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

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 localized 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 localization of order parameters potentially reduces the computation time by several orders of magnitude.

Influence of cumulative damage on synchronization of Kuramoto oscillators on networks

“…We use the fourth order Runge-Kutta method [68] implemented in C++ in the GNU Scientific Library [69]. Here, it is important to highlight that several works have shown that in the numerical solution of equation ( 2), the results do not depend significantly on the integration method implemented or on the time step ∆t used to obtain numerical solutions [23,70,71].…”

Community Integration Algorithms (CIAs) for Dynamical Systems on Networks

“…However, not all community detection algorithms pursue to optimize on that feature. Therefore, we have analyzed and compared different community detection algorithms with respect to that feature [8].…”

“…For a very specialized and particular case, we have demonstrated recently that employing simple variants of the steps (E1)-(E2) and (P1)-(P2) can work potentially work [8]. In this work, we develop the general CIA framework and show that it works in an extremely broad class of network dynamics applications, that it is robustness with regard to real data sets, that the general method does yield linear computational complexity with respect to the dimensions of the systems, and that the steps naturally extend to higher-order/polyadic dynamics.…”