Laplacian-based matrix design for finite-time average consensus in digraphs

In this paper, we consider the problem of assigning time-varying weights on the links of a time-invariant digraph, such that average consensus is reached in a finite number of steps. More specifically, we derive a finite set of weight matrices that are based on the Laplacian and the Laplacian eigenvalues of the given digraph, such that the product of these weight matrices (in any order) leads to a rank-one matrix. Using the weights associated with this sequence of weight matrices, the nodes run two linear iterations (each with its own initial conditions) and, after a finite number of steps, can calculate the average of the initial values by taking the ratio of the two values they possess at the end of the iteration process. As in the case of undirected graphs, we show that the set of matrices depends on the number of nonzero distinct eigenvalues of the Laplacian matrix. However, unlike the case for undirected graphs, the Laplacian matrix is no longer symmetric, and the number of steps depends not only on the number of distinct eigenvalues but also on their algebraic multiplicities. Illustrative examples demonstrate the validity of the derived results.