Coherence and convergence rate in networked dynamical systems

We study two metrics in stochastic consensus dynamics with leaders or stubborn agents: network coherence (defined in terms of the system ℋ₂ and ℋ_∞ norms), and convergence rate. We allow each agent to maintain an individual level of stubbornness in deviating from its initial values. We give bounds on the convergence rate and present sufficient conditions under which the bounds become tight. Moreover we study the effect of the level of stubbornness of the agents on network coherence and convergence rate. We then characterize these two metrics in random regular graphs and Erdos-Renyi random graphs. From a leader selection point of view, we show that maximizing ℋ_∞ coherence is equivalent to maximizing convergence rate. Moreover we study conditions under which the optimal leader for maximizing ℋ₂ coherence differs from the optimal leader for maximizing convergence rate, and conversely, provide sufficient conditions on the network for a single leader to maximize both metrics simultaneously.