DeepAI AI Chat
Log In Sign Up

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

by   Run Wang, et al.

For a connected graph 𝒒=(V,E) with n nodes, m edges, and Laplacian matrix 𝐿, a grounded Laplacian matrix 𝐿(S) of 𝒒 is a (n-k) Γ— (n-k) principal submatrix of 𝐿, obtained from 𝐿 by deleting k rows and columns corresponding to k selected nodes forming a set SβŠ† V. The smallest eigenvalue Ξ»(S) of 𝐿(S) plays a pivotal role in various dynamics defined on 𝒒. 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 β‰ͺ n nodes, in order to maximize the smallest eigenvalue Ξ»(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 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.

βˆ™ 10/15/2021

Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities

In this paper, we give a spectral approximation result for the Laplacian...
βˆ™ 06/11/2021

Maximizing Influence of Leaders in Social Networks

The operation of adding edges has been frequently used to the study of o...
βˆ™ 11/03/2016

Fast Eigenspace Approximation using Random Signals

We focus in this work on the estimation of the first k eigenvectors of a...
βˆ™ 02/16/2022

Flat tori with large Laplacian eigenvalues in dimensions up to eight

We consider the optimization problem of maximizing the k-th Laplacian ei...
βˆ™ 06/01/2020

Least-squares regressions via randomized Hessians

We consider the least-squares regression problem with a finite number of...
βˆ™ 03/04/2019

Designing Optimal Multiplex Networks for Certain Laplacian Spectral Properties

We discuss the design of interlayer edges in a multiplex network, under ...
βˆ™ 09/15/2022

The Controllability and Structural Controllability of Laplacian Dynamics

In this paper, classic controllability and structural controllability un...