Reaction spreading on graphs

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 $M(t) \sim t^{d_l}$. In the case of Erd\"{o}s-Renyi random graphs, the reaction-product is characterized by an exponential growth $M(t) \sim e^{\alpha t}$ with $\alpha$ proportional to $\ln \lra{k}$, where $\lra{k}$ is the average degree of the graph.