大规模紧密连接网络的动力学集中
Dynamics Concentration of Large-Scale Tightly-Connected Networks
AI总结：
本文研究了大规模紧密连接网络中节点响应趋于一致的现象，通过高维概率工具证明了在网络规模和连接度增加时，节点响应趋于相同，提出了动力学集中的概念，并讨论了其在模型降阶和鲁棒性中的应用。
摘要：
实现网络化系统中协调行为（工程或涌现）的能力在多个领域引起了广泛关注。这导致了在理解在何种条件下网络中的智能体能够达成一致（共识）或发展协调行为如同步方面取得了显著进展。然而，对于紧密连接网络系统中另一个常见现象的解释却较少：尽管节点的个体动力学存在异质性，网络中节点的输出响应却几乎相同。本文利用高维概率工具，为这一现象提供了一个初步答案。更具体地说，我们证明了对于具有节点随机转移函数的线性网络，当网络规模和连接度增加时，网络中的每个节点都遵循相同的输入或扰动响应——无论输入来源如何。我们将这种行为称为动力学集中，因为它源于网络转移矩阵以概率方式均匀收敛，即集中，到一个由每个节点随机转移函数分布决定的唯一动态响应。我们进一步讨论了我们的分析在模型降阶和鲁棒性中的影响，并提供了数值证据，证明在适当定义的频率带内，小确定性网络中也出现类似现象。
英文摘要：
The ability to achieve coordinated behavior --engineered or emergent-- on networked systems has attracted widespread interest over several fields. This has led to remarkable advances on the development of a theoretical understanding of the conditions under which agents within a network can reach agreement (consensus) or develop coordinated behaviors such as synchronization. However, fewer advances have been made toward explaining another commonly observed phenomena in tightly-connected networks systems: output responses of nodes in the networks are almost identical to each other despite heterogeneity in their individual dynamics. In this paper, we leverage tools from high-dimensional probability to provide an initial answer to this phenomena. More precisely, we show that for linear networks of nodal random transfer functions, as the network size and connectivity grows, every node in the network follows the same response to an input or disturbance --irrespectively of the source of this input. We term this behavior as dynamics concentration since it stems from the fact that the network transfer matrix uniformly converges in probability, i.e., it concentrates, to a unique dynamic response determined by the distribution of the random transfer function of each node. We further discuss the implications of our analysis in the context of model reduction and robustness, and provide numerical evidence that similar phenomena occur in small deterministic networks over a properly defined frequency band.
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