Robustness of synchronization to additive noise: How vulnerability depends on dynamics

From biological to technological networks, scientists and engineers must face the question of vulnerability to understand evolutionary processes or design-resilient systems. Here, we examine the vulnerability of a network of coupled dynamical units to failure or malfunction of one of its nodes. More specifically, we study the effect of additive noise that is injected at one of the network sites on the overall synchronization of the coupled dynamical systems. In the context of mean square stochastic stability, we present a mathematically principled approach to illuminate the interplay between dynamics and topology on network robustness. Through the new theoretical construct of robust metric, we uncover a complex and often counterintuitive effect of dynamics. While networks are more robust to noise injected at their hubs for a classical consensus problem, these hubs could become the most vulnerable nodes for higher order dynamics, such as second-order consensus and Rössler chaos. From the exact treatment of star networks and the systematic application of perturbation techniques, we offer a mechanistic explanation of these surprising results and lay the foundation for a theory of dynamic robustness of networks.