TY - JOUR
T1 - Path Laplacian operators and superdiffusive processes on graphs. I. one-dimensional case
AU - Estrada, Ernesto
AU - Hameed, Ehsan
AU - Hatano, Naomichi
AU - Langer, Matthias
PY - 2017/6/15
Y1 - 2017/6/15
N2 - 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.
AB - 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.
KW - k-path Laplacian
KW - anomalous diffusion
UR - https://www.sciencedirect.com/journal/linear-algebra-and-its-applications
U2 - 10.1016/j.laa.2017.02.027
DO - 10.1016/j.laa.2017.02.027
M3 - Article
VL - 523
SP - 307
EP - 334
JO - Linear Algebra and its Applications
JF - Linear Algebra and its Applications
SN - 0024-3795
ER -