MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland.
Department of Mathematics, Duke University, Durham, NC, 27708, USA.
Nat Commun. 2017 Oct 31;8(1):1227. doi: 10.1038/s41467-017-01212-0.
An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the temporal profiles of avalanches of fixed duration. At the critical point of the dynamics, the rescaled average avalanche shapes for different durations collapse onto a single universal curve. We apply Markov branching process theory to derive an equation governing the average avalanche shape for cascade dynamics on networks. Analysis of the equation at criticality demonstrates that nonsymmetric average avalanche shapes (as observed in some experiments) occur for certain combinations of dynamics and network topology. We give examples using numerical simulations of models for information spreading, neural dynamics, and behavior adoption and we propose simple experimental tests to quantify whether cascading systems are in the critical state.
当一个事件导致一个或多个后续事件，而这些后续事件又可能在链式反应中引发进一步的事件时，就会发生雪崩或级联现象。在许多学科中都研究了雪崩动力学，最近的研究重点是平均雪崩形状，即在固定持续时间内的雪崩的时间分布。在动力学的临界点，不同持续时间的归一化平均雪崩形状会坍塌到一个单一的通用曲线上。我们应用马尔可夫分支过程理论，推导出一个用于描述网络上级联动力学的平均雪崩形状的方程。在临界点处对该方程的分析表明，对于某些动力学和网络拓扑的组合，会出现非对称的平均雪崩形状（如在一些实验中观察到的那样）。我们通过对信息传播、神经动力学和行为采用模型的数值模拟给出了示例，并提出了简单的实验测试来量化级联系统是否处于临界状态。
