  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.
