Winners don't take all: Characterizing the competition for links on the web

Abstract

As a whole, the World Wide Web displays a striking "rich get richer" behavior, with a relatively small number of sites receiving a disproportionately large share of hyperlink references and traffic. However, hidden in this skewed global distribution, we discover a qualitatively different and considerably less biased link distribution among subcategories of pages-for example, among all university homepages or all newspaper homepages. Although the connectivity distribution over the entire web is close to a pure power law, we find that the distribution within specific categories is typically unimodal on a log scale, with the location of the mode, and thus the extent of the rich get richer phenomenon, varying across different categories. Similar distributions occur in many other naturally occurring networks, including research paper citations, movie actor collaborations, and United States power grid connections. A simple generative model, incorporating a mixture of preferential and uniform attachment, quantifies the degree to which the rich nodes grow richer, and how new (and poorly connected) nodes can compete. The model accurately accounts for the true connectivity distributions of category-specific web pages, the web as a whole, and other social networks.

Figure 1

Diamonds plot the empirically observed connectivity distribution for company homepages. Circles display the histogram resulting from a simulation of the model, with parameters t = 4,923, m = 1,356, and α = 0.950 set to match the company data. The dashed line marks the analytic solution (Eq. 3) instantiated with the same parameters.

Figure 2

Diamonds display log–log histograms of inbound connectivities for category-specific homepages, and inbound and outbound connectivities for random web pages. Circles mark the connectivity distributions, with m₀ = 0, t set equal to the number of web pages, 2m set equal to the average number of inbound links per page, and α chosen according to a nonlinear least-squares fit. Dashed lines indicate the analytic solutions (Eq. 3).