An Exact, Linear Time Barab\'asi-Albert Algorithm

TL;DR

This paper introduces an exact, linear-time algorithm for generating scale-free graphs based on the Barabási-Albert model, improving accuracy and efficiency over previous approximate methods.

Contribution

The paper presents a novel class of algorithms that exactly implement preferential attachment in linear time, ensuring proportional vertex selection probabilities.

Findings

Algorithms are exact in replicating BA model properties.

Methods operate in linear time relative to graph size.

They include options for controlling joint vertex inclusion.

Abstract

This paper presents the development of a new class of algorithms that accurately implement the preferential attachment mechanism of the Barab\'asi-Albert (BA) model to generate scale-free graphs. Contrary to existing approximate preferential attachment schemes, our methods are exact in terms of the proportionality of the vertex selection probabilities to their degree and run in linear time with respect to the order of the generated graph. Our algorithms utilize a series of precise, diverse, weighted and unweighted random sampling steps to engineer the desired properties of the graph generator. We analytically show that they obey the definition of the original BA model that generates scale-free graphs and discuss their higher-order properties. The proposed methods additionally include options to manipulate one dimension of control over the joint inclusion of groups of vertices.