# Feedback arcs and node hierarchy in directed networks

arXiv: https://arxiv.org/abs/1612.05347 · HTML: https://mlsanity.com/paper/1612.05347/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2016.

## Built on by (most cited first, 2 of 2)

- [Belief propagation for permutations, rankings, and partial orders](https://mlsanity.com/paper/2110.00513/) (2021)
- [Feedforward Ordering in Neural Connectomes via Feedback Arc Minimization](https://mlsanity.com/paper/2506.13799/) (2025)

## Abstract (arXiv)

Directed networks such as gene regulation networks and neural networks are connected by arcs (directed links). The nodes in a directed network are often strongly interwound by a huge number of directed cycles, which lead to complex information-processing dynamics in the network and make it highly challenging to infer the intrinsic direction of information flow. In this theoretical paper, based on the principle of minimum-feedback, we explore the node hierarchy of directed networks and distinguish feedforward and feedback arcs. Nearly optimal node hierarchy solutions, which minimize the number of feedback arcs from lower-level nodes to higher-level nodes, are constructed by belief-propagation and simulated-annealing methods. For real-world networks, we quantify the extent of feedback scarcity by comparison with the ensemble of direction-randomized networks and identify the most important feedback arcs. Our methods are also useful for visualizing directed networks.