  The ability to reroute and control flow is vital to the function of venation
networks across a wide range of organisms. By modifying individual edges in
these networks, either by adjusting edge conductances or creating and
destroying edges, organisms can robustly control the propagation of inputs to
perform specific tasks. However, a fundamental disconnect exists between the
structure and function of these networks: networks with different local
architectures can perform the same functions. Here we answer the question of
how structural changes at the microscopic level are able to collectively create
functionality at the scale of an entire network. Using persistent homology, we
analyze networks tuned to perform complex multifunctional tasks. We find that
the responses of such networks encode a hidden topological structure composed
of sectors of uniform pressure. Although these sectors are not apparent in the
underlying network architectures, we find that they nonetheless correlate
strongly with the tuned function. We conclude that the connectivity of these
sectors, rather than that of the individual nodes, provides a quantitative
relationship between structure and function in flow networks. Finally, we use
this topological description to place a bound on the limits of task complexity.
