We consider how a shared data network can effectively support connectivity-centric applications. Such applications include command dissemination and emergency alerts. These applications have low data rate requirements but depend critically on maximizing the fraction of messages sent that are delivered on time, which is called the connectivity of the network. Stresses that impair connectivity include low-quality wireless links, adversary action, network management errors, and high load. We argue that connectivity-centric applications require dedicated support at multiple layers. The network should provide two separate communications services, high-connectivity and high-capacity, with independent protocol stacks. The highconnectivity service trades resource consumption for improved connectivity at all layers and hence cannot be used for more than a small fraction of the overall traffic load. In addition to application data it carries control plane traffic critical to network operation and crisis recovery. The high-capacity service is a more traditional efficiency-optimized stack. We consider architectures for the high-connectivity service in both tactical wireless and core fiber networks. Within the overall architecture, we investigate in more detail path-diversity mechanisms for overcoming connectivity failures, and network segmentation to mitigate the effect of cyber attacks on the control plane.