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

TL;DR

This paper investigates the mathematical properties of non-local random walks on graphs, highlighting the incompatibility between original and transformed graphs, and proposes a regularization method to ensure compatibility while maintaining desirable properties.

Contribution

It demonstrates the incompatibility of transformed complete graphs with original graph dynamics and introduces a regularization technique to restore compatibility.

Findings

G' is generally incompatible with G for non-local random walks.

Normalized graphs rom G and G' cannot be embedded into each other.

A regularization method is proposed to ensure compatibility while preserving properties.

Abstract

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 $G'$.