The topological basis of function in flow networks.

The ability to reroute and control flow is vital to the function of many biological transport networks. By tuning the conductance of edges, many flow networks robustly control the propagation of inputs in order to achieve a wide variety of specific tasks. However, recent evidence suggests that network architectures used to achieve such tasks demonstrate significant design flexibility, blurring the relationship between structure and function. Here we seek to identify the structural features responsible for function in tuned networks. By adding and removing links, we optimize flow networks to perform a specific function, namely to achieve a minimum pressure difference along a chosen target edge when a pressure difference is applied to another predetermined source edge in the network. Using persistent homology, we show that networks tuned to perform such functions develop characteristic large-scale topological features that are similar for different networks that perform the same function, regardless of differences in the local link connectivity. These features correlate strongly with the tuned response, providing a clear topological relationship between structure and function.