Broken detailed balance and entropy production in directed networks

TL;DR

This paper investigates how the directionality and hierarchy of complex networks influence their non-equilibrium dynamics, entropy production, and detailed balance breaking, with applications to real-world systems like brain activity and finance.

Contribution

It introduces a method to infer broken detailed balance from time-series data and demonstrates the impact of network structure on entropy production across various systems.

Findings

Directed and hierarchical networks exhibit higher entropy production.

Real-world networks show significant deviation from detailed balance.

The proposed inference method successfully detects non-equilibrium dynamics.

Abstract

The structure of a complex network plays a crucial role in determining its dynamical properties. In this work, we show that the the degree to which a network is directed and hierarchically organised is closely associated with the degree to which its dynamics break detailed balance and produce entropy. We consider a range of dynamical processes and show how different directed network features affect their entropy production rate. We begin with an analytical treatment of a 2-node network followed by numerical simulations of synthetic networks using the preferential attachment and Erd\"os-Renyi algorithms. Next, we analyse a collection of 97 empirical networks to determine the effect of complex real-world topologies. Finally, we present a simple method for inferring broken detailed balance and directed network structure from multivariate time-series and apply our method to identify non-equilibrium dynamics and hierarchical organisation in both human neuroimaging and financial time-series. Overall, our results shed light on the consequences of directed network structure on non-equilibrium dynamics and highlight the importance and ubiquity of hierarchical organisation and non-equilibrium dynamics in real-world systems.