We investigate the impact of attractive-repulsive interaction in networks of limit cycle oscillators. Mainly we focus on the design principle for generating an antiphase state between adjacent nodes in a complex network. We establish that a partial negative control throughout the branches of a spanning tree inside the positively coupled limit cycle oscillators works efficiently well in comparison with randomly chosen negative links to establish zero frustration (antiphase synchronization) in bipartite graphs. Based on the emergence of zero frustration, we develop a universal $0\text{−}\pi$ rule to understand the antiphase synchronization in a bipartite graph. Further, this rule is used to construct a nonbipartite graph for a given nonzero frustrated value. We finally show the generality of $0\text{−}\pi$ rule by implementing it in arbitrary undirected nonbipartite graphs of attractive-repulsively coupled limit cycle oscillators and successfully calculate the nonzero frustration value, which matches with numerical data. The validation of the rule is checked through the bifurcation analysis of small networks. Our work may unveil the underlying mechanism of several synchronization phenomena that exist in a network of oscillators having a mixed type of coupling.

中文翻译：

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排斥连接对有吸引力耦合网络中的挫败感的影响

我们研究了极限循环振荡器网络中的排斥排斥相互作用的影响。我们主要关注于在复杂网络中相邻节点之间生成反相状态的设计原理。我们确定，与在二部图中建立零挫折（反相同步）的随机选择的负链接相比，在正耦合极限周期振荡器内部的生成树的整个分支中的局部负控制都可以很好地工作。基于零挫折的出现，我们开发了通用$0\text{-}\pi$规则以了解二部图中的反相同步。此外，该规则用于为给定的非零受挫值构造非二分图。我们终于证明了$0\text{-}\pi$通过在吸引-排斥耦合极限循环振荡器的任意无向非二分图中实现该规则，成功计算出与数值数据匹配的非零挫折值。通过小型网络的分叉分析来检查规则的有效性。我们的工作可能会揭示具有混合类型耦合的振荡器网络中存在的几种同步现象的潜在机制。