  We produce a characterization of finite metric spaces which are given by the
effective resistance of a graph. This characterization is applied to the more
general context of resistance metrics defined by Kigami. A countably infinite
resistance metric gives rise to a sequence of finite, increasing graphs with
invariant effective resistance. We show that these graphs have a unique limit
graph in terms of the convergence of edge weights and that their associated
random walks converge weakly to the random walk on the limit graph. If the
limit graph is recurrent, its effective resistance is identified as the initial
resistance metric.

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