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