Multiscale Gossip for Efficient Decentralized Averaging in Wireless Packet Networks

This paper describes and analyzes a hierarchical algorithm called Multiscale Gossip for solving the distributed average consensus problem in wireless sensor networks. The algorithm proceeds by recursively partitioning a given network. Initially, nodes at the finest scale gossip to compute local averages. Then, using multi-hop communication and geographic routing to communicate between nodes that are not directly connected, these local averages are progressively fused up the hierarchy until the global average is computed. We show that the proposed hierarchical scheme with $k=\Theta (\log \log n)$ levels of hierarchy is competitive with state-of-the-art randomized gossip algorithms in terms of message complexity, achieving $\epsilon$-accuracy with high probability after $O\big (n \log \log n \log {{1} \over {\epsilon}} \big)$ single-hop messages. Key to our analysis is the way in which the network is recursively partitioned. We find that the above scaling law is achieved when subnetworks at scale $j$ contain $O(n^{(2/3)^{j}})$ nodes; then the message complexity at any individual scale is $O(n \log {{1} \over {\epsilon}})$. Another important consequence of the hierarchical construction is that the longest distance over which messages are exchanged is $O(n^{1/3})$ hops (at the highest scale), and most messages (at lower scales) travel shorter distances. In networks that use link-level acknowledgements, this results in less congestion and resource usage by reducing message retransmissions. Simulations illustrate that the proposed scheme is more efficient than state-of-the-art randomized gossip algorithms based on averaging along paths.