On the Emergence of Linear Behavior in Large-Scale Dynamical Systems via Spatial Averaging

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 $\mathit{\text{in}}\mathit{\text{a}}\mathit{\text{broad}}\mathit{\text{class}}\mathit{\text{of}}\mathit{\text{autonomous}}\mathit{\text{nonlinear}}\mathit{\text{networked}}\mathit{\text{systems,}}\mathit{\text{the}}\mathit{\text{dynamics}}\mathit{\text{of}}\mathit{\text{the}}\mathit{\text{average}}\mathit{\text{of}}\mathit{\text{all}}\mathit{\text{subsystems’}}\mathit{\text{states}}\mathit{\text{becomes}}\mathit{\text{asymptotically}}\mathit{\text{linear}}\mathit{\text{as}}\mathit{\text{the}}\mathit{\text{number}}\mathit{\text{of}}\mathit{\text{subsystems}}\mathit{\text{grows}}\mathit{\text{to}}\mathit{\text{infinity,}}\mathit{\text{provided}}\mathit{\text{that,}}\mathit{\text{in}}\mathit{\text{addition}}\mathit{\text{to}}\mathit{\text{technical}}\mathit{\text{assumptions,}}\mathit{\text{pairwise}}\mathit{\text{correlations}}\mathit{\text{between}}\mathit{\text{subsystems}}\mathit{\text{decay}}\mathit{\text{to}}\mathit{\text{0}}\mathit{\text{as}}\mathit{\text{their}}\mathit{\text{pairwise}}\mathit{\text{distance}}\mathit{\text{grows}}\mathit{\text{to}}\mathit{\text{infinity}}$. 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.