  In this paper, we study nonlocal random walk strategies generated with the
fractional Laplacian matrix of directed networks. We present a general approach
to analyzing these strategies by defining the dynamics as a discrete-time
Markovian process with transition probabilities between nodes expressed in
terms of powers of the Laplacian matrix. We analyze the elements of the
transition matrices and their respective eigenvalues and eigenvectors, the mean
first passage times and global times to characterize the random walk
strategies. We apply this approach to the study of particular local and
nonlocal ergodic random walks on different directed networks; we explore
circulant networks, the biased transport on rings and the dynamics on random
networks. We study the efficiency of a fractional random walker with bias on
these structures. Effects of ergodicity loss which occur when a directed
network is not any more strongly connected are also discussed.
