  The paper presents higher dimension consensus (HDC) for large-scale networks.
HDC generalizes the well-known average-consensus algorithm. It divides the
nodes of the large-scale network into anchors and sensors. Anchors are nodes
whose states are fixed over the HDC iterations, whereas sensors are nodes that
update their states as a linear combination of the neighboring states. Under
appropriate conditions, we show that the sensor states converge to a linear
combination of the anchor states. Through the concept of anchors, HDC captures
in a unified framework several interesting network tasks, including distributed
sensor localization, leader-follower, distributed Jacobi to solve linear
systems of algebraic equations, and, of course, average-consensus. In many
network applications, it is of interest to learn the weights of the distributed
linear algorithm so that the sensors converge to a desired state. We term this
inverse problem the HDC learning problem. We pose learning in HDC as a
constrained non-convex optimization problem, which we cast in the framework of
multi-objective optimization (MOP) and to which we apply Pareto optimality. We
prove analytically relevant properties of the MOP solutions and of the Pareto
front from which we derive the solution to learning in HDC. Finally, the paper
shows how the MOP approach resolves interesting tradeoffs (speed of convergence
versus quality of the final state) arising in learning in HDC in resource
constrained networks.
