  We study the probability that a random walk started inside a subgraph of a
larger graph exits that subgraph (or, equivalently, hits the exterior boundary
of the subgraph). Considering the chance a random walk started in the subgraph
never leaves the subgraph leads to a notion we call "survival" transience, or
$S$-transience. In the case where the heat kernel of the larger graph satisfies
two-sided Gaussian estimates, we prove an upper bound on the probability of
hitting the boundary of the subgraph. Under the additional hypothesis that the
subgraph is inner uniform, we prove a two-sided estimate for this probability.
The estimate depends upon a harmonic function in the subgraph. We also provide
two-sided estimates for related probabilities, such as the harmonic measure
(the chance the walk exits the subgraph at a particular point on its boundary).
