Looplessness in networks is linked to trophic coherence

TL;DR

This paper demonstrates that trophic coherence in networks determines the number of feedback loops and eigenvalues, explaining why many natural systems are loopless and stable, and introduces a null model for assessing feedback significance.

Contribution

It introduces the concept of trophic coherence as a structural property that explains looplessness and feedback suppression in diverse networks, linking structure to stability.

Findings

Trophic coherence classifies networks into high and low feedback regimes.

The theory accurately predicts feedback properties in various real-world networks.

A null model assesses the significance of feedback and coherence measures.

Abstract

Many natural, complex systems are remarkably stable thanks to an absence of feedback acting on their elements. When described as networks, these exhibit few or no cycles, and associated matrices have small leading eigenvalues. It has been suggested that this architecture can confer advantages to the system as a whole, such as `qualitative stability', but this observation does not in itself explain how a loopless structure might arise. We show here that the number of feedback loops in a network, as well as the eigenvalues of associated matrices, are determined by a structural property called trophic coherence, a measure of how neatly nodes fall into distinct levels. Our theory correctly classifies a variety of networks -- including those derived from genes, metabolites, species, neurons, words, computers and trading nations -- into two distinct regimes of high and low feedback, and provides a null model to gauge the significance of related magnitudes. Since trophic coherence suppresses feedback, whereas an absence of feedback alone does not lead to coherence, our work suggests that the reasons for `looplessness' in nature should be sought in coherence-inducing mechanisms.