Looplessness in networks is linked to trophic coherence

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, is 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. Because 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.