Balanced colorings of networks classify robust synchrony patterns — those that are defined by subspaces that are flow-invariant for all admissible ODEs. In symmetric networks, the obvious balanced colorings are orbit colorings, where colors correspond to orbits of a subgroup of the symmetry group. All other balanced colorings are said to be exotic. We analyze balanced colorings for two closely related types of network encountered in applications: trained Wilson networks, which occur in models of binocular rivalry, and opinion networks, which occur in models of decision making. We give two examples of exotic colorings which apply to both types of network, and prove that Wilson networks with at most two learned patterns have no exotic colorings. We discuss in general terms how exotic colorings affect the existence and stability of branches for local bifurcations of the corresponding model ODEs, both to equilibria and to periodic states.

中文翻译：

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竞争和意见网络中的平衡色彩和分歧

网络的平衡着色对稳健的同步模式进行分类——这些模式由对所有可接受的 ODE 都是流不变的子空间定义的。在对称网络中，明显的平衡着色是轨道着色，其中颜色对应于对称群的子群的轨道。据说所有其他平衡的颜色都是异国情调的。我们分析了在应用中遇到的两种密切相关的网络类型的平衡着色：训练有素的威尔逊网络，出现在双眼竞争模型中，以及意见网络，出现在决策模型中。我们给出了两个适用于这两种网络类型的奇异着色的例子，并证明具有最多两种学习模式的威尔逊网络没有奇异着色。