Applications of Geometric and Group-theoretic Methods to Network Dynamics

Boston College, Chestnut Hill MA

Investigators

Abstract

A fundamental challenge in science is to understand how collective behavior emerges in complex systems. For example, how do the thousands of heart cells that form the heart's natural pacemaker network manage to fire an electrical pulse simultaneously? How does a flock of birds or school of fish organize so as to fly or swim together? What causes a population of fireflies to flash in unison? How can we prevent a single component failure in a large power grid from cascading through the entire network? The investigators' particular interest is in networks of oscillators, in which the individual components cycle periodically in time. Oscillator networks can exhibit a variety of collective behaviors, ranging in simplicity from complete synchronization, when all the oscillators behave in unison, to more complex patterns such as chimera states, where part of the network is synchronized while the rest exhibits asynchronous behavior. The investigators develop mathematical tools, using geometry, group theory, and dimensional reduction, to greatly simplify the analysis of oscillator networks and their stability. The virtue of this approach is that the behavior of a large network with many variables can be precisely described using only a small number of equations. They use these methods to design and analyze networks with applications to reservoir computing and machine learning, an important and developing area of modern data analysis. An important ramification of this work is to forge interdisciplinary collaborations between the more applied study of oscillator dynamics and the pure mathematical study of low-dimensional geometry. Graduate students are engaged in the research of the project. The central theme of this project is the application of geometric and group-theoretic techniques to the study of networks of Kuramoto and more general types of oscillators. The application of geometric techniques makes possible the explicit description of the dynamics on group orbits. Building on on their earlier work using this methodology, the investigators take up four topics. The first classifies higher degree order parameter functions that have a geometric (gradient) structure similar to the classic Kuramoto model with first degree order parameter. Topic 2 uses geometric techniques to understand the dynamics of multi-population models, particularly to clarify the dynamics of chimera states, which are states with some but not all of the populations completely synchronized. Topic 3 extends this structure to networks of higher-dimensional oscillators, with states on spheres of dimension 2 or higher, that are used to model flocks, swarms and other social networks. Topic 4 applies this technology to machine learning and reservoir computing. Graduate students are engaged in the research of the project. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.