Diffusion-annihilation processes in complex networks

Abstract

We present a detailed analytical study of the $A+A\to \varnothing$ 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.

9 pages, 5 EPS figures

References in corpus (23)

• Emergence of scaling in random networks
• Statistical mechanics of complex networks
• Epidemic spreading in scale-free networks
• Assortative mixing in networks
• The large-scale organization of metabolic networks
• Evolution of networks
• Classes of behavior of small-world networks
• Network robustness and fragility: Percolation on random graphs
• Resilience of the Internet to random breakdowns
• Evolution of the social network of scientific collaborations
• Epidemic dynamics and endemic states in complex networks
• Dynamical and correlation properties of the Internet
• Breakdown of the Internet under intentional attack
• The Web of Human Sexual Contacts
• Generation of uncorrelated random scale-free networks
• Large-scale topological and dynamical properties of Internet
• Epidemic dynamics in finite size scale-free networks
• Epidemic spreading in correlated complex networks
• Pattern Detection in Complex Networks: Correlation Profile of the Internet
• Cut-offs and finite size effects in scale-free networks
• Renormalization Group Calculation for the Reaction $kA\to \varnothing$
• The origin of degree correlations in the Internet and other networks
• Absence of kinetic effects in reaction-diffusion processes in scale-free networks

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• Phase transitions with infinitely many absorbing states in complex networks
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• Hierarchical invasion of cooperation in complex networks
• Segregation in the annihilation of two-species reaction-diffusion processes on fractal scale-free networks
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• Scaling in local to global condensation of wealth on sparse networks
• A mechanism to synchronize fluctuations in scale free networks using growth models
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