Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Abstract

For a connected graph $\mathcal{G}=(V,E)$ with $n$ nodes, $m$ edges, and Laplacian matrix $L$, a grounded Laplacian matrix $L(S)$ of $\mathcal{G}$ is a $(n-k)\times (n-k)$ principal submatrix of $L$, obtained from $L$ by deleting $k$ rows and columns corresponding to $k$ selected nodes forming a set $S\subseteq V$. The smallest eigenvalue $\hat{I}»(S)$ of $L(S)$ plays a pivotal role in various dynamics defined on $\mathcal{G}$. For example, $\hat{I}»(S)$ characterizes the convergence rate of leader-follower consensus, as well as the effectiveness of a pinning scheme for the pinning control problem, with larger $\hat{I}»(S)$ 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 $k\ll n$ nodes, in order to maximize the smallest eigenvalue $\hat{I}»(S)$ 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 na\"ıve 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 na\"ıve heuristic algorithm takes $\tilde{O}(knm)$ time, while the fast greedy heuristic has a nearly linear time complexity of $\tilde{O}(km)$. 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.