Broken detailed balance and entropy production in directed networks

The structure of a complex network plays a crucial role in determining its dynamical properties. In this work, we show that the directed, hierarchical organisation of a network causes the system to break detailed balance and dictates the production of entropy through non-equilibrium dynamics. We consider a wide range of dynamical processes and show how different directed network features govern their thermodynamics. Next, we analyse a collection of 97 empirical networks and show that strong directedness and non-equilibrium dynamics are both ubiquitous in real-world systems. 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 and hierarchical organisation in both human neuroimaging and financial time-series. Overall, our results shed light on the thermodynamic consequences of directed network structure and indicate the importance and ubiquity of hierarchical organisation and non-equilibrium dynamics in real-world systems.