Multiscale Gossip for Efficient Decentralized Averaging in Wireless Packet Networks

TL;DR

This paper introduces a hierarchical gossip algorithm for wireless sensor networks that efficiently computes the average consensus with reduced message complexity and network congestion, outperforming existing methods.

Contribution

The paper presents a novel multiscale hierarchical gossip scheme with optimized network partitioning, achieving near-optimal message complexity and reduced message distances compared to prior algorithms.

Findings

Achieves $oxed{ ext{O}(n ext{ log } ext{log } n)}$ message complexity for $oxed{ ext{epsilon}}$-accuracy.

Optimal subnetwork size scales as $O(n^{(2/3)^j})$, balancing message complexity and hierarchy depth.

Most messages travel shorter distances, reducing congestion and resource usage.

Abstract

This paper describes and analyzes a hierarchical gossip algorithm for solving the distributed average consensus problem in wireless sensor networks. The network is recursively partitioned into subnetworks. Initially, nodes at the finest scale gossip to compute local averages. Then, using geographic routing to enable gossip 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 $k$ 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 $O\big(n \log \log n \log \frac{kn}{\epsilon} \big)$ messages. Key to our analysis is the way in which the network is recursively partitioned. We find that the optimal 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 $O(n \log \frac{kn}{\epsilon})$, and the total number of scales in the hierarchy grows slowly, as $\Theta(\log \log n)$. Another important consequence of 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 message-efficient than existing state-of-the-art randomized gossip algorithms based on averaging along paths.