  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.
