Toward resource-optimal averaging consensus over the wireless medium

We carry out a comprehensive study of the resource costs of distributed averaging consensus in wireless sensor networks. In particular, we consider two metrics appropriate to the wireless medium: total transmit energy and time-bandwidth product. Most previous approaches, such as gossip algorithms, suppose a graphical network, which abstracts away crucial features of the wireless medium, and measure resource consumption only in terms of the total number of transmissions required to achieve consensus. Under a path-loss dominated protocol interference model, we study the performance of several popular gossip algorithms, showing that they are nearly order-optimal with respect to transmit energy but strictly sub-optimal with respect to time-bandwidth product. We also propose a new scheme, termed hierarchical averaging, which is tailored to the wireless medium, and show that in general this approach is nearly order-optimal with respect to time-bandwidth product but strictly sub-optimal with respect to transmit energy. For the special case of free-space propagation, however, the proposed hierarchical scheme is approximately order-optimal with respect to both metrics.