Dynamic graphs

Dynamic graphs are defined in a linear space as a one-parameter group of transformations of the graph space into itself. Stability of equilibrium graphs is formulated in the sense of Lyapunov to study motions of positive graphs in the nonnegative orthant of the graph space. Relying on the isomorphism of graphs and adjacency matrices, a new concept of dynamic connective stability of complex systems is introduced. A dynamic interaction coordinator is added to complex interconnected system to ensure that the desired level of interconnections between subsystems is preserved as a connectively stable equilibrium of the overall system despite uncertain structural perturbations. It is shown how the coordinator can be designed to adaptively adjust the interconnection levels in order to assign a prescribed state of the complex multi-agent system as a stable equilibrium point. The equilibrium assignment is achieved by the action of the coordinator which solves an optimization problem involving a two-time-scale system; the coordinator action is slow compared to the fast dynamics of the agents. Polytopic connective stability of the multi-agent systems with a coordinator is established by the concept of vector Lyapunov functions and the theory of M -matrices.