# A Multi-type Preferential Attachment Tree

## TL;DR

This paper introduces a multi-type preferential attachment tree model, providing a framework for analyzing its growth and degree distribution, which follows a power law under linear attachment rates, extending understanding of complex network structures.

## Contribution

It develops a general framework for multi-type preferential attachment trees and derives power law degree distributions for linear attachment rates, advancing network modeling techniques.

## Key findings

- Degree distribution follows a power law under linear preferential attachment.
- Framework applies to multiple types with complex attachment functions.
- Asymptotic composition of vertex types is characterized.

## Abstract

A multi-type preferential attachment tree is introduced, and studied using general multi-type branching processes. For the $p$-type case we derive a framework for studying the tree where a type $i$ vertex generates new type $j$ vertices with rate $w_{ij}(n_1,n_2,\ldots, n_p)$ where $n_k$ is the number of type $k$ vertices previously generated by the type $i$ vertex, and $w_{ij}$ is a non-negative function from $\mathbb{N}^p$ to $\mathbb{R}$. The framework is then used to derive results for trees with more specific attachment rates. In the case with linear preferential attachment---where type $i$ vertices generate new type $j$ vertices with rate $w_{ij}(n_1,n_2,\ldots, n_p)=\gamma_{ij}(n_1+n_2+\dots +n_p)+\beta_{ij}$, where $\gamma_{ij}$ and $\beta_{ij}$ are positive constants---we show that under mild regularity conditions on the parameters $\{\gamma_{ij}\}, \{\beta_{ij}\}$ the asymptotic degree distribution of a vertex is a power law distribution. The asymptotic composition of the vertex population is also studied.