Dynamical network models are a flexible frame-work to describe groups of dynamical systems interacting through a network and have been widely used in several applications to model real-world systems, including transportation, communication, and biology. In this paper, we investigate the resilience of dynamical network models under structured perturbations of their edges. Given a linear dynamical network with the property that poles are confined to a prescribed region, we ask whether it is possible to compromise this property by perturbing a single communication edge. We prove that only a subset of the edges, if perturbed, could compromise stability and we provide a graph-theoretic characterization to determine these edges. Interestingly, our results show that only edges that belong to some cycles of the communication graph play a role in the considered measure of resilience, thus identifying cycles as the basic element that determines resilience in dynamical networks. The theoretical guarantees are illustrated through simulations applied to a nonlinear epidemic model.