Evolving Shelah-Spencer Graphs

TL;DR

This paper studies an evolving random graph model inspired by Shelah-Spencer graphs, analyzing its properties and limits, and identifying which classical axioms hold or fail in this dynamic setting.

Contribution

It extends the Shelah-Spencer graph framework to an evolving process, showing that some axioms persist while others are replaced by weaker properties.

Findings

'Generic Extension' axiom continues to hold in the evolving model.

'No Dense Subgraphs' axiom fails, replaced by 'Few Rigid Subgraphs'.

Analysis of the infinite limit graph and its model-theoretic properties.

Abstract

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.