We define the anti-communicability function for the nodes of a simple graph as the nondiagonal entries of exp(-A). We prove that it induces an embedding of the nodes into a Euclidean space. The anti-communicability angle is then defined as the angle spanned by the position vectors of the corresponding nodes in the anti-communicability Euclidean space. We prove analytically that in a given k-partite graph, the anti-communicability angle is larger than 90 for every pair of nodes in different partitions and smaller than 90 for those in the same partition. This angle is then used as a similarity metric to detect the best k-partitions in networks where certain level of edge frustration exists. We apply this method to detect the best k-partitions in 15 real-world networks, finding partitions with a very low level of edge frustration. Most of these partitions correspond to bipartitions but tri- and pentapartite structures of real-world networks are also reported.

中文翻译：

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反连通性欧氏空间中的网络划分

我们将简单图的节点的反通信功能定义为exp（-A）的非对角项。我们证明它诱导了节点到欧几里得空间的嵌入。然后将反通信性角度定义为反通信欧几里德空间中相应节点的位置矢量所跨越的角度。我们通过分析证明，在给定的k部分图中，不同分区中每对节点的抗连通性角均大于90，而相同分区中的对角则小于90。然后将该角度用作相似性度量，以检测存在一定程度的边缘挫折感的网络中的最佳k分区。我们应用此方法来检测15个实际网络中的最佳k分区，查找边缘挫折感非常低的分区。