Finding Non-Zero Stable Fixed Points of the Weighted Kuramoto model is NP-hard

TL;DR

This paper proves that determining the existence of multiple stable fixed points in the weighted Kuramoto model is NP-hard, highlighting the computational complexity of analyzing synchronization phenomena in complex networks.

Contribution

It establishes the NP-hardness of finding non-zero stable fixed points in the weighted Kuramoto model, revealing the problem's computational intractability.

Findings

Determining multiple stable fixed points is NP-hard for weighted Kuramoto models.

Unweighted case is at least as hard as the number partition problem.

Stable fixed points cannot be characterized by simple network invariants.

Abstract

The Kuramoto model when considered over the full space of phase angles [$0,2\pi$) can have multiple stable fixed points which form basins of attraction in the solution space. In this paper we illustrate the fundamentally complex relationship between the network topology and the solution space by showing that determining the possibility of multiple stable fixed points from the network topology is NP-hard for the weighted Kuramoto Model. In the case of the unweighted model this problem is shown to be at least as difficult as a number partition problem, which we conjecture to be NP-hard. We conclude that it is unlikely that stable fixed points of the Kuramoto model can be characterized in terms of easily computable network invariants.