The asymptotical error of broadcast gossip averaging algorithms

In problems of estimation and control which involve a network, efficient distributed computation of averages is a key issue. This paper presents theoretical and simulation results about the accumulation of errors during the computation of averages by means of iterative "broadcast gossip" algorithms. Using martingale theory, we prove that the expectation of the accumulated error can be bounded from above by a quantity which only depends on the mixing parameter of the algorithm and on few properties of the network: its size, its maximum degree and its spectral gap. Both analytical results and computer simulations show that in several network topologies of applicative interest the accumulated error goes to zero as the size of the network grows large.Comment: 10 pages, 3 figures. Based on a draft submitted to IFACWC201

On the contraction method with degenerate limit equation

A class of random recursive sequences (Y_n) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X\stackrel{L}{=}X. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the ``unique'' solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000017

Heuristics for Network Coding in Wireless Networks

Multicast is a central challenge for emerging multi-hop wireless architectures such as wireless mesh networks, because of its substantial cost in terms of bandwidth. In this report, we study one specific case of multicast: broadcasting, sending data from one source to all nodes, in a multi-hop wireless network. The broadcast we focus on is based on network coding, a promising avenue for reducing cost; previous work of ours showed that the performance of network coding with simple heuristics is asymptotically optimal: each transmission is beneficial to nearly every receiver. This is for homogenous and large networks of the plan. But for small, sparse or for inhomogeneous networks, some additional heuristics are required. This report proposes such additional new heuristics (for selecting rates) for broadcasting with network coding. Our heuristics are intended to use only simple local topology information. We detail the logic of the heuristics, and with experimental results, we illustrate the behavior of the heuristics, and demonstrate their excellent performance