Wolter Justine, Lünsmann Benedict, Zhang Xiaozhu, Schröder Malte, Timme Marc
Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany.
Max Planck Institute for the Physics of Complex Systems (MPIPKS), 01187 Dresden, Germany.
Chaos. 2018 Jun;28(6):063122. doi: 10.1063/1.5000996.
Spreading phenomena on networks are essential for the collective dynamics of various natural and technological systems, from information spreading in gene regulatory networks to neural circuits and from epidemics to supply networks experiencing perturbations. Still, how local disturbances spread across networks is not yet quantitatively understood. Here, we analyze generic spreading dynamics in deterministic network dynamical systems close to a given operating point. Standard dynamical systems' theory does not explicitly provide measures for arrival times and amplitudes of a transient spreading signal because it focuses on invariant sets, invariant measures, and other quantities less relevant for transient behavior. We here change the perspective and introduce formal expectation values for deterministic dynamics to work out a theory explicitly quantifying when and how strongly a perturbation initiated at one unit of a network impacts any other. The theory provides explicit timing and amplitude information as a function of the relative position of initially perturbed and responding unit as well as depending on the entire network topology.
网络上的传播现象对于各种自然和技术系统的集体动力学至关重要，从基因调控网络中的信息传播到神经回路，从流行病到遭受扰动的供应网络。然而，局部干扰如何在网络中传播尚未得到定量理解。在这里，我们分析确定性网络动力系统中接近给定工作点的一般传播动力学。标准动力系统理论没有明确提供瞬态传播信号到达时间和幅度的度量，因为它关注不变集、不变测度以及其他与瞬态行为相关性较小的量。我们在此改变视角，为确定性动力学引入形式期望值，以建立一个明确量化网络中一个单元发起的扰动何时以及以多强的程度影响其他任何单元的理论。该理论根据初始受扰单元和响应单元的相对位置以及整个网络拓扑结构提供明确的时间和幅度信息。
