Balanced colorings of networks correspond to flow-invariant synchrony spaces. It is known that the coarsest balanced coloring is equivalent to nodes having isomorphic infinite input trees, but this condition is not algorithmic. We provide an algorithmic characterization: two nodes have the same color for the coarsest balanced coloring if and only if their [Formula: see text]th input trees are isomorphic, where [Formula: see text] is the number of nodes. Here [Formula: see text] is the best possible. The proof is analogous to that of Leighton’s theorem in graph theory, using the universal cover of the network and the notion of a symbolic adjacency matrix to set up a partition refinement algorithm whose output is the coarsest balanced coloring. The running time of the algorithm is cubic in [Formula: see text].

中文翻译：

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网络最粗平衡着色的有限表征

网络的平衡着色对应于流不变的同步空间。已知最粗略的平衡着色等价于具有同构无限输入树的节点，但这种条件不是算法性的。我们提供了一种算法表征：当且仅当它们的第 [Formula: see text] 输入树是同构的，其中 [Formula: see text] 是节点的数量时，两个节点对于最粗略的平衡着色具有相同的颜色。这里 [公式：见正文] 是最好的。证明类似于图论中的 Leighton 定理，利用网络的普遍覆盖和符号邻接矩阵的概念来建立分区细化算法，其输出是最粗糙的平衡着色。该算法的运行时间是[公式：见正文]中的三次方。