Diffusion-annihilation processes in complex networks

TL;DR

This paper analytically investigates the $A+A o ext{empty}$ diffusion-annihilation process in complex networks, revealing how particle density decay varies with network heterogeneity and size, supported by numerical simulations.

Contribution

It derives a set of rate equations for particle density in complex networks and solves them for uncorrelated networks, highlighting differences between homogeneous and scale-free networks.

Findings

Homogeneous networks show inverse time decay of particle density.

Scale-free networks exhibit power-law decay with an exponent depending on degree distribution.

Finite size effects lead to mean-field behavior in finite scale-free networks.

Abstract

We present a detailed analytical study of the $A+A\to\emptyset$ diffusion-annihilation process in complex networks. By means of microscopic arguments, we derive a set of rate equations for the density of $A$ particles in vertices of a given degree, valid for any generic degree distribution, and which we solve for uncorrelated networks. For homogeneous networks (with bounded fluctuations), we recover the standard mean-field solution, i.e. a particle density decreasing as the inverse of time. For heterogeneous (scale-free networks) in the infinite network size limit, we obtain instead a density decreasing as a power-law, with an exponent depending on the degree distribution. We also analyze the role of finite size effects, showing that any finite scale-free network leads to the mean-field behavior, with a prefactor depending on the network size. We check our analytical predictions with extensive numerical simulations on homogeneous networks with Poisson degree distribution and scale-free networks with different degree exponents.