Ernesto Estrada - University of Strathclyde, Glasgow -"Long-range influences and dynamics on networks"

I will start with a motivation for the necessity of including long-range influences
on the study of dynamical processes on networks. I will provide some experimental evidences
supporting the existence of long-range hops in the diffusion of atoms and molecules
adsorbed on metallic surfaces. Then, I will generalise the Laplacian operator of a network
to account for long-range hops in graphs. I will define this operator on a Hilbert space
and prove that it is bounded and self-adjoint. At this point I will make a generalisation of
the diffusion equation by using Laplace and Mellin transformations of the d-path Laplacian
operators. I will prove analytically that this generalised diffusion equation produces
super-diffusive processes when certain parameters in the Mellin transform are used. Finally,
I will illustrate the generalisation of other well-known equations for networks, which
includes the Kuramoto model, and the epidemic spreading models. In this last case I will
show how epidemic propagating on plants are subject to long-range dispersals and how
this influences dramatically the epidemic threshold and spatial patterns of the disease.