  The first-passage time (FPT) is the time it takes a system variable to cross
a given boundary for the first time. In the context of Markov networks, the FPT
is the time a random walker takes to reach a particular node (target) by
hopping from one node to another. If the walker pauses at each node for a
period of time drawn from a continuous distribution, the FPT will be a
continuous variable; if the pauses last exactly one unit of time, the FPT will
be discrete and equal to the number of hops. We derive an exact analytical
expression for the discrete first-passage time (DFPT) in Markov networks. Our
approach is as follows: first, we divide each edge (connection between two
nodes) of the network into $h$ unidirectional edges connecting a cascade of $h$
fictitious nodes and compute the continuous FPT (CFPT). Second, we set the
transition rates along the edges to $h$, and show that as $h\to\infty$, the
distribution of travel times between any two nodes of the original network
approaches a delta function centered at 1, which is equivalent to pauses
lasting 1 unit of time. Using this approach, we also compute the
joint-probability distributions for the DFPT, the target node, and the node
from which the target node was reached. A comparison with simulation confirms
the validity of our approach.
