Generating Preferential Attachment Graphs via a P\'olya Urn with Expanding Colors

TL;DR

This paper introduces a new preferential attachment graph model using a Pólya urn with expanding colors, allowing later vertices to achieve high degrees, and analyzes its degree distribution compared to classical models.

Contribution

The paper presents a novel preferential attachment model based on a Pólya urn with expanding colors, capturing influential opinions and degree dynamics not modeled by Barabási-Albert.

Findings

Derived the probability distribution of vertex degrees in the new model.

Demonstrated the model's ability to produce high-degree vertices later in the process.

Compared the new model's properties with the Barabási-Albert network through simulations.

Abstract

We introduce a novel preferential attachment model using the draw variables of a modified P\'olya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Similar to the Barab\'asi-Albert model, the generated graph grows in size by one vertex at each time instance; in contrast however, each vertex of the graph is uniquely characterized by a color, which is represented by a ball color in the P\'olya urn. More specifically at each time step, we draw a ball from the urn and return it to the urn along with a number (potentially time-varying and non-integer) of reinforcing balls of the same color; we also add another ball of a new color to the urn. We then construct an edge between the new vertex (corresponding to the new color) and the existing vertex whose color ball is drawn. Using color-coded vertices in conjunction with the time-varying reinforcing parameter allows for vertices added (born) later in the process to potentially attain a high degree in a way that is not captured in the Barab\'asi-Albert model. We study the degree count of the vertices by analyzing the draw vectors of the underlying stochastic process. In particular, we establish the probability distribution of the random variable counting the number of draws of a given color which determines the degree of the vertex corresponding to that color in the graph. We further provide simulation results presenting a comparison between our model and the Barab\'asi-Albert network.