Many natural, complex systems are remarkably stable thanks to an
absence of feedback acting on their elements. When described as net-
works, these exhibit few or no cycles, and associated matrices have small
leading eigenvalues. It has been suggested that this architecture can con-
fer 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 net-
work, 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, neu-
rons, words, computers and trading nations – into two distinct regimes
of high and low feedback, and provides a null model to gauge the signifi-
cance 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.