Spatially balanced topological interaction grants optimal cohesion in flocking models

Models of self-propelled particles (SPPs) are an indispensable tool to investigate collective animal behaviour. Originally, SPP models were proposed with metric interactions, where each individual coordinates with neighbours within a fixed metric radius. However, recent experiments on bird flocks indicate that interactions are topological: each individual interacts with a fixed number of neighbours, irrespective of their distance. It has been argued that topological interactions are more robust than metric ones against external perturbations, a significant evolutionary advantage for systems under constant predatory pressure. Here, we test this hypothesis by comparing the stability of metric versus topological SPP models in three dimensions. We show that topological models are more stable than metric ones. We also show that a significantly better stability is achieved when neighbours are selected according to a spatially balanced topological rule, namely when interacting neighbours are evenly distributed in angle around the focal individual. Finally, we find that the minimal number of interacting neighbours needed to achieve fully stable cohesion in a spatially balanced model is compatible with the value observed in field experiments on starling flocks.