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Most coordinated tasks performed by teams of mobile robots require reliable communications between team members. Therefore, task accomplishment requires that robots navigate their environment with their collective movement restricted to formations that guarantee integrity of the communication network. Maintaining this communication capability induces physical constraints on trajectories but also requires determination of communication variables like routes and transmitted powers. This problem is addressed here using a distributed hybrid approach. Continuous distributed motion controllers based on potential fields interact with discrete distributed optimization of communication variables to result in a multi-robot network ensuring communication integrity. The definition of network integrity differs from existing approaches in that it is not based only on the topology of the network but also on metrics that are of interest to the performance of communication between robots and a fixed infrastructure. Specifically, integrity is defined as the ability of the network to support desired communication rates. The ability of the hybrid controller to guarantee communication integrity while robots move to accomplish their task is studied theoretically and numerically.