  Continuous-time random walks (CTRWs) on discrete state spaces, ranging from
regular lattices to complex networks, are ubiquitous across physics, chemistry,
and biology. Models with coarse-grained states, for example those employed in
studies of molecular kinetics, and models with spatial disorder can give rise
to memory and non-exponential distributions of waiting times and first-passage
statistics. However, existing methods for analyzing CTRWs on complex energy
landscapes do not address these effects. We therefore use statistical mechanics
of the nonequilibrium path ensemble to characterize first-passage CTRWs on
networks with arbitrary connectivity, energy landscape, and waiting time
distributions. Our approach is valuable for calculating higher moments (beyond
the mean) of path length, time, and action, as well as statistics of any
conservative or non-conservative force along a path. For homogeneous networks
we derive exact relations between length and time moments, quantifying the
validity of approximating a continuous-time process with its discrete-time
projection. For more general models we obtain recursion relations, reminiscent
of transfer matrix and exact enumeration techniques, to efficiently calculate
path statistics numerically. We have implemented our algorithm in PathMAN, a
Python script that users can easily apply to their model of choice. We
demonstrate the algorithm on a few representative examples which underscore the
importance of non-exponential distributions, memory, and coarse-graining in
CTRWs.
