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. Lower 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 specific  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.
['Weiguo Xia', 'Ming Cao']
Analysis and applications of spectral properties of grounded Laplacian matrices for directed networks