关于图上Kuramoto模型及相关动力系统的可靠有效的数值积分
On the reliable and efficient numerical integration of the Kuramoto model and related dynamical systems on graphs
摘要：
在这项工作中，一种新的方法，可靠和有效的数值积分的仓本模型的图进行了研究。为了这个目的，序参数的概念被重新审视的经典仓本模型描述的所有对所有的相互作用的一组振子。第一数值实验证实，某些总和的预先计算显着降低了计算成本的右手边的评估，因此，使高维系统的模拟。为了设计数值积分方法，是有利的背景下，有关的动力系统的网络图，局部序参数的概念提出。此外，复杂图的社区检测和底层邻接矩阵到块结构的转换是进一步改进的必要组成部分。它表明，对于一个子矩阵，包括相对较少的系数等于零，预先计算的总和是有利的，而简单的求和是适当的互补的情况下。总结理论的考虑和数值比较表明，有效的社区检测算法与序参数的本地化相结合的策略，可能会减少计算时间的几个数量级。
英文摘要：
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.
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