Predicted and Verified Deviation from Zipf's Law in Growing Social Networks

TL;DR

This paper empirically verifies a theory linking Zipf's law deviations to non-stationary growth in social networks, using data to predict and confirm the power law exponent without adjustable parameters.

Contribution

It provides the first detailed empirical validation of a theory predicting Zipf's law deviations based on stochastic growth and death processes in social networks.

Findings

Predicted power law exponent μ = 0.75 ± 0.05 matches empirical estimates.

Deviation from Zipf's law indicates non-stationary growth in social systems.

Empirical parameters r, σ, h effectively predict the size distribution exponent.

Abstract

Zipf's power law is a general empirical regularity found in many natural and social systems. A recently developed theory predicts that Zipf's law corresponds to systems that are growing according to a maximally sustainable path in the presence of random proportional growth, stochastic birth and death processes. We report a detailed empirical analysis of a burgeoning network of social groups, in which all ingredients needed for Zipf's law to apply are verifiable and verified. We estimate empirically the average growth $r$ and its standard deviation $\sigma$ as well as the death rate $h$ and predict without adjustable parameters the exponent $\mu$ of the power law distribution $P(s)$ of the group sizes $s$. The predicted value $\mu = 0.75 \pm 0.05$ is in excellent agreement with maximum likelihood estimations. According to theory, the deviation of $P(s)$ from Zipf's law (i.e., $\mu < 1$) constitutes a direct statistical quantitative signature of the overall non-stationary growth of the social universe.