  Active biological flow networks pervade nature and span a wide range of
scales, from arterial blood vessels and bronchial mucus transport in humans to
bacterial flow through porous media or plasmodial shuttle streaming in slime
molds. Despite their ubiquity, little is known about the self-organization
principles that govern flow statistics in such non-equilibrium networks. Here
we connect concepts from lattice field theory, graph theory, and transition
rate theory to understand how topology controls dynamics in a generic model for
actively driven flow on a network. Our combined theoretical and numerical
analysis identifies symmetry-based rules that make it possible to classify and
predict the selection statistics of complex flow cycles from the network
topology. The conceptual framework developed here is applicable to a broad
class of non-biological far-from-equilibrium networks, including actively
controlled information flows, and establishes a new correspondence between
active flow networks and generalized ice-type models.
