Bounds on the Smallest Eigenvalue of a Pinned Laplacian Matrix

In this note, we study a networked system with single/multiple pinning. Given a weighted and undirected network, we derive lower and upper bounds on its algebraic connectivity with respect to the reference signal. The bounds are derived by partitioning the network in terms of distance of each node from the pinning set. Upper and lower bounds for two networks with differing topologies are computed to demonstrate the tightness of the derived results. It is shown, using the derived bounds, how requirements on the number of pinning nodes and pinning gain required for achieving stability or a specified convergence rate for the network can be easily obtained.