HRL Laboratories, LLC, Malibu, California 90265, USA.
Phys Rev E. 2018 Feb;97(2-1):022313. doi: 10.1103/PhysRevE.97.022313.
The general mechanisms behind self-organized criticality (SOC) are still unknown. Several microscopic and mean-field theory approaches have been suggested, but they do not explain the dependence of the exponents on the underlying network topology of the SOC system. Here, we first report the phenomena that in the Bak-Tang-Wiesenfeld (BTW) model, sites inside an avalanche area largely return to their original state after the passing of an avalanche, forming, effectively, critically arranged clusters of sites. Then, we hypothesize that SOC relies on the formation process of these clusters, and present a model of such formation. For low-dimensional networks, we show theoretically and in simulation that the exponent of the cluster-size distribution is proportional to the ratio of the fractal dimension of the cluster boundary and the dimensionality of the network. For the BTW model, in our simulations, the exponent of the avalanche-area distribution matched approximately our prediction based on this ratio for two-dimensional networks, but deviated for higher dimensions. We hypothesize a transition from cluster formation to the mean-field theory process with increasing dimensionality. This work sheds light onto the mechanisms behind SOC, particularly, the impact of the network topology.
自组织临界性（SOC）背后的一般机制尚不清楚。已经提出了几种微观和平均场理论方法，但它们不能解释指数对 SOC 系统基础网络拓扑的依赖性。在这里，我们首先报告了这样的现象，即在 Bak-Tang-Wiesenfeld（BTW）模型中，在一个雪崩区内部的站点在雪崩经过后很大程度上会回到其原始状态，从而形成实际上是临界排列的站点簇。然后，我们假设 SOC 依赖于这些簇的形成过程，并提出了一个这样的形成模型。对于低维网络，我们从理论和模拟上表明，簇大小分布的指数与簇边界的分形维数与网络维度的比值成正比。对于 BTW 模型，在我们的模拟中，雪崩区分布的指数与我们基于此比值对二维网络的预测大致匹配，但对于更高的维度则存在偏差。我们假设随着维度的增加，从簇的形成到平均场理论过程的转变。这项工作揭示了 SOC 背后的机制，特别是网络拓扑的影响。
