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

2025-09-29

AI Summary

How do complex nonlinear network systems exhibit emergent macroscopic linear behavior arising from microscopic nonlinear dynamics? Method: We propose and rigorously prove that the spatially averaged dynamics of large-scale heterogeneous networks asymptotically linearize when inter-subsystem spatial correlations decay with distance. Our analysis integrates mixed-sequence theory, asymptotic analysis, finite-sample convergence rate estimation, and spatial embedding modeling to systematically characterize both the conditions for linearization and the resulting dynamical properties. Contribution: We establish the first general theoretical framework for macroscopic linear emergence, providing rigorous sufficient conditions for linearization. The framework encompasses linear time-invariant limiting dynamics, explicit finite-sample convergence rate bounds, and broad applicability to diverse real-world networks—including neural, ecological, and multi-agent systems—thereby overcoming classical limitations of local approximations and homogeneity assumptions in conventional linearization approaches.

Technology Category

• Multiagent Systems: Agent-Based Simulation and Emergent Behavior
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Social Networks and Social Media: Applications and emergent phenomena
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• Explaining emergent linearity from nonlinear microscopic dynamics
• Analyzing spatial averaging and vanishing correlation effects
• Developing theoretical framework for macroscopic behavior in networked systems

Innovation

• Spatial averaging enables emergent linear macroscopic dynamics
• Vanishing correlations ensure asymptotic linearity in large systems
• Mixing sequence theory proves linear time-invariant limit behavior