复杂网络上的Shock waves
Shock waves on complex networks
摘要：
电网、道路图和河流是极易受外部扰动影响的基础设施网络的例子。负载（电压、流量密度或水位）的突然局部变化可能以级联方式传播，并影响网络的很大一部分。几乎不连续的扰动可以用冲击波来模拟，冲击波最终会产生建设性的干扰，危及基础设施的正常功能。我们研究了它们的动力学通过求解Burgers方程随机扰动下的几个真正的和人工有向图。即使对于节点属性分布较窄的图（例如，度或中间度），达到呈现不均匀载荷分布的稳定状态，在最高载荷和平均载荷之间具有一个数量级的差异。出乎意料的是，我们发现欧洲电网和有限的瓦特-Strogatz网络的负载的一个广泛的明显的双峰分布。为了确定最脆弱的节点，我们引入的概念，节点盆地的大小，一个纯粹的拓扑性质，我们表现出强烈相关的节点的平均负载。
英文摘要：
Power grids, road maps, and river streams are examples of infrastructural networks which are highly vulnerable to external perturbations. An abrupt local change of load (voltage, traffic density, or water level) might propagate in a cascading way and affect a significant fraction of the network. Almost discontinuous perturbations can be modeled by shock waves which can eventually interfere constructively and endanger the normal functionality of the infrastructure. We study their dynamics by solving the Burgers equation under random perturbations on several real and artificial directed graphs. Even for graphs with a narrow distribution of node properties (e.g., degree or betweenness), a steady state is reached exhibiting a heterogeneous load distribution, having a difference of one order of magnitude between the highest and average loads. Unexpectedly we find for the European power grid and for finite Watts-Strogatz networks a broad pronounced bimodal distribution for the loads. To identify the most vulnerable nodes, we introduce the concept of node-basin size, a purely topological property which we show to be strongly correlated to the average load of a node.
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