On spectral properties of the grounded Laplacian matrix
Linear consensus and opinion dynamics in networks that contain stubborn agents are studied in this thesis. Previous works have 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 those works, we study the smallest eigenvalue of grounded Laplacian matrices, and provide bounds on this eigenvalue in terms of the number of edges between the grounded nodes and the rest of the network, bottlenecks in the network, and the smallest component of the eigenvector for the smallest eigenvalue. We show that these bounds are tight when the smallest eigenvector component is close to the largest component, and provide graph-theoretic conditions that cause the smallest component to converge to the largest component. An outcome of our analysis is a tight bound for Erdos-Renyi random graphs and $d$-regular random graphs. Moreover, we define a new 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. Finally we interprete this centrality via the notion of absorption time in a random walk on the graph.