  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.
