Spectral dimension reduction of complex dynamical networks

TL;DR

This paper introduces a spectral graph theory-based dimension reduction method for complex dynamical networks, enabling prediction of critical points and collective dynamics, thus linking network structure to system resilience.

Contribution

The novel spectral dimension reduction approach predicts critical points and multiple activations in complex networks, improving understanding of network resilience and dynamics.

Findings

Predicts critical points in arbitrary degree distribution networks

Enables low-dimensional modeling of complex network dynamics

Provides insights into network resilience and structural influence

Abstract

Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems. They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve (nodes' dynamics). Despite substantial progress, little is known about why some subtle changes in the network structure, at the so-called critical points, can provoke drastic shifts in its dynamics. We tackle this challenging problem by introducing a method that reduces any network to a simplified low-dimensional version. It can then be used to describe the collective dynamics of the original system. This dimension reduction method relies on spectral graph theory and, more specifically, on the dominant eigenvalues and eigenvectors of the network adjacency matrix. Contrary to previous approaches, our method is able to predict the multiple activation of modular networks as well as the critical points of random networks with arbitrary degree distributions. Our results are of both fundamental and practical interest, as they offer a novel framework to relate the structure of networks to their dynamics and to study the resilience of complex systems.