  We introduce a formalism of fractional diffusion on networks based on a
fractional Laplacian matrix that can be constructed directly from the
eigenvalues and eigenvectors of the Laplacian matrix. This fractional approach
allows random walks with long-range dynamics providing a general framework for
anomalous diffusion and navigation, and inducing dynamically the small-world
property on any network. We obtained exact results for the stationary
probability distribution, the average fractional return probability and a
global time, showing that the efficiency to navigate the network is greater if
we use a fractional random walk in comparison to a normal random walk. For the
case of a ring, we obtain exact analytical results showing that the fractional
transition and return probabilities follow a long-range power-law decay,
leading to the emergence of L\'evy flights on networks. Our general fractional
diffusion formalism applies to regular, random and complex networks and can be
implemented from the spectral properties of the Laplacian matrix, providing an
important tool to analyze anomalous diffusion on networks.
