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Reaction spreading on graphs

Dipartimento di Fisica and INFN, Università di Parma, Parco Area delle Scienze 7/A, 43100 Parma, Italy

Institut D'Alembert University Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France and CNRS UMR 7190, 4 Place Jussieu, 75252 Paris Cedex 05, France

Istituto Applicazioni Calcolo, CNR, Viale Manzoni 30, I-00185 Rome, Italy

Dipartimento di Fisica, Università “La Sapienza” and ISC-CNR, Piazzale Aldo Moro 2, I-00185 Rome, Italy

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Phys. Rev. E 86, 055101(R) – Published 26 November, 2012

DOI: https://doi.org/10.1103/PhysRevE.86.055101

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 $d_{s}$ , 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 $M (t) \sim t^{d_{l}}$ . In the case of Erdös-Renyi random graphs, the reaction product is characterized by an exponential growth $M (t) \sim e^{α t}$ with $α$ proportional to $\ln 〈 k 〉$ , where $〈 k 〉$ is the average degree of the graph.

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Reaction spreading on graphs

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