  Large unweighted directed graphs are commonly used to capture relations
between entities. A fundamental problem in the analysis of such networks is to
properly define the similarity or dissimilarity between any two vertices.
Despite the significance of this problem, statistical characterization of the
proposed metrics has been limited. We introduce and develop a class of
techniques for analyzing random walks on graphs using stochastic calculus.
Using these techniques we generalize results on the degeneracy of hitting times
and analyze a metric based on the Laplace transformed hitting time (LTHT). The
metric serves as a natural, provably well-behaved alternative to the expected
hitting time. We establish a general correspondence between hitting times of
the Brownian motion and analogous hitting times on the graph. We show that the
LTHT is consistent with respect to the underlying metric of a geometric graph,
preserves clustering tendency, and remains robust against random addition of
non-geometric edges. Tests on simulated and real-world data show that the LTHT
matches theoretical predictions and outperforms alternatives.
