AI 中文总结
该研究针对耦合振子网络，推导了临界耦合强度对网络结构变化的一阶公式，提出布雷斯悖论发生的局部判据，并在随机网络中验证了其预测效果，建立了网络拓扑变化产生反直觉同步结果的预测框架。
AI 中文摘要
我们研究耦合振子网络中的布雷斯型悖论，在这类网络中，向网络添加一条边会提高存在稳定锁相同步解所需的临界耦合强度。通过引入连续边权重参数并构建增广系统以应用隐函数定理，我们推导出了临界耦合强度对网络结构微小变化的一阶显式公式，即$K'_c(0)$。该公式表明，若被耦合振子的排序与鞍结分岔处临界特征向量中各分量的排序相反，则该边是局部布雷斯边，这为布雷斯悖论的发生提供了简单判据。随后，我们将该局部近似作为边完全纳入网络时布雷斯悖论发生的预测因子，在几类随机网络中对该预测进行数值测试，结果显示其兼具高预测保真度和局部近似的非线性局限性。我们还检验了若干具体实例，凸显了布雷斯型悖论发生的可能基元模式。综上，这些结果建立了一个预测框架，用于理解网络拓扑的增量变化如何产生反直觉的同步结果。
英文摘要
We study Braess-type paradoxes in coupled oscillator networks where the addition of an edge to the network leads to an increase in the critical coupling required for the existence of a stable phase-locked synchronized solution. By introducing a continuous edge-weight parameter and constructing an augmented system that allows application of the Implicit Function Theorem, we derive an explicit first-order formula for the sensitivity of the critical coupling, $K'_c(0)$, to small changes in the network structure. This formula reveals that an edge is locally Braess if the ordering of the oscillators being coupled is opposite the ordering of their components in the critical eigenvector at the saddle-node bifurcation, providing a simple criterion for Braess paradox to occur. We then employ this local approximation as a predictor for the occurrence of Braess paradox when the edge is fully incorporated in the network. This prediction is tested numerically in several classes of random networks, demonstrating both high predictive fidelity and the nonlinear limitations of the local approximation. We also examine several explicit examples, highlighting possible motifs by which Braess-type paradoxes occur. Together, these results establish a predictive framework for understanding how incremental changes in network topology can produce counterintuitive synchronization outcomes.
