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In this paper, we explore different Markovian random walk strategies onnetworks with transition probabilities between nodes defined in terms offunctions of the Laplacian matrix. We generalize random walk strategies withlocal information in the Laplacian matrix, that describes the connections of anetwork, to a dynamics determined by functions of this matrix. The resultingprocesses are non-local allowing transitions of the random walker from one nodeto nodes beyond its nearest neighbors. We find that only two types of Laplacianfunctions are admissible with distinct behaviors for long-range steps in theinfinite network limit: type (i) functions generate Brownian motions, type (ii)functions L\'evy flights. For this asymptotic long-range step behavior only thelowest non-vanishing order of the Laplacian function is relevant, namely firstorder for type (i), and fractional order for type (ii) functions.  In the first part, we discuss spectral properties of the Laplacian matrix anda series of relations that are maintained by a particular type of functionsthat allow to define random walks on any type of undirected connected networks.Once described general properties, we explore characteristics of random walkstrategies that emerge from particular cases with functions defined in terms ofexponentials, logarithms and powers of the Laplacian as well as relations ofthese dynamics with non-local strategies like L\'evy flights and fractionaltransport. Finally, we analyze the global capacity of these random walkstrategies to explore networks like lattices and trees and different types ofrandom and complex networks.

Random walks with long-range steps generated by functions of Laplacian matrices