Journey Through the World of Dynamical Systems on Networks

TL;DR

This paper provides a broad overview of methods for analyzing dynamical systems on networks, highlighting their interrelations and practical applications across various mathematical and computational approaches.

Contribution

It offers a comprehensive synthesis of diverse analytical techniques applied to network-based dynamical systems, emphasizing their connections and practical insights.

Findings

Network-based methods reveal interrelations between different dynamical systems theories.

Visualizations help understand the strengths and limitations of each approach.

Applied problems demonstrate practical utility of the methods.

Abstract

We present a subjective selection of methods for complex systems analysis ranging from statistical tools through numerical methods based on AI to both linear and non-linear ODEs and PDEs. All the notions apply the network structure and are presented in the context of applied problems to visualise the strengths and drawbacks of the approach. The major aim of capturing such a broad overview is to understand the interrelations between network theories that seem to be distant from the mathematical perspective.