Dynamics of Averaging Systems: Convergence and Cyclical Trends

Abstract

Averaging dynamics drives countless processes in biology, social science, physics, and engineering. Notable examples include opinion dynamics, bird flocking, and synchronization processes.These processes involve agents interacting across a dynamic network by averaging their state variables with those of their neighbors. The dynamics of averaging systems therefore depend on the structure of the underlying network sequences. Under mild conditions, such systems are known to converge to fixed-point attractors but are hard to analyze because of the time-varying nature of networks. The s-energy, a generating function over inter-agent distances, has emerged as a powerful tool to deal with this issue. In this thesis, we refine the analysis of the s-energy by highlighting the impact of network connectivity. This allows us to address an intriguing exponential gap in the convergence rate of averaging systems. We also apply this new s-energy bound to resolve open questions in several areas including bird flocking, where we establish sufficient conditions for the first polynomial bound in bird flocking convergence. Additionally, we investigate the emergence of cyclical trends in opinion dynamics. Augmenting the model with a separation rule reveals that the systems either converge to a nonconsensual equilibrium or are attracted to periodic or quasi-periodic orbits. Our analysis includes exploring geometric properties such as dimensionality, periodicity, and conditions for various behaviors. Later, we shift our focus from averaging systems to explore compelling examples of dynamic multi-agent systems, such as epidemiological and traffic network models. Furthermore, our emphasis is not on convergence rates or dynamic evolution but rather on practical applications: approximating the relevant values within these systems.