The topological basis of function in flow and mechanical networks

Recently, both flow networks and mechanical networks have been shown to be remarkably tunable. By tuning the local node connectivity, it is possible to robustly control the propagation of inputs in order to achieve a wide variety of specific tasks. However, the 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. Using persistent homology, we show that networks develop large-scale topological features when they are tuned, which are similar for different networks with the same function, regardless of the details of the local link topology. These features correlate strongly with the tuned response, providing a clear relationship between structure and function.