Antifragility in the synchronization of oscillators on networks with communities

Antifragility refers to the counterintuitive phenomenon whereby a system improves its global performance in response to damage. Although antifragile behavior has been reported in a variety of complex systems, the structural and dynamical mechanisms underlying this response remain poorly understood. In this work, we investigate antifragility in the synchronization dynamics of identical oscillators on networks, focusing on the effect of damage defined as the reduction in the weight of an edge connecting pairs of nodes in the network. Using a linearized description of the Kuramoto model with normalized couplings, we introduce a spectral measure of antifragility based on the response of the second eigenvalue of the normalized Laplacian. This framework enables an analytical and numerical characterization of how local reductions in coupling strength can accelerate global synchronization. Through first-order perturbation theory, we show that the sign and magnitude of the response are controlled by the structure of the slowest relaxation mode. In networks with modular organization, this mechanism leads to a systematic distinction between intra- and inter-community links: damage to intra-community edges is typically antifragile, whereas damage to inter-community edges is fragile. These predictions are validated by numerical simulations on deterministic, random, and real-world networks with community structure. Overall, our results establish a direct connection between spectral properties, modular topology, and antifragility in synchronization processes.