Average consensus in sensor networks via broadcast multi-gossip algorithms Huiwei Wang a,n,1 , Xiaofeng Liao a , Tingwen Huang b a State Key Laboratory of Power Transmission Equipment & System Security and New Technology, College of Computer Science, Chongqing University, Chongqing 400030, PR China b Texas A&M University at Qatar, PO Box 23874, Doha, Qatar article info Article history: Received 24 July 2012 Received in revised form 18 January 2013 Accepted 21 January 2013 Communicated by Y. Liu Available online 26 February 2013 Keywords: Broadcasting Distributed average consensus Gossip algorithms Push-sum protocol Weak ergodicity Wireless sensor networks abstract Motivated by applications to wireless sensor, peer-to-peer, and ad hoc networks, we propose a distributed algorithm called broadcast based multi-gossiping algorithm (BMGA), which is designed for exchanging information and computing in an arbitrarily connected network of nodes. Unlike traditional randomized gossip algorithms, push-sum mechanism based BMGA preserves the sums and weights, and admits stochastic diffusion matrices which need not be doubly stochastic. Based on the theory of weak ergodicity and message spreading, we derive a lower bound on the weight, and give an approximate value for this bound. By introducing a potential function, we show that BMGA converges almost surely to the average of initial node measurements with probability one. Specifically, we further provide the upper bounds on the diffusion speed, E-convergence time and the number of radio transmissions. Finally, we present a numerical example to assess and compare the communication cost with several gossip-based algorithms to achieve a given performance. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Consensus problems have a long history in computer science due to their extensive application on distributed decision-making and parallel computing, and further form the foundation of the field of distributed algorithms [1]. Distributed consensus, which was formally discussed in the pioneering work by Borkar and Varaiya [2] and Tsitsiklis [3], has been identified as a canonical problem in signal processing and cooperative control. For more details, the readers may refer to the monograph [1], some recent surveys [4–6] and the references therein. Consensus framework can be roughly classified as synchronous and asynchronous [7]. At each iteration of the synchronous algorithms, all the nodes of the network update uniformly their current estimate by computing a weighted estimates of their neighbors. Unlike the known synchronous algorithms, the main type of asynchronous consensus are gossip-based algorithms, where at each iteration, only one random node wakes up and randomly chooses another one or a few nodes from its neighbors, and exchanges information with these nodes. In the past few years, synchronous [4] and gossip-based asynchronous consensus algorithms [8–10] have recently received significant attention, mainly because they constitute simple and robust algorithms for distributed information processing over networks. Both approaches have merit. The major advantages of gossip-based protocols include threefold: (1) they usually do not require error recovery mechanisms; (2) the guar- antees obtained from gossiping are usually probabilistic in nature, which will ensure to achieve high stability under stress and disruptions; (3) gossip-based protocols can solve gracefully distributed consensus over networks with a huge number of nodes [8]. In comparison, synchronous algorithms have absolute guarantees, but they are unstable or fail to make progress during periods of even modest disruption. This paper takes the gossip- based algorithm and focuses on the distributed average consensus problem in wireless communication networks. 1.1. Related work There has been considerable interest recently in developing algorithms for distributing information among the members of a group of sensors or mobile autonomous agents via local interac- tions. Notable among these is those algorithms intended to cause such a group to reach a consensus in a distributed manner. Gossip-based average consensus algorithm was initially intro- duced by Tsitsiklis [3] and attracting considerable recent atten- tion from other researchers [5,6,8–14,21,22,26]. Sum-preserving gossip: Uniform gossip algorithm (UGA) based on push-sum protocol was described and analyzed in [8], where in each round, each node maintains the sums and weights, then chooses a node from its neighbors uniformly at random and Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/neucom Neurocomputing 0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.01.013 n Corresponding author. Tel.: þ86 23 65106393. E-mail address: huiwei.wang@gmail.com (H. Wang). 1 Part of this work was done when H. Wang was with the Science Program, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar. Neurocomputing 117 (2013) 150–160