This work presents a model-based quasi-decentralized networked control structure with a state-dependent communication policy for plants with interconnected units that exchange measurements over a shared, resource-constrained communication network. The objective is to find a strategy for establishing and terminating communication between the local control systems in a way that minimizes network resource utilization without jeopardizing closed-loop stability. To this end, a Lyapunov-based controller that enforces closed-loop stability in the absence of communication suspensions is initially designed. A set of dynamic models are included within each local control system to provide estimates of the states of the neighboring units when measurements are not transmitted through the network. To determine when communication must be re-established, the evolution of each Lyapunov function is monitored locally within each unit such that if it begins to breach a certain stability threshold at any time, the sensor suites of the neighboring units are prompted to send their data over the network to update their corresponding models. Communication is then suspended for as long as the Lyapunov function continues to decay. The underlying idea is to use the Lyapunov stability constraint for each unit as the basis for switching on or off the communication between a given unit and its neighbors. This formulation, which leads to a state-dependent time-varying communication rate, allows the plant to respond adaptively to changes in operating conditions. Finally, the results are illustrated through an application to a chemical plant example.