Quantifying Transient Spreading Dynamics on Networks

TL;DR

This paper develops a theoretical framework to quantitatively analyze how local disturbances spread through networks, providing explicit timing and amplitude predictions based on network topology.

Contribution

It introduces effective expectation values for deterministic network dynamics to quantify transient spreading behavior, a novel approach beyond traditional invariant set analysis.

Findings

Provides explicit timing and amplitude predictions for spreading signals.

Quantifies impact of initial perturbations based on network topology.

Offers a new theoretical perspective on transient network dynamics.

Abstract

Spreading phenomena on networks are essential for the collective dynamics of various natural and technological systems, from information spreading in gene regulatory networks to neural circuits or from epidemics to supply networks experiencing perturbations. Still, how local disturbances spread across networks is not yet quantitatively understood. Here we analyze generic spreading dynamics in deterministic network dynamical systems close to a given operating point. Standard dynamical systems' theory does not explicitly provide measures for arrival times and amplitudes of a transient, spreading signal because it focuses on invariant sets, invariant measures and other quantities less relevant for transient behavior. We here change the perspective and introduce effective expectation values for deterministic dynamics to work out a theory explicitly quantifying when and how strongly a perturbation initiated at one unit of a network impacts any other. The theory provides explicit timing and amplitude information as a function of the relative position of initially perturbed and responding unit as well as on the entire network topology.