  We develop a path-based approach to continuous-time random walks on networks
with arbitrarily weighted edges. We describe an efficient numerical algorithm
for calculating statistical properties of the stochastic path ensemble. After
demonstrating our approach on two reaction rate problems, we present a
biophysical model that describes how proteins evolve new functions while
maintaining thermodynamic stability. We use our methodology to characterize
dynamics of evolutionary adaptation, reproducing several key features observed
in directed evolution experiments. We find that proteins generally fall into
two qualitatively different regimes of adaptation depending on their binding
and folding energetics.
