# Agglomeration in a preferential attachment random graph with edge-steps

## TL;DR

This paper studies the geometric properties of a preferential attachment graph model with edge-steps, analyzing how clustering and clique sizes evolve over time depending on the probability of adding vertices or edges.

## Contribution

It introduces a model combining vertex and edge additions, and derives asymptotic behaviors of clustering coefficient and clique number based on the parameter p.

## Key findings

- Global clustering coefficient decays as t^{-}

- Clique number scales approximately as t^{(1-p)/(2-p)}
- Results depend on the probability p of vertex vs. edge addition

## Abstract

In this paper we investigate geometric properties of graphs generated by a preferential attachment random graph model with edge-steps. More precisely, at each time $t\in\mathbb{N}$, with probability $p$ a new vertex is added to the graph (a vertex-step occurs) or with probability $1-p$ an edge connecting two existent vertices is added (an edge-step occurs). We prove that the global clustering coefficient decays as $t^{-\gamma(p)}$ for a positive function $\gamma$ of $p$. We also prove that the clique number of these graphs is, up to sub-polynomially small factors, of order~$t^{(1-p)/(2-p)}$.