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

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 $1<s<3$.

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• Compatibility, embedding and regularization of non-local random walks on graphs
• Asymptotic spectra of large (grid) graphs with a uniform local structure
• Nonlocal diffusion of variable order on complex networks
• Dynamical systems associated with adjacency matrices
• Indirect Influence on Network Diffusion
• Markov chain approach to anomalous diffusion on Newman-Watts networks