U. A. Khan, "High-dimensional consensus in large-scale networks: Theory and
applications," *PhD Thesis*, ECE Department,
Carneigie Mellon University, Aug. 2009.

## Abstract

In this thesis, we develop the theory of \emph{High Dimensional Consensus} (HDC), a general class of distributed algorithms in large-scale networks. HDC relies only on (i) local information, (ii) local communication, and (iii) low-order computation, and, hence, is ideally suited to implement network tasks under resource constraints, e.g., in sparse networks with a limited computation budget. HDC, in general, is iterative because the underlying sparsity of the network limits the information flow. Each HDC iteration is a simple (linear or non-linear) update at each node in the network. In this context, HDC partitions the network nodes into two classes: (i) sensors, nodes that update their state as some function of their neighboring nodes; and (ii) anchors, nodes whose states are fixed. HDC includes as special cases several existing well-known algorithms, for example, average-consensus and the Jacobi algorithm. We show also how to cast a banded matrix inversion algorithm in the HDC framework.

Using HDC, we derive a novel sensor localization algorithm that is distributed, iterative, and linear. With this algorithm, each sensor (with unknown location) updates its location estimate as a convex combination of its neighbors, where the coefficients of the convex combination are the barycentric coordinates computed locally by Cayley-Menger determinants. We show that this localization algorithm converges to exact sensor locations, if all the sensors lie in the convex hull of a minimal number, $m+1$, of anchors (with known locations), where $m$ is the dimension of the space.

We divide our theoretical contributions regarding HDC into two parts: (i) analysis of HDC; and (ii) synthesis of HDC. The analysis part studies the convergence of HDC, establishes the conditions under which HDC converges, and derives its convergence rate. It shows that the HDC converges under very general conditions, in particular, linear non-convex updates, where the updating coefficients may be negative. The synthesis part designs the HDC, for example, its parameters and weights, under network constraints, such that it converges to a pre-specified state. The thesis presents practical applications of HDC to very diverse areas including: (i) average-consensus with non-linear updates; (ii) distributed sensor localization; (iii) distributed banded matrix inversion; (iv) distributed estimation in complex dynamical systems; and (v) modeling, estimation, and inference in electrical power grids.