On the Smallest Eigenvalue of Grounded Laplacian Matrices

We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for Erdos-Renyi random graphs, we show that when a (sufficiently small) set $S$ of rows and columns is removed from the Laplacian, and the probability $p$ of adding an edge is sufficiently large, the smallest eigenvalue of the grounded Laplacian asymptotically almost surely approaches $|S|p$. We also show that for random $d$-regular graphs with a single row and column removed, the smallest eigenvalue is $\Theta(\frac{d}{n})$. Our bounds have applications to the study of the convergence rate in continuous-time and discrete-time consensus dynamics with stubborn or leader nodes

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 dynam- ics 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 small- est 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 de ne 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 interpret this centrality via the notion of absorption time in a random walk on the graph

Analysis and applications of spectral properties of grounded Laplacian matrices for directed networks

In-depth understanding of the spectral properties of grounded Laplacian matrices is critical for the analysis of convergence speeds of dynamical processes over complex networks, such as opinion dynamics in social networks with stubborn agents. We focus on grounded Laplacian matrices for directed graphs and show that their eigenvalues with the smallest real part must be real. Power and upper bounds for such eigenvalues are provided utilizing tools from nonnegative matrix theory. For those eigenvectors corresponding to such eigenvalues, we discuss two cases when we can identify the vertex that corresponds to the smallest eigenvector component. We then discuss an application in leader-follower social networks where the grounded Laplacian matrices arise naturally. With the knowledge of the vertex corresponding to the smallest eigenvector component for the smallest eigenvalue, we prove that by removing or weakening specic directed couplings pointing to the vertex having the smallest eigenvector component, all the states of the other vertices converge faster to that of the leading vertex. This result is in sharp contrast to the well-known fact that when the vertices are connected together through undirected links, removing or weakening links does not accelerate and in general decelerates the converging process