A General Theory of Emergent Linearity in Complex Dynamical Systems: The Role of Spatial Averaging and Vanishing Correlations

Various natural and engineered systems, from urban traffic flow to the human brain, have been described by large-scale networked dynamical systems. Despite their vast differences, these systems are often similar in being comprised of numerous microscopic subsystems with complex nonlinear dynamics and interactions that give rise to diverse emergent macroscopic behaviors. As such, a long-standing question across various fields has been to understand why and how various forms of macroscopic behavior emerge from underlying microscopic dynamics. Motivated by a growing body of empirical observations, in this work we focus on linearity as one of the most fundamental aspects of system dynamics, and develop a general theoretical framework for the interplay between spatial averaging, decaying microscopic correlations, and emergent macroscopic linearity. Using and extending the theory of mixing sequences, we show that 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 systems, and provides a solid foundation for further theoretical and empirical analyses in various domains of science and engineering.