Path Laplacian operators and superdiffusive processes on graphs. I. One-dimensional case

TL;DR

This paper introduces a generalized diffusion equation on one-dimensional graphs using $k$-path Laplacian operators, demonstrating conditions under which superdiffusive processes occur, expanding understanding of diffusion dynamics on graphs.

Contribution

It proves the self-adjointness of $k$-path Laplacians and analyzes their transforms, showing Mellin-transformed operators induce superdiffusion for certain parameters.

Findings

Laplace- and factorial-transformed operators produce normal diffusion.

Mellin-transformed $k$-path Laplacians induce superdiffusion when 1<s<3.

Self-adjointness of $k$-path Laplacians established.

Abstract

We consider a generalization of the diffusion equation on graphs. This generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one-dimensional graphs. The generalization is based on the $k$-path Laplacian operators $L_{k}$, which account for the hop of a diffusive particle to non-nearest neighbours in a graph. We first prove that the $k$-path Laplacian operators are self-adjoint. Then, we study the transformed $k$-path Laplacian operators using Laplace, factorial and Mellin transforms. We prove that the generalized diffusion equation using the Laplace- and factorial-transformed operators always produce normal diffusive processes independently of the parameters of the transforms. More importantly, the generalized diffusion equation using the Mellin-transformed $k$-path Laplacians $\sum_{k=1}^{\infty}k^{-s}L_{k}$ produces superdiffusive processes when $1<s<3$.