  We introduce a non-equilibrium discrete-time random walk model on multiplex
networks, in which at each time step the walker first undergoes a random jump
between neighboring nodes in the same layer, and then tries to hop from one
node to one of its replicas in another layer. We derive the so-called
supra-Markov matrix that governs the evolution of the occupation probability of
the walker. The occupation probability at stationarity is different from the
weighted average over the counterparts on each layer, unless the transition
probabilities between layers vanish. However, they are approximately equal when
the transition probabilities between layers are very small, which is given by
the first-order degenerate perturbation theory. Moreover, we compute the mean
first passage time (MFPT) and the graph MFPT (GrMFPT) that is the average of
the MFPT over all pairs of distinct nodes. Interestingly, we find that the
GrMFPT can be smaller than that of any layer taken in isolation. The result
embodies the advantage of global search on multiplex networks.
