# Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

For a connected graph π’=(V,E) with n nodes, m edges, and Laplacian matrix πΏ, a grounded Laplacian matrix πΏ(S) of π’ is a (n-k) Γ (n-k) principal submatrix of πΏ, obtained from πΏ by deleting k rows and columns corresponding to k selected nodes forming a set Sβ V. The smallest eigenvalue Ξ»(S) of πΏ(S) plays a pivotal role in various dynamics defined on π’. For example, Ξ»(S) characterizes the convergence rate of leader-follower consensus, as well as the effectiveness of a pinning scheme for the pinning control problem, with larger Ξ»(S) corresponding to smaller convergence time or better effectiveness of a pinning scheme. In this paper, we focus on the problem of optimally selecting a subset S of fixed k βͺ n nodes, in order to maximize the smallest eigenvalue Ξ»(S) of the grounded Laplacian matrix πΏ(S). We show that this optimization problem is NP-hard and that the objective function is non-submodular but monotone. Due to the difficulty to obtain the optimal solution, we first propose a naΓ―ve heuristic algorithm selecting one optimal node at each time for k iterations. Then we propose a fast heuristic scalable algorithm to approximately solve this problem, using derivative matrix, matrix perturbations, and Laplacian solvers as tools. Our naΓ―ve heuristic algorithm takes Γ(knm) time, while the fast greedy heuristic has a nearly linear time complexity of Γ(km). We also conduct numerous experiments on different networks sized up to one million nodes, demonstrating the superiority of our algorithm in terms of efficiency and effectiveness.

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