  Given the variable-speed random walk on a weighted graph and a metric adapted
to the structure of the random walk, we construct a Brownian motion on a
closely related metric graph which behaves similarly to the VSRW and for which
the associated intrinsic metric has certain desirable properties. Jump
probabilities and moments of jump times for Brownian motion on metric graphs
with varying edge lengths, jump conductances, and edge densities are computed.
We use these results together with a theorem of Sturm for stochastic
completeness, or non-explosiveness, on local Dirichlet spaces to prove sharp
volume growth criteria in adapted metrics for stochastic completeness of
graphs.
