  Several variants of the graph Laplacian have been introduced to model
non-local diffusion processes, which allow a random walker to
{\textquotedblleft jump\textquotedblright} 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 $\hat{G}$ can
not be embedded into the normalized graph $\hat{G}'$. Eventually, we provide a
regularization method to guarantee such compatibility and preserving at the
same time all the nice properties granted by $G'$.
