E. Estrada, E. Hameed, N. Hatano and M. Langer,
Path Laplacian operators and superdiffusive processes on graphs. I.
One-dimensional case,
Linear Algebra Appl. 523 (2017), 307–334
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.