Weibull Processes in Network Degree Distributions

TL;DR

The paper tests whether Weibull distributions better describe the degree distributions of two century-spanning collaboration networks (MAG and IMDb) than traditional power-law or log-normal models. Using yearly degree distributions, adaptive logarithmic binning, and χ^2 goodness-of-fit across three candidate models, the study finds Weibull fits superior in mature phases, with stable shape parameters around 0.8–1.1 despite enormous network growth. Early epochs show a transition from approximate power-law behavior to Weibull-like forms, accompanied by characteristic low-degree flattening, particularly in MAG. The results imply constraint-based growth processes, rather than pure preferential attachment, underpin collaboration networks and point to universal mechanisms shaping social organization across domains and time scales.

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 $χ^2$ 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.

Figures (16)

• Figure 1: Fitting parameters for power-law, log-normal, and Weibull fits to the MAG and IMDb network degree distributions in red, green, and black lines respectively.
• Figure 2: $\chi^{2}$ for power-law, log-normal, and Weibull fits to the MAG and IMDb network degree distributions in red, green, and black lines respectively.
• Figure 3: Degree distributions for cohorts of authors who first published in a given year, 1800 to 1824 (red line). Power-law, Log-normal, and Weibull fits are shown with black, blue, and green lines respectively.
• Figure 4: Degree distributions for cohorts of authors who first published in a given year, 1825 to 1849 (red line). Power-law, Log-normal, and Weibull fits are shown with black, blue, and green lines respectively.
• Figure 5: Degree distributions for cohorts of authors who first published in a given year, 1850 to 1874 (red line). Power-law, Log-normal, and Weibull fits are shown with black, blue, and green lines respectively.