$\gamma$-variable first-order logic of uniform attachment random graphs

TL;DR

This paper investigates the logical properties of uniform attachment random graphs, proving they follow a convergence law for certain first-order logical sentences, revealing insights into their asymptotic structure.

Contribution

It establishes a new convergence law for first-order logic with limited variables in the context of uniform attachment random graphs.

Findings

Convergence law holds for first-order sentences with up to m-2 variables.

Vertices and edges are added recursively, influencing the graph's structure.

The model provides a framework for understanding logical limits in growing random graphs.

Abstract

We study logical limit laws for uniform attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time $n+1$, the vertex $n+1$ is introduced together with $m$ edges joining the new vertex with $m$ different vertices chosen uniformly at random from $1,\ldots,n$. We prove that this random graph obeys convergence law for first-order sentences with at most $m-2$ variables.