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 productM(t). At variance with pure diffusive processes, characterized by the spectral dimension d(s), the important quantity for reaction spreading 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(dt). In the case of Erdos-Renyi random graphs, the reaction product is characterized by an exponential growth M(t) similar to e(alpha t) with alpha proportional to ln < k >, where < k > is the average degree of the graph.