Hidden Topological Structure of Flow Network Functionality.

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 robustly control the propagation of inputs to perform specific tasks. However, a fundamental disconnect exists between the structure and function: networks with different local architectures can perform the same functions. Here, we answer the question of how changes at the level of individual edges collectively create functionality at the scale of an entire network. Using persistent homology, we analyze networks tuned to perform complex tasks. We find that the responses of such networks encode a hidden topological structure composed of sectors of nearly uniform pressure. Although these sectors are not apparent in the underlying network structure, they correlate strongly with the tuned function. The connectivity of these sectors, rather than that of individual nodes, provides a quantitative relationship between structure and function in flow networks.