Convergence Analysis of Distributed Consensus Algorithms

Inspired by new emerging technologies and networks of devices with high collective computational power, I focus my work on the problematics of distributed algorithms. While each device runs a relatively simple algorithm with low complexity, the group of interconnected units (agents) determines a behavior of high complexity. Typically, such units have their own memory and processing unit, and are interconnected and capable to exchange information with each other. More specifically, this work is focused on the distributed consensus algorithms. Such algorithms allow the agents to coordinate their behaviour and to distributively find a common agreement (consensus). To understand and analyze their behaviour, it is necessary to analyze the convergence of the consensus algorithm, i.e., under which conditions the algorithm reaches a consensus and under which it does not. Naturally, the communication channel can change and the agents may function asynchronously and improperly. All such errors may lead to severe problems and influence the reached consensus. Note that the target platform of the algorithms presented in this thesis is a wireless sensor network. Nevertheless, since the area of potential applications of distributed algorithms is, in general, much broader, the results of this thesis may be applicable also elsewhere, without significant changes. The work can be divided into two main parts. First, I focus on the convergence analysis of the distributed consensus algorithms. At the beginning, I review the spectral graph theory and classical results on the convergence of the average consensus algorithm. Next, I propose a unifying framework for describing distributed algorithms. In this framework, I analyze the behaviour of a quantized consensus algorithm and show bounds on the quantization error. Furthermore, I discuss the asynchronous consensus algorithms and derive necessary conditions on the convergence. Later on, I derive also bounds on the mixing weights of such asynchronous algorithms. The asynchronous consensus algorithms are analyzed by the concept of so-called “state-transition” matrices. The first part of the thesis is concluded with the analysis of a dynamic consensus algorithm, where I prove novel bounds on the convergence time and rate, and propose a generalized dynamic consensus algorithm. In the second part, I propose two novel distributed algorithms which directly utilize the consensus algorithms discussed in the first part. The “likelihood consensus” algorithm allows to distributively compute a joint-likelihood function, and thus, to distributively solve statistical inference problems (e.g., target tracking). The distributed Gram-Schmidt orthogonalization algorithm, on the other hand, can find a set of orthogonal vectors from general vectors stored at the nodes. As an application of the orthogonalization algorithm, I further propose algorithms for estimating the size of a network.