Graph Laplacian based matrix design for finite-time distributed average consensus

In this paper, we consider the problem of finding a linear iteration scheme that yields distributed average consensus in a finite number of steps D. By modeling interactions between the nodes in the network by means of a time-invariant undirected graph, the problem is solved by deriving a set of D Laplacian based consensus matrices. We show that the number of steps is given by the number of nonzero distinct eigenvalues of the graph Laplacian matrix. Moreover the inverse of these eigenvalues constitute the step-sizes of the involved Laplacian based consensus matrices. When communications are made through an additive white Gaussian noise channel, based on an ensemble averaging method, we show how average consensus can be asymptotically reached. Performance analysis of the suggested protocol is given along with comparisons with other methods in the literature.