  We generalize the weak Omori-Yau maximum principle to the setting of strongly
local Dirichlet forms. As an application, we obtain an analytic approach to
compare the stochastic completeness of a weighted graph with that of an
associated metric graph. This comparison result played an essential role in the
volume growth criterion of Folz \cite{FOLZSC}, who first proved it via a
probabilistic approach. We also give an alternative analytic proof based on a
criterion in Fukushima, Oshima, and Takeda \cite{FOT}.
