Static Pairwise Annihilation in Complex Networks

TL;DR

This paper develops a mean-field formalism to analyze static pairwise annihilation in complex networks, accurately predicting the dynamics and highlighting the critical role of highly connected nodes in network disintegration.

Contribution

It introduces an exact mean-field approach for disordered networks and applies it to various network types, revealing the impact of node connectivity on annihilation dynamics.

Findings

Higher connectivity leads to faster annihilation.

Hubs in scale-free networks accelerate network disintegration.

The formalism agrees well with numerical simulations.

Abstract

We study static annihilation on complex networks, in which pairs of connected particles annihilate at a constant rate during time. Through a mean-field formalism, we compute the temporal evolution of the distribution of surviving sites with an arbitrary number of connections. This general formalism, which is exact for disordered networks, is applied to Kronecker, Erd\"os-R\'enyi (i.e. Poisson) and scale-free networks. We compare our theoretical results with extensive numerical simulations obtaining excellent agreement. Although the mean-field approach applies in an exact way neither to ordered lattices nor to small-world networks, it qualitatively describes the annihilation dynamics in such structures. Our results indicate that the higher the connectivity of a given network element, the faster it annihilates. This fact has dramatic consequences in scale-free networks, for which, once the ``hubs'' have been annihilated, the network disintegrates and only isolated sites are left.