[Paper Review] A Multi-type Preferential Attachment Model.

Sebastian Rosengren|arXiv (Cornell University)|Apr 11, 2017

Stochastic processes and statistical mechanics12 references3 citations

TL;DR

This paper introduces a multi-type preferential attachment model where vertices of different types generate offspring of other types based on dynamic rates dependent on prior counts. For linear preferential attachment with positive constants, it proves the asymptotic degree distribution follows a power law under mild regularity conditions, and characterizes the long-term composition of the vertex population.

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.

Motivation & Objective

• To develop a general framework for multi-type preferential attachment trees using multi-type branching processes.
• To analyze the asymptotic degree distribution in multi-type networks with dynamic attachment rates.
• To study the long-term composition of vertex populations in such networks.
• To establish conditions under which the degree distribution converges to a power law in the linear preferential attachment case.

Proposed method

• Modeling vertex generation via type-specific rates $ w_{ij}(n_1, n_2, \ldots, n_p) $, where $ n_k $ is the count of type $ k $ vertices generated by a type $ i $ vertex.
• Using general multi-type branching processes to analyze the stochastic evolution of the tree structure.
• Focusing on the linear preferential attachment case: $ w_{ij}(n_1, \ldots, n_p) = \gamma_{ij}(n_1 + \cdots + n_p) + \beta_{ij} $, with $ \gamma_{ij}, \beta_{ij} > 0 $.
• Applying regularity conditions on $ \{\gamma_{ij}\} $ and $ \{\beta_{ij}\} $ to ensure convergence properties.
• Deriving the asymptotic degree distribution through limit analysis of the branching process.
• Studying the asymptotic proportion of each vertex type in the population using the branching process framework.

Experimental results

Research questions

• RQ1 Under what conditions does the degree distribution of a multi-type preferential attachment tree converge to a power law?
• RQ2 How does the population composition of different vertex types evolve asymptotically in such a model?
• RQ3 What role do the parameters $ \gamma_{ij} $ and $ \beta_{ij} $ play in shaping the long-term structure of the network?
• RQ4 Can the framework handle non-linear attachment rates, and what are the implications for network topology?
• RQ5 How does the dynamic dependence of attachment rates on prior counts affect the emergence of scale-free properties?

Key findings

• Under mild regularity conditions on $ \{\gamma_{ij}\} $ and $ \{\beta_{ij}\} $, the asymptotic degree distribution of a vertex in the linear preferential attachment model is a power law.
• The power-law exponent is determined by the parameters $ \gamma_{ij} $ and $ \beta_{ij} $, though the exact value is not specified in the abstract.
• The model establishes the existence of a well-defined asymptotic population composition across different vertex types.
• The framework successfully generalizes single-type preferential attachment to multi-type settings with dynamic, count-dependent attachment rates.
• The use of multi-type branching processes enables rigorous analysis of both degree distribution and population dynamics.
• The results confirm that linear preferential attachment in multi-type networks leads to scale-free behavior in the limit.