Reaction Spreading on Graphs

TL;DR

This paper investigates reaction spreading on graphs, revealing that the growth of reaction products depends on the connectivity dimension, with different behaviors on random graphs compared to pure diffusion.

Contribution

It introduces a reaction-diffusion framework on graphs and identifies the connectivity dimension as key to reaction spreading, providing analytical and numerical insights.

Findings

Reaction spreading scales as t^{d_l} with the connectivity dimension.

On Erdős-Rényi graphs, reaction products grow exponentially with a rate proportional to ln<k>.

Numerical data agree with analytical estimates based on random walk features.

Abstract

We study reaction-diffusion processes on graphs through an extension of the standard reaction-diffusion equation starting from first principles. We focus on reaction spreading, i.e. on the time evolution of the reaction product, M(t). At variance with pure diffusive processes, characterized by the spectral dimension, d_s, for reaction spreading the important quantity is found to be the connectivity dimension, d_l. Numerical data, in agreement with analytical estimates based on the features of n independent random walkers on the graph, show that M(t) ~ t^{d_l}. In the case of Erdos-Renyi random graphs, the reaction-product is characterized by an exponential growth M(t) ~ e^{a t} with a proportional to ln<k>, where <k> is the average degree of the graph.