Laplacian Eigenvalue Allocation Through Asymmetric Weights in Acyclic Leader-Follower Networks

Abstract

This paper tackles the eigenvalue allocation problem through an appropriate choice of the edge weights for the class of combinatorially-symmetric Laplacian matrices of acyclic graphs, namely Laplacian matrices showing symmetric zero/nonzero values in its entries according to a tree graph pattern. The mathematical setting of the problem is remarkably suited for several current multi-agent systems engineering applications, when the communication graph is bidirectional but each agent can set the weight of each incoming neighbor value. The resulting algorithm is inherently iterative and it requires a finite time execution, so that it is well fit for real-world applications as a preliminary routine. For this reason, a special focus is devoted to a distributed implementation of the main algorithm. As a final theoretical result, it is proved that, under the strict interlacing property, the solution is positive, and the algorithm can be iterated. An illustrative example closes the paper, showing how the algorithm works in practice.

Keywords

• Laplacian eigenvalue allocation
• consensus networks
• tree graphs
• spectral graph theory