On the degree of synchrony

In this paper, we introduce a measure quantifying the degree of synchrony for consensus networks in terms of the angle between the network state vector and the agreement subspace of ℝⁿ. The desynchronization of a consensus network is then examined by adopting a simple two-stubborn-input strategy, through which the network synchrony can be alleviated when the inputs are symmetric with respect to the origin. In particular, for tree graphs, we show that the proposed measure of synchrony is characterized by the so-called backbone path and can be computed efficiently by examining graph-theoretic properties of the network.