Average Consensus in Dense Wireless Networks with Long-Range Connectivity

We consider the effect of interference on the convergence rate of a class of distributed averaging algorithms called consensus algorithms, which iteratively compute the measurement average by message passing among nodes. It is usually assumed that these algorithms converge faster with a greater exchange of information (i.e., by increased network connectivity) in every iteration. However, when these topologies are realized on wireless networks, the rate at which they can be formed is in general limited by interference. It is thus not clear if the rate of convergence always increases with network connectivity. We study this problem for randomly-placed consensusseeking nodes that are connected through a dense (interference-limited) wireless network. We investigate the following questions: (a) How does the rate of convergence vary with increasing communication range of each node, and (b) How does this result change when each node is allowed to additionally exchange messages with a few selected far-off nodes? When nodes schedule their transmissions to avoid interference, we show that with increasing network dimension, the benefit from increased connectivity remains the same while the time needed to realize these topologies becomes larger. In particular, while increased connectivity improves the convergence rate in onedimensional networks, it is exactly offset by the longer schedule lengths in two-dimensions. In higher dimensions, greater connectivity actually degrades the rate of convergence. Our results thus provide new insights into the MAC protocol design for this family of distributed averaging algorithms. Keywords–Average Consensus, Wireless Networks, Scaling Laws, MAC Protocols. *Corresponding author. The work of the first two authors was partially supported by NSF (grants CNS 04-47869 and CCF 728763). The work of the third author was supported partially by the NSF award 0846631.