Synchronization among a set of networked nodes has attracted much attention in different fields. This paper thoroughly investigates linear formulation of the Kuramoto model, with and without frustration, for an arbitrarily weighted undirected network where all nodes may have different intrinsic frequencies. We develop a mathematical framework to estimate errors of the linear approximation for globally and locally coupled networks. We mathematically prove that the eigenvector corresponding to the largest eigenvalue of the network’s Laplacian matrix is enough for examining synchrony alignment and that the functionality of this vector depends on the corresponding eigenvalue. Moreover, we prove that if a globally coupled network with frustration has perfect phase synchronization when its coupling strength tends to infinity, it is a regular network. Finally, the effect of correlation between frustration values and degrees (or frequencies) on the synchronizability of the network is investigated.

一组联网节点之间的同步在不同领域引起了广泛关注。本文彻底研究了 Kuramoto 模型的线性公式，无论有没有挫折，对于任意加权的无向网络，其中所有节点可能具有不同的固有频率。我们开发了一个数学框架来估计全局和局部耦合网络的线性近似误差。我们在数学上证明了对应于网络拉普拉斯矩阵的最大特征值的特征向量足以检查同步对齐，并且该向量的功能取决于相应的特征值。此外，我们证明，如果一个具有挫折的全局耦合网络在其耦合强度趋于无穷大时具有完美的相位同步，则它是一个规则网络。