Modeling Network Contagion Via Interacting Finite Memory Pólya Urns

Abstract

We introduce a novel preferential attachment model using the draw variables of a modified Pólya 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 Barabási-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 Pólya 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 Barabási-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 Barabási-Albert network.

Generating preferential attachment graphs via a Pólya urn with expanding colors

“…In particular, in the context of randomly growing graphs, the reinforcements favor a high-degree vertex in getting an even higher degree as the network grows. Indeed, Pólya urn models have been widely used to model reinforcements, for instance, in communication channels (Alajaji and Fuja, 1994), image segmentation (Banerjee et al, 1999), social and epidemic networks (Avin et al, 2020;Kim and Jo, 2010;Toivonen et al, 2006;Hayhoe et al, 2018;Singh et al, 2022c, a, b;Jadbabaie et al, 2023;He et al, 2023), citation networks (Jeong et al, 2003;Milojević, 2010;Newman, 2001), actor collaborations (Albert and Barabási, 2000;Jeong et al, 2003), mechanics (Albert and Barabási, 2002), and polymer formations (Bhutani et al, 2022). Naturally, Pólya urns have been used to model preferential attachment graphs in the literature.…”