Responsible eigenvalue concept for the stability of a class of single-delay consensus dynamics with fixed topology

This study investigates the stability of a class of single-delay interconnected dynamics where dynamics share only delayed information with each other while seeking consensus in a large scale network with fixed topology. The coupled dynamics, which are infinite dimensional because of the presence of delays, can remain stable for at most a certain amount of delays called the delay margin. This margin is intricately determined by the nature of the dynamics, the network connectivity and coupling strengths among the dynamics. Here we present a systematic approach to correlate the finite dimensional network graph properties to the delay margin associated with an infinite dimensional eigenvalue problem. In particular, the developed mathematical approach leads to the responsible eigenvalue concept, which becomes the one and only one eigenvalue that directly determines the delay margin of the entire network. Case studies are provided to demonstrate the effectiveness of the approach as well as the connections between delay margin and the eigenvalues of the corresponding graph Laplacian.