Application of semidefinite programming to maximize the spectral gap produced by node removal

TL;DR

This paper introduces semidefinite programming techniques to optimize the spectral gap of a network by strategically removing nodes, enhancing network properties related to dynamics and stability.

Contribution

It presents a novel mathematical programming approach, using relaxations via semidefinite programming, to maximize the spectral gap through node removal.

Findings

Effective semidefinite programming relaxations for the problem

Application to example networks demonstrates practical utility

Potential improvements in network robustness and dynamics

Abstract

The smallest positive eigenvalue of the Laplacian of a network is called the spectral gap and characterizes various dynamics on networks. We propose mathematical programming methods to maximize the spectral gap of a given network by removing a fixed number of nodes. We formulate relaxed versions of the original problem using semidefinite programming and apply them to example networks.