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In this paper, we study nonlocal random walk strategies generated with thefractional Laplacian matrix of directed networks. We present a general approachto analyzing these strategies by defining the dynamics as a discrete-timeMarkovian process with transition probabilities between nodes expressed interms of powers of the Laplacian matrix. We analyze the elements of thetransition matrices and their respective eigenvalues and eigenvectors, the meanfirst passage times and global times to characterize the random walkstrategies. We apply this approach to the study of particular local andnonlocal ergodic random walks on different directed networks; we explorecirculant networks, the biased transport on rings and the dynamics on randomnetworks. We study the efficiency of a fractional random walker with bias onthese structures. Effects of ergodicity loss which occur when a directednetwork is not any more strongly connected are also discussed.

Nonlocal biased random walks and fractional transport on directed networks