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

Tools

Estrada, Ernesto and Hameed, Ehsan and Hatano, Naomichi and Langer, Matthias (2017) Path Laplacian operators and superdiffusive processes on graphs. I. one-dimensional case. Linear Algebra and its Applications, 523. pp. 307-334. ISSN 0024-3795 (https://doi.org/10.1016/j.laa.2017.02.027)

[thumbnail of Estrada-etal-LAA-2017-Path-Laplacian-operators-and-superdiffusive-processes]
Preview
 Text. Filename: Estrada_etal_LAA_2017_Path_Laplacian_operators_and_superdiffusive_processes.pdf
Accepted Author Manuscript
License: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 logo
Download (598kB)| Preview

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 Lk, 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 Σ∞k=1k-sLk produces superdiffusive processes when 1 < s < 3.

ORCID iDs

Estrada, Ernesto ORCID logoORCID: https://orcid.org/0000-0002-3066-7418, Hameed, Ehsan ORCID logoORCID: https://orcid.org/0000-0002-4630-3483, Hatano, Naomichi and Langer, Matthias ORCID logoORCID: https://orcid.org/0000-0001-8813-7914;
CORE (COnnecting REpositories)