First-order logic of uniform attachment random graphs with a given degree

TL;DR

This paper establishes a first-order convergence law for a specific class of uniform attachment random graphs where most vertices share the same degree, using Markov chain analysis to describe their logical structure.

Contribution

It introduces the first first-order convergence law for uniform attachment graphs with uniform degree constraints, employing Markov chains to analyze their logical equivalence classes.

Findings

Proves first-order convergence law for the model.

Describes the graph dynamics via Markov chains.

Establishes limit distribution for the Markov chain.

Abstract

In this paper, we prove the first-order convergence law for the uniform attachment random graph with almost all vertices having the same degree. In the considered model, vertices and edges are introduced recursively: at time $m+1$ we start with a complete graph on $m+1$ vertices. At step $n+1$ the vertex $n+1$ is introduced together with $m$ edges joining the new vertex with $m$ vertices chosen uniformly from those vertices of $1,\ldots,n$, whom degree is less then $d=2m$. To prove the law, we describe the dynamics of the logical equivalence class of the random graph using Markov chains. The convergence law follows from the existence of a limit distribution of the considered Markov chain.