Weibull Processes in Network Degree Distributions

Abstract

This study examines degree distributions in two large collaboration networks: the Microsoft Academic Graph (1800-2020) and Internet Movie Database (1900-2020), comprising $2.72 \times 10^8$ and $1.88 \times 10^6$ nodes respectively. Statistical comparison using $\chi^2$ measures showed that Weibull distributions fit the degree distributions better than power-law or log-normal models, especially at later stages in the network evolution. The Weibull shape parameters exhibit notable stability ($k \approx 0.8$-$1.0$ for academic, $k \approx 0.9$-$1.1$ for entertainment collaborations) despite orders of magnitude growth in network size. While early-stage networks display approximate power-law scaling, mature networks develop characteristic flattening in the low-degree region that Weibull distributions appear to capture better. In the academic network, the cutoff between the flattened region and power-law tail shows a gradual increase from $5$ to $9$ edges over time, while the entertainment network maintains a distinctive degree structure that may reflect storytelling and cast-size constraints. These patterns suggest the possibility that collaboration network evolution might be influenced more by constraint-based growth than by pure preferential attachment or multiplicative processes.