Multiscale Gossip for Efficient Decentralized Averaging in Wireless\n Packet Networks

This paper describes and analyzes a hierarchical gossip algorithm for solving\nthe distributed average consensus problem in wireless sensor networks. The\nnetwork is recursively partitioned into subnetworks. Initially, nodes at the\nfinest scale gossip to compute local averages. Then, using geographic routing\nto enable gossip between nodes that are not directly connected, these local\naverages are progressively fused up the hierarchy until the global average is\ncomputed. We show that the proposed hierarchical scheme with $k$ levels of\nhierarchy is competitive with state-of-the-art randomized gossip algorithms, in\nterms of message complexity, achieving $\\epsilon$-accuracy with high\nprobability after $O\\big(n \\log \\log n \\log \\frac{kn}{\\epsilon} \\big)$\nmessages. Key to our analysis is the way in which the network is recursively\npartitioned. We find that the optimal scaling law is achieved when subnetworks\nat scale $j$ contain $O(n^{(2/3)^j})$ nodes; then the message complexity at any\nindividual scale is $O(n \\log \\frac{kn}{\\epsilon})$, and the total number of\nscales in the hierarchy grows slowly, as $\\Theta(\\log \\log n)$. Another\nimportant consequence of hierarchical construction is that the longest distance\nover which messages are exchanged is $O(n^{1/3})$ hops (at the highest scale),\nand most messages (at lower scales) travel shorter distances. In networks that\nuse link-level acknowledgements, this results in less congestion and resource\nusage by reducing message retransmissions. Simulations illustrate that the\nproposed scheme is more message-efficient than existing state-of-the-art\nrandomized gossip algorithms based on averaging along paths.\n

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