Restrictions and stability of time-delayed dynamical networks

In this paper we present a criteria for the global stability of general time-delayed dynamical networks. We show that under our criteria a network's stability is invariant with respect to the removal of time delays and the addition of single type time delays. As modifying a network's delays can have a destabilizing effect on the system's dynamics, this introduces a new and stronger form of global stability, which we call intrinsic stability. To carry out this analysis we introduce a family of graph transformations that can be used to maintain or modify the spectral radius of a graph. We then introduce the notion of an implicit delay and show that by removing a network's implicit delays the result is a lower dimensional system, which we term a network restriction. We demonstrate that such restrictions can be used to obtain improved estimates of a network's global stability. The effectiveness of our approach is illustrated by applications to various classes of Cohen-Grossberg neural networks.