Feedback arcs and node hierarchy in directed networks

TL;DR

This paper develops methods to identify hierarchical structures and feedback arcs in directed networks, aiding understanding of information flow and network visualization, with applications to biological and neural systems.

Contribution

It introduces belief-propagation and simulated-annealing techniques to find nearly optimal node hierarchies that minimize feedback arcs in directed networks.

Findings

Nearly optimal hierarchies can be constructed using proposed methods.

Feedback scarcity in real networks can be quantified and compared.

The methods help identify key feedback arcs and improve network visualization.

Abstract

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.