Location-Aided Fast Distributed Consensus in Wireless Networks

Abstract

Existing works on distributed consensus explore linear iterations based on reversible Markov chains, which contribute to the slow convergence of the algorithms. It has been observed that by overcoming the diffusive behavior of reversible chains, certain nonreversible chains lifted from reversible ones mix substantially faster than the original chains. In this paper, we investigate the idea of accelerating distributed consensus via lifting Markov chains, and propose a class of Location-Aided Distributed Averaging (LADA) algorithms for wireless networks, where nodes' coarse location information is used to construct nonreversible chains that facilitate distributed computing and cooperative processing. First, two general pseudo-algorithms are presented to illustrate the notion of distributed averaging through chain-lifting. These pseudo-algorithms are then respectively instantiated through one LADA algorithm on grid networks, and one on general wireless networks. For a k × k grid network, the proposed LADA algorithm achieves an ǫ-averaging time of O(k log(ǫ −1 )). Based on this algorithm, in a wireless network with transmission range r, an ǫ-averaging time of O(r −1 log(ǫ −1 )) can be attained through a centralized algorithm. Subsequently, we present a fully-distributed LADA algorithm for wireless networks, which utilizes only the direction information of neighbors to construct nonreversible chains. It is shown that this distributed LADA algorithm achieves the same scaling law in averaging time as the centralized scheme in wireless networks for all r satisfying the connectivity requirement. The constructed chain attains the optimal scaling law in terms of an important mixing metric, the fill time, among all chains lifted from one with an approximately uniform stationary distribution on geometric random graphs. Finally, we propose a cluster-based LADA (C-LADA) algorithm, which, requiring no central coordination, provides the additional benefit of reduced message complexity compared with the distributed LADA algorithm.

Polynomial Filtering for Fast Convergence in Distributed Consensus

“…We build a network of sensors and apply polynomial filtering on the maximum-degree weight matrix , given in (26). We use and solve the optimization problem OPT2 using the maximum-degree matrix as input.…”

“…The lifted graph is then "projected" back to the original graph, where the dynamics of the lifted Markov chain are simulated subject to the original graph topology. We mention the work in [26], which proposes a fast distributed averaging algorithm for geographic random graphs. In particular, the location information of the sensors is assumed to be known and is used in order to construct a nonreversible lifted Markov chain that mixes faster than corresponding reversible chains.…”

Fast Decentralized Averaging via Multi-scale Gossip

“…However, the only known scaling laws for this approach are for a deterministic, synchronous variant of gossip [10], leading to Θ( n 1.5 √ log n log −1 ) communication cost. Gossip algorithms based on lifted Markov chains have been proposed that achieve similar scaling laws [11,12].…”

Order-Optimal Consensus Through Randomized Path Averaging

“…Exact localization can be hard to achieve and approximate algorithms (e.g., [2]) can be used to give partial location information. It is probable that partial location is sufficient (as it is for geographic gossip, e.g., [17]) for efficient path averaging gossip but the analysis remains as an open problem.…”