Evolving Shelah‐Spencer graphs

Abstract

We define an evolving Shelah‐Spencer process as one by which a random graph grows, with at each time τ∈N a new node incorporated and attached to each previous node with probability τ−α, where α∈(0,1)∖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.

Preferential attachment processes approaching the Rado multigraph

“…One might hope somewhere within this regime to identify a connection to (a multigraph analogue of) the theory of Shelah-Spencer sparse random graphs as elucidated in [16]. The author recently established a connection between Shelah-Spencer graphs and the limits of finitary random processes in [8], albeit in a context rather simpler than preferential attachment.…”