Dimension reduction of dynamics on modular and heterogeneous directed networks
Marina Vegué1,2, Vincent Thibeault1,2, Patrick Desrosiers1,2,3
1Département de physique, de génie physique et d'optique, Université Laval, 2325 rue de l'Université, G1V 0A6 Québec, Canada.
PNAS Nexus
|May 22, 2023
Summary
We developed a new dimension reduction method for complex dynamical systems on networks. This approach simplifies large systems by grouping similar units, enabling easier prediction of network dynamics and revealing key structural drivers.
Area of Science:
- Complex Systems
- Network Science
- Dynamical Systems Theory
Background:
- Dimension reduction is crucial for analyzing large nonlinear dynamical systems.
- Simplifying complex systems while preserving key dynamics is challenging.
- Network structure significantly influences system behavior.
Purpose of the Study:
- To develop a dimension reduction method for dynamics on weighted directed networks.
- To specifically address modular and heterogeneous network structures.
- To enable systematic comparison of structural properties' effects on network dynamics.
Main Methods:
- A two-step dimension reduction technique based on adjacency matrix properties.
- Partitioning network units into groups with similar connectivity profiles.
- Deriving and approximately solving equations for group observables (weighted averages of node activities).
Main Results:
- A reduced adjacency matrix and an approximate system of ordinary differential equations (ODEs) for observable evolution.
- Successful prediction of characteristic dynamics for synthetic and real-world networks (neuronal, ecological, social).
- Demonstration of the method's applicability across diverse network structures.
Conclusions:
- The proposed method effectively reduces complexity in network dynamical systems.
- It facilitates the identification of structural drivers influencing system evolution.
- This formalism supports systematic analysis of how network architecture shapes dynamics.
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