Information processing with only locally connected networks such as cellular neural networks is advantageous for integrated circuit implementations. Adding long range connections can often enhance considerably their performance. It is sufficient to activate these connections randomly from time to time (blinking connections). This can be realized by sending packets on a communication network underlying the information processing network that is needed anyway for bringing information in and out of the locally connected network. We prove for the case of multi-stable networks that if the long-range connections are switched on and off sufficiently fast, the behavior of the blinking network is with high probability the same as the behavior of the time-averaged network. In the averaged network the blinking connections are replaced by fixed connections with low (average) coupling strength.