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 network is . We consider a range of dynamical processes and show how different directed network features . 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 and the importance and ubiquity of hierarchical organisation and non-equilibrium dynamics in real-world systems.