We define an evolving Shelah-Spencer process as one by which a random graph grows, with at each time $\tau \in \mathbb{N}$ a new node incorporated and attached to each previous node with probability ${\tau }^{-\alpha }$, where $\alpha \in (0,1)\setminus \mathbb{Q}$ is fixed. We analyse the graphs that result from this process, including the infinite limit, in comparison to Shelah-Spencer sparse random graphs discussed in [21] and throughout the model-theoretic literature. The first order axiomatisation for classical Shelah-Spencer graphs comprises a Generic Extension axiom scheme and a No Dense Subgraphs axiom scheme. We show that in our context Generic Extension continues to hold. While No Dense Subgraphs fails, a weaker Few Rigid Subgraphs property holds.

中文翻译：

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进化的 Shelah-Spencer 图

我们将不断发展的 Shelah-Spencer 过程定义为随机图增长的过程，每次$\tau \in \mathbb{N}$ 一个新节点以概率合并并附加到每个先前的节点 ${\tau }^{-\alpha }$， 在哪里 $\alpha \in (0,1)\setminus \mathbb{问}$是固定的。与 [21] 和整个模型理论文献中讨论的 Shelah-Spencer 稀疏随机图相比，我们分析了由该过程产生的图，包括无限极限。经典 Shelah-Spencer 图的一阶公理化包括通用扩展公理方案和无密集子图公理方案。我们表明，在我们的上下文中，通用扩展继续成立。虽然No Dense Subgraphs失败了，但较弱的Three Rigid Subgraphs属性仍然成立。