Although it is well-known that some exponential family random graph model (ERGM) families exhibit phase transitions (in which small parameter changes lead to qualitative changes in graph structure), the behavior of other models is still poorly understood. Recently, Krivitsky and Morris have reported a previously unobserved phase transition in the edge/concurrent vertex family (a simple starting point for models of sexual contact networks). Here, we examine this phase transition, showing it to be a first order transition with respect to an order parameter associated with the fraction of concurrent vertices. This transition stems from weak cooperativity in the recruitment of vertices to the concurrent phase, which may not be a desirable property in some applications.

中文翻译：

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边/并发顶点模型中的相变

尽管众所周知，一些指数族随机图模型 (ERGM) 族表现出相变（其中小的参数变化会导致图结构发生质的变化），但对其他模型的行为仍知之甚少。最近，Krivitsky 和 ​​Morris 报告了边缘/并发顶点族（性接触网络模型的一个简单起点）中以前未观察到的相变。在这里，我们检查这个相变，表明它是关于与并发顶点的分数相关联的顺序参数的一阶转换。这种转变源于将顶点招募到并发阶段的弱协同性，这在某些应用程序中可能不是理想的特性。