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Minimization of power consumption is a critical design goal for wireless-relay networks comprised of battery-powered sensor devices. Traditionally, wireless communication is optimized to minimize the total transmission power. In the paper, we further include the detection power and the reception power in order to minimize the overall network power consumption for information exchange in wireless-relay sensor networks (WSNs). Specifically, after the physical attributes of relay nodes are given, we wish to maximize the network lifetime by a proper selection of the transmission range, the sleep period, and the participation density. We propose a random gossip network (RGN) as a framework for our analysis. We find simple rules govern the optimal settings of these three parameters for all-to-all broadcast in the RGN. One key relationship is that the optimal number of nodes broadcasting messages in a time epoch within the transmission range depends only on the path loss exponent and the network dimensionality. This optimal value does not depend on factors such as the physical network size, the reception cost, and the message origination rate. Moreover, it is shown that neither increasing nor decreasing the physical network size will affect the optimal value of these design parameters. The optimal setting for power consumption minimization is a scalable solution. Finally, we demonstrate that our results can be well applied to predict the optimal value of operational parameters for information exchange as well as information dissemination in actual WSNs.