Various natural and engineered systems, from urban traffic flow to the human brain, can be described by large-scale networked dynamical systems. These systems are similar in being comprised of a large number of microscopic subsystems, each with complex nonlinear dynamics and interactions, that collectively give rise to different forms of macroscopic dynamics. Despite significant research, why and how various forms of macroscopic dynamics emerge from underlying micro-dynamics remains largely unknown. In this work we focus on linearity as one of the most fundamental aspects of system dynamics. By extending the theory of mixing sequences, we show that \textit{in a broad class of autonomous nonlinear networked systems, the dynamics of the average of all subsystems' states becomes asymptotically linear as the number of subsystems grows to infinity, provided that, in addition to technical assumptions, pairwise correlations between subsystems decay to 0 as their pairwise distance grows to infinity}. We prove this result when the latter distance is between subsystems' linear indices or spatial locations, and provide extensions to linear time-invariant (LTI) limit dynamics, finite-sample analysis of rates of convergence, and networks of spatially-embedded subsystems with random locations. To our knowledge, this work is the first rigorous analysis of macroscopic linearity in large-scale heterogeneous networked dynamical systems, and provides a solid foundation for further theoretical and empirical analyses in various domains of science and engineering.

翻译：从城市交通流到人脑，各种自然与工程系统均可描述为大规模网络化动力系统。这些系统的共同特征在于由大量微观子系统构成，每个子系统具有复杂的非线性动力学特性及相互作用，并共同涌现出不同形式的宏观动力学行为。尽管已有大量研究，但各类宏观动力学形式为何以及如何从底层微观动力学中涌现，至今仍不甚明晰。本文聚焦于线性这一系统动力学最基本特征之一。通过拓展混合序列理论，我们证明：\textit{在满足技术性假设的前提下，若子系统间的两两相关性随其距离趋于无穷而衰减至零，则在一大类自治非线性网络化系统中，所有子系统状态的平均值动力学将随着子系统数量趋于无穷而渐近线性化}。我们在子系统线性指标距离或空间位置距离两种情形下证明了该结论，并进一步拓展至线性时不变极限动力学、收敛速率的有限样本分析，以及随机空间嵌入子系统网络等场景。据我们所知，本研究首次对大规模异质网络化动力系统中的宏观线性现象进行了严格分析，为科学与工程多个领域的理论与实证研究奠定了坚实基础。