# Broken detailed balance and entropy production in directed networks

arXiv: https://arxiv.org/abs/2402.19157 · HTML: https://mlsanity.com/paper/2402.19157/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Feb 2024.

## Built on by (most cited first, 3 of 3)

- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Drift-Diffusion Matching: Embedding dynamics in latent manifolds of asymmetric neural networks](https://mlsanity.com/paper/2602.14885/) (2026)
- [Conditional Entropy of Heat Diffusion on Temporal Networks](https://mlsanity.com/paper/2605.21514/) (2026)

## Abstract (arXiv)

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.