Optimization of convergence rate via algebraic connectivity

The algebraic connectivity of a network characterizes the lower-bound of the exponential convergence rate of consensus processes. This paper investigates the problem of accelerating the convergence of consensus processes by adding links to the network. Based on a perturbation formula of the algebraic connectivity, we propose a greedy strategy for undirected networks and give a lower bound of its performance through an approximation of submodularity. We further extend our investigation to directed networks, where the second smallest real part among all the eigenvalues of the non-Hermitian Laplacian matrix, i.e., the generalized algebraic connectivity, indicates the expected convergence rate. We propose the metrics to evaluate the impact of an adding subgraph on the generalized algebraic connectivity, and apply a modified greedy strategy to optimize the generalized algebraic connectivity. Numerical results in empirical networks exhibit that our proposed methods outperform some other methods based on traditional topological metrics. Our research also verifies the dramatic differences between the optimization in undirected networks and that in directed networks.