  Consider a symmetric Markovian jump process $\{X_t\}$ on a metric measure
space $(M, d, \mu)$. Chen, Kumagai, and Wang recently showed that two-sided
heat kernel estimates and the parabolic Harnack inequality are both stable
under bounded perturbations of the jumping measure, assuming $(M, d, \mu)$
satisfies the volume-doubling and reverse-volume-doubling conditions. These
results do not apply if $(M, d, \mu)$ is a graph (or more generally, if $M$
contains any atoms $x$ such that $\mu(x)>0$) because it is impossible for
reverse-volume-doubling to hold on a space with atoms. We generalize the
results of Chen, Kumagai, and Wang to a larger class of metric measure spaces,
including all infinite graphs with volume-doubling. Our main tool is the
construction of an "auxiliary space" that smooths out the atoms. We show that
many properties transfer from $(M, d, \mu)$ to the auxiliary space, and vice
versa, including heat kernel estimates, the parabolic Harnack inequality, and
their stable characterizations.
