On the reliable and efficient numerical integration of the Kuramoto model and related dynamical systems on graphs

TL;DR

This paper introduces a new numerical integration approach for the Kuramoto model on graphs, leveraging localised order parameters and community detection to significantly improve computational efficiency in simulating large-scale dynamical systems.

Contribution

It proposes the concept of localised order parameters and combines community detection with precomputation strategies to enhance numerical integration of network-based dynamical systems.

Findings

Precomputation of sums reduces computational cost.

Community detection and block structure transformation improve efficiency.

Combining these methods can reduce computation time by several orders of magnitude.

Abstract

In this work, a novel approach for the reliable and efficient numerical integration of the Kuramoto model on graphs is studied. For this purpose, the notion of order parameters is revisited for the classical Kuramoto model describing all-to-all interactions of a set of oscillators. First numerical experiments confirm that the precomputation of certain sums significantly reduces the computational cost for the evaluation of the right-hand side and hence enables the simulation of high-dimensional systems. In order to design numerical integration methods that are favourable in the context of related dynamical systems on network graphs, the concept of localised order parameters is proposed. In addition, the detection of communities for a complex graph and the transformation of the underlying adjacency matrix to block structure is an essential component for further improvement. It is demonstrated that for a submatrix comprising relatively few coefficients equal to zero, the precomputation of sums is advantageous, whereas straightforward summation is appropriate in the complementary case. Concluding theoretical considerations and numerical comparisons show that the strategy of combining effective community detection algorithms with the localisation of order parameters potentially reduces the computation time by several orders of magnitude.