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The algebraic connectivity of a network characterizes the lower-bound of theexponential convergence rate of consensus processes. This paper investigatesthe problem of accelerating the convergence of consensus processes by addinglinks to the network. Based on a perturbation formula of the algebraicconnectivity, we propose a greedy strategy for undirected networks and give alower bound of its performance through an approximation of submodularity. Wefurther extend our investigation to directed networks, where the secondsmallest real part among all the eigenvalues of the non-Hermitian Laplacianmatrix, i.e., the generalized algebraic connectivity, indicates the expectedconvergence rate. We propose the metrics to evaluate the impact of an addingsubgraph on the generalized algebraic connectivity, and apply a modified greedystrategy to optimize the generalized algebraic connectivity. Numerical resultsin empirical networks exhibit that our proposed methods outperform some othermethods based on traditional topological metrics. Our research also verifiesthe dramatic differences between the optimization in undirected networks andthat in directed networks.

Optimization of convergence rate via algebraic connectivity