On the Optimal Laplacian Jordan Structure for Synchronizability

Abstract

In this work, for a network of differential equations with a prescribed graph structure, our goal is to show how to select the Laplacian of the network in order to obtain the most favorable outcome insofar as synchronizability. That is, we will want to: (i) guarantee asymptotic stability of a synchronous solution (as measured by a negative value of the master stability function), (ii) minimize the normalized spread of the Laplacian eigenvalues, and (iii) have a transient as short as possible. Within the class of tridiagonal Laplacians, we give both necessary and sufficient conditions for satisfying our three criteria above, and give extension to banded Laplacians as well. Finally, we give extensive numerical results to elucidate our theoretical results and to compare to existing works.