From weighted to unweighted graphs in Synchronizing Graph Theory

TL;DR

This paper introduces a method to derive unweighted graphs from weighted ones that preserve stable equilibria in the Kuramoto model, proving the NP-Hardness of stable equilibrium existence and establishing a new synchronization bound.

Contribution

It presents a novel approach to convert weighted graphs into unweighted graphs while maintaining stable equilibria in the Kuramoto model, and proves the NP-Hardness of equilibrium existence.

Findings

Stable equilibria are preserved under the new graph association method.

Existence of linearly stable equilibria is NP-Hard to determine.

A new lower bound for minimum degree ensuring synchronization is established.

Abstract

A way to associate unweighted graphs from weighted ones is presented, such that linear stable equilibria of the Kuramoto homogeneous model associated to both graphs coincide, i.e., equilibria of the system $\dot\theta_i = \sum_{j \sim i} \sin(\theta_{j}-\theta_j)$, where $i\sim j$ means vertices $i$ and $j$ are adjacent in the corresponding graph. As a consequence, the existence of linearly stable equilibrium is proved to be NP-Hard as conjectured by R. Taylor in 2015 and a new lower bound for the minimum degree that ensures synchronization is found.