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.
Highlights
Critical breakdowns generally arise unexpectedly in complex dynamical systems [1]
Using the dimension reduction procedure, we investigate the critical structural parameter αglobal, or an equivalent parameter depending on the reduction approach, for which the global state at equilibrium RÃ undergoes a critical transition characterized by d2RÃðαCÞ dα2
We have built systematic methods of dimension reduction adapted to different families of networks
Summary
Critical breakdowns generally arise unexpectedly in complex dynamical systems [1]. Noteworthy examples are financial crises [2,3], epileptic seizures [4], and species extinctions [5]. While much effort has been devoted to forecast breakdowns [6], no simple and universal method has yet been found This is mostly due to the inherent complexity of the problem: Real systems are composed of multiple units that participate in the global state in highly complicated patterns of interactions. In an attempt to determine the critical points of 59 bipartite mutualistic ecosystems [10], Jiang et al proposed a two-dimensional reduction that divides each original ecosystem into two populations for which they obtain the average interaction strength From numerical explorations, they conclude that the degree may not always be the key predictive property of a network.
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