Evolving Shelah-Spencer Graphs

An \emph{evolving Shelah-Spencer process} is one by which a random graph grows, with at each time $\tau \in {\bf N}$ a new node incorporated and attached to each previous node with probability $\tau^{-\alpha}$, where $\alpha \in (0,1) \setminus {\bf Q}$ 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 [Spencer, J., 2013. The strange logic of random graphs (Vol. 22). Springer Science & Business Media.] and throughout the model-theoretic literature. The first order axiomatisation for classical Shelah-Spencer graphs comprises a 'Generic Extension' axiom and a 'No Dense Subgraphs' axiom. We show that in our context 'Generic Extension' continues to hold. While 'No Dense Subgraphs' fails, a weaker 'Few Rigid Subgraphs' property holds.