大规模网络中同步问题的限制与权衡
Limitations and tradeoffs in synchronization of large-scale networks with uncertain links
AI总结：
本文研究了具有不确定连接的线性网络中标量非线性系统的同步问题，提出了一种基于非线性动态参数、网络拓扑和不确定性统计的同步条件，并揭示了内部动态、网络拓扑和不确定性统计之间的权衡关系。
摘要：
我们研究了具有不确定连接的线性网络中标量非线性系统的同步问题。我们提供了同步网络系统的充分条件，该条件以非线性标量动态的参数、均连接拉普拉斯矩阵的第二和最大特征值以及随机不确定性的方差来表达。该充分条件与网络规模无关，因此适合验证大规模网络的同步性。本文的主要贡献是提供对内部动态、网络拓扑和不确定性统计在同步中的相互作用的分析表征。我们展示了在实现同步时，这些网络参数之间存在重要的权衡关系。我们证明，在具有随机不确定性的最近邻网络中，存在一个最优的邻居数量，以实现最大的同步边际。这证明了在存在交互不确定性的情况下，过多的网络组件连接对同步是同样有害的，就像缺乏连接一样。我们还提供了一个分析公式，用于计算达到最大同步边际所需的最优增益，从而允许我们比较各种复杂网络拓扑的同步特性。
英文摘要：
We study synchronization in scalar nonlinear systems connected over a linear network with stochastic uncertainty in their interactions. We provide a sufficient condition for the synchronization of such network systems expressed in terms of the parameters of the nonlinear scalar dynamics, the second and largest eigenvalues of the mean interconnection Laplacian, and the variance of the stochastic uncertainty. The sufficient condition is independent of network size thereby making it attractive for verification of synchronization in a large size network. The main contribution of this paper is to provide analytical characterization for the interplay of roles played by the internal dynamics of the nonlinear system, network topology, and uncertainty statistics in network synchronization. We show there exist important tradeoffs between these various network parameters necessary to achieve synchronization. We show for nearest neighbor networks with stochastic uncertainty in interactions there exists an optimal number of neighbors with maximum margin for synchronization. This proves in the presence of interaction uncertainty, too many connections among network components is just as harmful for synchronization as the lack of connection. We provide an analytical formula for the optimal gain required to achieve maximum synchronization margin thereby allowing us to compare various complex network topology for their synchronization property.
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