Towards a Global Agreement: the Persistent Graph

This paper investigates the role which the persistent links play for a network to reach a global agreement under discrete-time or continuous-time consensus dynamics. Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function which describes the strength of the information flow between the from one node to another. Then an arc is said to be persistent if its weight function has infinite $\mathscr{L}_1$ or $\ell_1$ norm, and the subgraph which contains all persistent arcs is called the persistent graph. A series of sufficient and necessary conditions on global agreement or $\epsilon$-agreement are established, by which we clearly address that the persistent graph fully determines the convergence to a collective agreement. Convergence rates are given explicitly relying on the diameter of the persistent graph. The results conclude that towards a global agreement, the links which last in a long period contribute fundamentally while the links which are only significant for a short time do not.