Accelerating Consensus Gossip Algorithms: Sparsifying Networks Can Be Good for You

Abstract

In this paper, we consider the problem of improving the convergence speed of an average consensus gossip algorithm by sparsifying a sufficiently dense network graph. Thus, instead of adding links, as usually proposed in the literature, or globally optimizing the mixing matrix of the gossip algorithm for a given network, which requires global knowledge at every node, we find a sparser network that has better spectral properties and faster convergence than the original denser one. This allows to reduce simultaneously both the convergence time and the communication cost involved in the execution of the gossip algorithm. We first show why it is possible to sparsify a network while increasing its convergence rate and also that there exists an optimal fraction of links to be removed. As a benchmark, we devise a centralized method that selects in an optimal way the set of links to be removed from the original network. Then, we propose a low complexity and scalable decentralized protocol requiring only local information at each node, which also generates a sparser network having a substantially better convergence rate. Simulation results are presented to verify and show clearly the efficiency of our approach.

Network Topology Optimization for Accelerating Consensus Algorithms under Power Constraints

“…6. Convergence time comparison between the two methods proposed in this manuscript and the works presented in [8][9][10] and [6]. Even the convergence time in [8] is slightly faster, the resulting lifetime of applying our methods is at least 20% longer.…”

“…Finally, our distributed method performs well when it is compared with the methods presented in the literature regarding the three parameters, and it is competitive when compared with our centralized approach in which global knowledge is used. original topology [9] sparsifying method [6] growing graphs [10] distributed approach centralized approach topology control [8] Fig. 6.…”

Distributed Adjacency Weight Design for Second‐order Consensus in Wireless Sensor and Actuator Networks

“…It is known that the algebraic connectivity of a graph will decrease by removing some links from it [18]. Therefore, for the sparsified matrices of W obtained from (14) and (15), the convergence factor of (6) based on the solution to (15) will certainly larger than the one based on the solution to (14).…”

“…Therefore Vecchio , Yang , Lin and Chamie investigated how to minimize the convergence time by properly designing the adjacency weights among the network. Gnecco and Asensio verified that the convergence speed can be speeded up by sparsifying a sufficiently dense network graph.…”

On identifying sparse representations of consensus networks

“…A sparse representation of a dense graph can identify valuable communication links or facilitate understanding of the underlying dynamics of the original graph. Recent related work in Fardad et al (2011) and Lin et al (2012) deals with designing a sparse network to minimize input-output variance amplification or focuses on improving the algebraic connectivity of an existing network by adding edges in Ghosh and Boyd (2006) or removing edges of uniformly weighted graphs in Asensio-Marco and Beferull-Lozano (2010). This paper deals with the problem of removing edges from an existing dense network and preserving its input-output behavior as measured by the H 2 norm.…”