Towards a complete characterization of vulnerability of networked synchronization processes

We extend our previous study of vulnerability of a linear network synchronization process to external intrusion at a single network component. Our previous efforts on the topic were limited to undirected graphs. The principal limitation in generalizing to all graph classes is the complexity introduced by the possibility of a non-diagonalizable graph Laplacian. Here, we take a step towards overcoming this hurdle by developing closed-form expressions for the inverse reachability Gramian in cases where the Laplacian has a single nontrivial Jordan block. Our approach for handling a single Jordan block represents a path forward for the more general case of multiple Jordan blocks of varying sizes, and hence towards a complete graphical characterization of vulnerability of networked synchronization processes.