Quite often real-world networks can be thought of as being symmetric, in the abstract sense that vertices can be found to have similar or equivalent structural roles. However, traditional measures of symmetry in graphs are based on their automorphism groups, which do not account for the similarity of local structures. We introduce the concept of local symmetry, which reflects the structural equivalence of the vertices’ egonets. We study the emergence of asymmetry in the Erdõs-Rényi random graph model and identify regimes of both asymptotic local symmetry and asymptotic local asymmetry. We find that local symmetry persists at least to an average degree of $n^{2/5}$ and local asymmetry emerges at an average degree not greater than $n^{1/2}$, which are regimes of much larger average degree than for traditional, global asymmetry.

中文翻译：

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随机图中的局部对称

现实世界中的网络通常可以被认为是对称的，抽象意义上可以发现顶点具有相似或等效的结构角色。然而，图的传统对称性度量是基于它们的自同构群，这不考虑局部结构的相似性。我们引入了概念局部对称，这反映了顶点的自我的结构等价性。我们研究了 Erdõs-Rényi 随机图模型中不对称的出现，并确定了渐近局部对称和渐近局部不对称的机制。我们发现局部对称性至少持续到平均程度$n^{2/5}$ 并且局部不对称出现的平均程度不大于 $n^{1/2}$，这是比传统的全局不对称性大得多的平均程度的制度。