Mean Square Convergence of Consensus Algorithms in Random WSNs

Abstract

Abstract-Distributed consensus algorithms for estimation of parameters or detection of events in wireless sensor networks have attracted considerable attention in recent years. A necessary condition to achieve a consensus on the average of the initial values is that the topology of the underlying graph is balanced or symmetric at every time instant. However, communication impairments can make the topology vary randomly in time, and instantaneous link symmetry between pairs of nodes is not guaranteed unless an acknowledgment protocol or an equivalent approach is implemented. In this paper, we evaluate the convergence of the consensus algorithm in the mean square sense in wireless sensor networks with random asymmetric topologies. For the case of links with equal probability of connection, a closed form expression for the mean square error of the state along with the dynamical range and the optimum value of the link weights that guarantee convergence are derived. For the case of links with different probabilities of connection, an upper bound for the mean square error of the state is derived. This upper bound can be computed for any time instant and can be employed to compute a link weight that reduces the convergence time of the algorithm.Index Terms-Asymmetric links, mean average consensus, mean square convergence, random topology, wireless sensor networks.

Implementing MAS Agreement Processes Based on Consensus Networks

“…There exist several well known techniques to analyze it in advance, but it is out of the scope of this paper. We refer the reader to [8] for a broader explanation of this topic. Figure 5 shows the real execution of the same consensus process in the MAGEN-TIX2 implementation.…”

Reaching consensus in asynchronous WSNs: Algebraic approach

“…Typically, results are known for convergence to the average consensus in the mean or mean square sense (e.g. [12], [17]). In [16] it was proposed to use the notion of the joint spectral radius of paracontracting matrices to compute the convergence rate.…”

“…In [17] such analysis has been applied to the standard average consensus algorithm obtained by ensemble ave-raging over a set of graphs representing switching networks. As the obtained MSE value is not zero, it shows that for such networks the individual result may be off from the desired average consensus.…”

Linear analysis of the vectorial network model in the presence of leaders

“…Porfiri (2014). Therein, the system dynamics is linearized for η ≪ 1, resulting in a linear stochastic consensus problem (Cao, Yu, Ren, & Chen, 2013;Hatano & Mesbahi, 2005;Huang & Manton, 2010;Kar & Moura, 2008;Patterson, Bamieh, & El Abbadi, 2010;Pereira & Pages-Zamora, 2010;Porfiri & Stilwell, 2007;Tahbaz-Salehi & Jadbabaie, 2008;Wu, 2006;Zhou & Wang, 2009). By adapting the algebraic analysis proposed for mean square consensus in Refs.…”