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.
- k-path Laplacian
- anomalous diffusion
Estrada, E., Hameed, E., Hatano, N., & Langer, M. (2017). Path Laplacian operators and superdiffusive processes on graphs. I. one-dimensional case. Linear Algebra and its Applications, 523, 307-334. https://doi.org/10.1016/j.laa.2017.02.027