Preferential attachment with choice

TL;DR

This paper analyzes how different choice mechanisms in preferential attachment models influence degree distributions, revealing conditions for power-law behavior, exponential decay, and condensation phenomena.

Contribution

It introduces a detailed analysis of degree distributions in preferential attachment models with choice, identifying conditions for various asymptotic behaviors including power laws and condensation.

Findings

Double exponential decay and condensation are possible with meek choice.

Power law with logarithmic correction occurs for greedy choice with r=2.

Condensation-like behavior emerges when r>2.

Abstract

We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$ vertices according to a preferential rule and connects to the vertex in the selection with the $s$th highest degree. For meek choice, where $s>1$, we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where $s=1$, we confirm that the degree distribution asympotically follows a power law with logarithmic correction when $r=2$ and shows condensation-like behaviour when $r>2$.