Multiple Timescale Dynamics of Network Adaptation with Constraints

TL;DR

This paper investigates how constraints and multiple timescales in adaptive network systems lead to low-dimensional dynamics, particularly in Kuramoto oscillator networks, revealing intrinsic dimension reduction mechanisms.

Contribution

It demonstrates that constraints and timescale separation induce low-dimensional behavior in high-dimensional adaptive networks, with specific insights into Kuramoto oscillator dynamics.

Findings

Constraints induce low-dimensional adaptation dynamics.

Multiple timescales shape the network's evolution.

Low-dimensional behavior emerges in high-dimensional systems.

Abstract

Adaptive network dynamical systems describe the co-evolution of dynamical quantities on the nodes as well as dynamics of the network connections themselves. For dense networks of many nodes, the resulting dynamics are typically high-dimensional. Here we consider adaptive dynamical systems subject to constraints on network adaptation: Asymptotically, the adaptive dynamics of network connections evolve on a low-dimensional subset of possible connectivity. Such dimension reduction may be intrinsic to the adaptation rule or arise from an additional dynamical mechanism acting on a timescale distinct from that of network adaptation. We illustrate how network adaptation with various constraints influences the dynamics of Kuramoto oscillator networks and elucidate the role of multiple timescales in shaping the dynamics. Our results shed light on why one may expect effective low-dimensional adaptation dynamics in generally high-dimensional adaptive network dynamical systems.