  Bow-tie or hourglass structure is a common architectural feature found in
biological and technological networks. A bow-tie in a multi-layered structure
occurs when intermediate layers have much fewer components than the input and
output layers. Examples include metabolism where a handful of building blocks
mediate between multiple input nutrients and multiple output biomass
components, and signaling networks where information from numerous receptor
types passes through a small set of signaling pathways to regulate multiple
output genes. Little is known, however, about how bow-tie architectures evolve.
Here, we address the evolution of bow-tie architectures using simulations of
multi-layered systems evolving to fulfill a given input-output goal. We find
that bow-ties spontaneously evolve when two conditions are met: (i) the
evolutionary goal is rank deficient, where the rank corresponds to the minimal
number of input features on which the outputs depend, and (ii) The effects of
mutations on interaction intensities between components are described by
product rule - namely the mutated element is multiplied by a random number.
Product-rule mutations are more biologically realistic than the commonly used
sum-rule mutations that add a random number to the mutated element. These
conditions robustly lead to bow-tie structures. The minimal width of the
intermediate network layers (the waist or knot of the bow-tie) equals the rank
of the evolutionary goal. These findings can help explain the presence of
bow-ties in diverse biological systems, and can also be relevant for machine
learning applications that employ multi-layered networks.
