We analyze ensembles of random networks with fixed numbers of edges, triangles, and nodes. In the limit as the number of nodes goes to infinity, this model is known to exhibit sharp phase transitions as the density of edges and triangles is varied. In this paper we study finite size effects in ensembles where the number of nodes is between 30 and 66. Despite the small number of nodes, the phases and phase transitions are clearly visible. Concentrating on 54 nodes and one particular phase transition, we use spectral data to examine the structure of a typical graph in each phase, and see that is it very similar to the graphon that describes the system as n diverges. We further investigate how graphs change their structure from that of one phase to the other under a natural edge flip dynamics in which the number of triangles is either forced downwards or allowed to drift upwards. In each direction, spectral data shows that the change occurs in three stages: a very fast stage of about 100 edge flips, in which the triangle count reaches the targeted value, a slower second stage of several thousand edge flips in which the graph adopts the qualitative features of the new phase, and a very slow third stage, requiring hundreds of thousands of edge flips, in which the fine details of the structure equilibrate.

中文翻译：

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有限随机网络中的相变

我们分析具有固定数量的边、三角形和节点的随机网络的集合。在节点数量趋于无穷大的极限中，已知该模型随着边缘和三角形的密度变化而表现出急剧的相变。在本文中，我们研究了节点数在 30 到 66 之间的集合中的有限尺寸效应。尽管节点数很少，但相和相变清晰可见。专注于 54 个节点和一个特定的相变，我们使用光谱数据来检查每个相中典型图的结构，并看到它与将系统描述为 n 发散的图非常相似。我们进一步研究了图如何在自然边缘翻转动态下将其结构从一个阶段的结构更改为另一个阶段，其中三角形的数量要么被迫向下要么允许向上漂移。在每个方向，光谱数据显示变化发生在三个阶段：大约 100 次边缘翻转的非常快的阶段，其中三角形计数达到目标值，数千次边缘翻转的较慢第二阶段，其中图形采用新阶段的定性特征，以及非常缓慢的第三阶段，需要数十万次边缘翻转，其中结构的细节达到平衡。