Antifragility and response to damage in the synchronization of oscillators on networks

In this paper, we introduce a mathematical framework to assess the impact of damage, defined as the reduction of weight in a specific link, on identical oscillator systems governed by the Kuramoto model and coupled through weighted networks. We analyze how weight modifications in a single link affect the system when its global function is to achieve the synchronization of coupled oscillators starting from random initial phases. We introduce different measures that allow the identification of cases where damage enhances synchronization (antifragile response), deteriorates it (fragile response), or has no significant impact. Using numerical solutions of the Kuramoto model, we investigate the effects of damage on network links where antifragility emerges. Our analysis includes lollipop graphs of varying sizes and a comprehensive evaluation and all the edges of 109 non-isomorphic graphs with six nodes. The approach is general and can be applied to study antifragility in other oscillator systems with different coupling mechanisms, offering a pathway for the quantitative exploration of antifragility in diverse complex systems.