Spectral Dimension Reduction of Complex Dynamical Networks

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.

The Emergence of Informative Higher Scales in Complex Networks

“…is result is similar to recent advances in spectral coarse-graining that also observe that the ideal coarsegraining of a star network is to collapse it into a two-node network, grouping all the spokes into a single macronode [34], which is what happens to star networks that are recast as macroscales.…”

Scale free topology as an effective feedback system

“…Alternatively, a degree-dependent mean-field approach can be developed [41,42]. Weighted averages of activity, guided by topology, can also serve to reduce the dimensionality of the dynamics [18,19]. Our approach can be thought of as splitting the network and then approximating each part as homogeneous.…”

“…Understanding how heterogeneous connectivity profiles relate to properties of the associated dynamical system is a non-trivial theoretical challenge [16][17][18][19][20]. In particular, heterogeneity makes these networks non-tractable by mean field methods [20].…”

Dimensionality reduction of complex dynamical systems

“…Furthermore, Laurence et al. ( Laurence et al., 2019 ) developed a polynomial approximation to reduce complex networks based on spectral graph theory and showed that the proposed reduction of Gao et al. ( Gao et al., 2016 ) is a special case of the general scheme when applied to uncorrelated random networks (see Transparent methods , section Dimension reduction based on spectral graph theory).…”

“…Unfortunately, these frameworks ( Gao et al., 2016 ; Laurence et al., 2019 ) can be applied only to the particular case where the local dynamics at every node (hereafter termed “self-dynamics”) as well as the pairwise dynamics (here called “coupling-dynamics”) are expressed by functions that are not node specific but are the same at all nodes. In fact, only in such a case, the one-dimensional effective equation ( Laurence et al., 2019 ; Gao et al., 2016 ; Tu et al., 2017 ) can be used to predict changes in resilience. Moreover, even when both self and coupling-dynamics are expressed by the same function at all nodes, the proposed framework works well only when the model parameters of the -dimensional system are not too heterogeneous (e.g.…”