater network connectivity. However, one must also account for the additional network resources that establishing such a connectivity would entail. In this paper, we study this problem in the context of randomly-placed consensusseeking nodes that are connected through a dense wireless network, i.e., whose capacity is interference-limited. By analyzing the outage of each communication link along with results from mixing times of Markov chains, we obtain scaling laws for the mixing times of fastest-converging consensus topologies over such networks.