Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?

The spectral theory of random walks on networks arbitrary topology can be readily extended to study and L\'evy flights subject resetting these structures. When a discrete-time process is stochastically brought back from time its starting node, the mean search needed reach another node network may significantly decreased. In other cases, however, detrimental search. Using eigenvalues eigenvectors transition matrix defining without resetting, we derive general criterion for finite that establishes when there exists non-zero probability minimizes first passage at target node. Right optimality, coefficient variation not unity, unlike in continuous processes with instantaneous but above 1 depends minimal time. approach applicable different ergodic Markov such as flights, where long-range dynamics introduced terms fractional Laplacian graph. We apply results optimal transport rings Cayley trees.