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

TL;DR

This paper investigates the logical properties of preferential attachment graphs, proving they follow a convergence law for certain first-order logical sentences, which advances understanding of their asymptotic logical behavior.

Contribution

It establishes a new logical limit law for preferential attachment graphs with a specific variable bound, expanding the theoretical framework of random graph logic.

Findings

Graphs obey the convergence law for first-order sentences with up to m-2 variables

Provides a formal logical characterization of preferential attachment graph limits

Enhances understanding of asymptotic properties of complex network models

Abstract

We study logical limit laws for preferential attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time $1$, we start with vertices $0,1$ and $m$ edges between them. At step $n+1$ the vertex $n+1$ is introduced together with $m$ edges joining the new vertex with $m$ vertices chosen from $1,\ldots,n$ independently with probabilities proportional to their degrees plus a positive parameter $\delta$. We prove that this random graph obeys the convergence law for first-order sentences with at most $m-2$ variables.