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.
翻译:在这项工作中,研究了一种在图表上可靠和高效数字整合仓本模型的新颖方法,为此,对古典仓本模型的顺序参数概念进行了重新审视,该模型描述了一组振动器的全面相互作用。第一次数字实验证实,某些金额的预估大大降低了对右侧评价的计算成本,从而可以模拟高维系统的计算成本。为了设计在网络图形相关动态系统的背景下有利的数字整合方法,提出了本地化顺序参数的概念。此外,为复杂的图表探测社区以及将基本相邻矩阵转换成块状结构是进一步改善的一个必要组成部分。事实证明,对于相对较少的等于零的系数构成的子矩阵来说,对金额的预估量是有利的,而对于补充性案例来说,直截了当的和对等表明,将有效的社区检测算法与定序参数的本地化相结合的战略可能使计算时间减少几个数量级。