Gradient and Hamiltonian coupled systems on undirected networks.

Area of Science:

• Complex Systems
• Network Science
• Dynamical Systems Theory

Background:

• Coupled systems on networks are crucial for modeling real-world applications.
• Gradient and Hamiltonian systems are key classes of coupled systems, exclusively found on undirected networks.
• Flow-invariant spaces (synchrony subspaces) exist in coupled systems, dependent on network topology, allowing for quotient networks.

Purpose of the Study:

• To characterize the conditions under which gradient or Hamiltonian properties are preserved in lift and quotient coupled systems.
• To determine necessary and sufficient conditions for quotient networks to retain undirected properties.
• To explore how gradient/Hamiltonian structures can be lost or gained during network simplification.

Main Methods:

• Analysis of network topology and its relation to system dynamics.
• Characterization of conditions for preserving gradient and Hamiltonian properties through network lifting and quotienting.
• Investigation of necessary and sufficient conditions for quotient networks to be undirected.

Main Results:

• The preservation of gradient or Hamiltonian structure in quotient systems depends on the network topology.
• Gradient/Hamiltonian properties can be lost when moving from an undirected network to a directed quotient network.
• Gradient/Hamiltonian dynamics can emerge in undirected quotient networks derived from directed or non-gradient/Hamiltonian undirected networks.

Conclusions:

• Network topology dictates the preservation or emergence of gradient and Hamiltonian dynamics in coupled systems.
• The study provides a framework for understanding structural dynamics in simplified network models.
• Illustrative example using a mutually coupled neural network highlights the practical implications of these findings.