Generalized network density matrices for analysis of multiscale functional diversity

TL;DR

This paper introduces a generalized framework for network density matrices that captures a wider range of dynamics and structures, enabling multiscale analysis of functional diversity in complex networks.

Contribution

It proposes a novel approach to derive density matrices from dynamical systems and information theory, extending analysis to directed, signed, and nonlinear networks.

Findings

Topological complexity does not necessarily imply functional diversity.

Functional diversity emerges independently of topological features.

The framework effectively analyzes neural and gene-regulatory networks.

Abstract

The network density matrix formalism allows for describing the dynamics of information on top of complex structures and it has been successfully used to analyze from system's robustness to perturbations to coarse graining multilayer networks from characterizing emergent network states to performing multiscale analysis. However, this framework is usually limited to diffusion dynamics on undirected networks. Here, to overcome some limitations, we propose an approach to derive density matrices based on dynamical systems and information theory, that allows for encapsulating a much wider range of linear and non-linear dynamics and richer classes of structure, such as directed and signed ones. We use our framework to study the response to local stochastic perturbations of synthetic and empirical networks, including neural systems consisting of excitatory and inhibitory links and gene-regulatory interactions. Our findings demonstrate that topological complexity does not lead, necessarily, to functional diversity -- i.e., complex and heterogeneous response to stimuli or perturbations. Instead, functional diversity is a genuine emergent property which cannot be deduced from the knowledge of topological features such as heterogeneity, modularity, presence of asymmetries or dynamical properties of a system.