Spectral properties of the grounded Laplacian matrix with applications to consensus in the presence of stubborn agents

We study linear consensus and opinion dynamics in networks that contain stubborn agents. Previous work has shown that the convergence rate of such dynamics is given by the smallest eigenvalue of the grounded Laplacian induced by the stubborn agents. Building on this, we define a notion of centrality for each node in the network based upon the smallest eigenvalue obtained by removing that node from the network. We show that this centrality can deviate from other well known centralities. We then characterize certain properties of the smallest eigenvalue and corresponding eigenvector of the grounded Laplacian in terms of the graph structure and the expected absorption time of a random walk on the graph.