Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Abstract

For a connected graph G = (V, E) with n nodes, m edges, and Laplacian matrix L, a grounded Laplacian matrix L(S) of G is a (n − k) × (n − k) principal submatrix of L, obtained from L by deleting k rows and columns corresponding to k selected nodes forming a set S ⊆ V . The smallest eigenvalue λ(S) of L(S) plays a pivotal role in various dynamics defined on G. For example, λ(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 λ(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 ≪ n nodes, in order to maximize the smallest eigenvalue λ(S) of the grounded Laplacian matrix L(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 naïve 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 naïve heuristic algorithm takes Õ(knm) time, while the fast greedy heuristic has a nearly linear time complexity of Õ(km), where Õ(•) notation suppresses the poly(log n) factors. 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 compared to baseline methods.

New methods for maximizing the smallest eigenvalue of the grounded Laplacian matrix

“…To demonstrate and compare the efficiency of the proposed methods we compare them with the two algorithms proposed by Wang et al [14]. We implemented all the algorithms in Julia 1.7.0 using the package JuMP 0.22.1.…”

“…As a simple demonstration, by applying the idea on the Laplacian matrix in Figure 1 with k = 2 we obtained λ(S) = 1.27, the output represented in Figure 2. In contrast, the greedy-type Naïve algorithm of [14] gives the set S = {2, 4} for which we obtain λ(S) = 1.2. Note that none of these methods were able to obtain the optimal solution, that is λ(S) = 1.47 with S = {1, 5}, for this simple problem.…”

“…Finding L(S) with the maximum possible λ(S) has been shown to be an NPhard problem [14]. Thus, solution methods based on heuristics are desired.…”

“…Thus, solution methods based on heuristics are desired. The authors in [14] introduced two greedy-type algorithms. The first one, referred as the Naïve algorithm, involves k iterations.…”