Compatibility, embedding and regularization of non-local random walks on graphs

Several variants of the graph Laplacian have been introduced to model non-local diffusion processes, which allow a random walker to “jump” to non-neighborhood nodes, most notably the transformed path graph Laplacians and the fractional graph Laplacian. From a rigorous point of view, this new dynamics is made possible by having replaced the original graph G with a weighted complete graph G′ on the same node-set, that depends on G and wherein the presence of new edges allows a direct passage between nodes that were not neighbors in G. We show that, in general, the graph G′ is not compatible with the dynamics characterizing the original model graph G: the random walks on G′ subjected to move on the edges of G are not stochastically equivalent, in the wide sense, to the random walks on G. From a purely analytical point of view, the incompatibility of G′ with G means that the normalized graph Gˆ can not be embedded into the normalized graph Gˆ′. Eventually, we provide a regularization method to guarantee such compatibility and preserving at the same time all the nice properties granted by G′.