**Title: **Dynamic Stability of Time-Delayed Networks

**Abstract:** Networks, including computer networks, social networks, and biological networks, have within the past few years attracted an enormous amount of interest. Due to both their physical size and finite processing speeds the dynamics of these networks are inherently time-delayed. In this talk we consider how the global stability of a dynamical network depends on its time-delays. We find that a network’s stability is unaffected by the introduction or removal of undistributed delays and that such delays can be characterized in terms of the network’s graph structure. By extending this technique we also describe how to remove the implicit delays in a network (or general dynamical system) to gain improved estimates of the network’s (system’s) stability. This approach of “restricting” a network is illustrated by applications to certain classes of Cohen-Grossberg neural networks.