Catanzaro Michele, Boguñá Marián, Pastor-Satorras Romualdo
Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 May;71(5 Pt 2):056104. doi: 10.1103/PhysRevE.71.056104. Epub 2005 May 10.
We present a detailed analytical study of the A+A --> 0 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.
我们对复杂网络中A + A→0扩散湮灭过程进行了详细的分析研究。通过微观论证，我们推导了一组关于给定度数顶点中A粒子密度的速率方程，该方程对任何一般度数分布均有效，并且我们求解了不相关网络的情况。对于均匀网络（波动有界），我们得到了标准的平均场解，即粒子密度随时间的倒数减小。对于无限网络规模极限下的异质（无标度网络），我们得到的密度随幂律减小，其指数取决于度数分布。我们还分析了有限尺寸效应的作用，表明任何有限的无标度网络都会导致平均场行为，其前置因子取决于网络规模。我们通过对具有泊松度数分布的均匀网络和具有不同度数指数的无标度网络进行广泛的数值模拟来检验我们的分析预测。
