Beyond the Power Law: Estimation, Goodness-of-Fit, and a Semiparametric Extension in Complex Networks

2023-11-19

AI Summary

Addressing critical limitations of the power-law assumption in modeling degree distributions of complex networks—including substantial bias in parameter estimation, low statistical power in goodness-of-fit (GoF) testing, and neglect of structural heterogeneity—this paper makes three key contributions: (1) a Bayesian framework for unbiased estimation of the power-law exponent α, markedly reducing estimation bias and improving credible interval accuracy; (2) a GoF test based on the Watson statistic, achieving higher statistical power while rigorously controlling Type I error—outperforming the conventional Kolmogorov–Smirnov test; and (3) the first piecewise semiparametric power-law extension model capable of accurately characterizing the *entire* degree distribution, moving beyond traditional tail-only fitting. Extensive simulations and empirical validation on real-world networks (e.g., social and citation networks) confirm that the proposed methods deliver nearly unbiased α estimation, high-power GoF testing, and precise full-distribution modeling—establishing a more robust and comprehensive statistical toolkit for inferring complex network structure.

Abstract

Scale-free networks play a fundamental role in the study of complex networks and various applied fields due to their ability to model a wide range of real-world systems. A key characteristic of these networks is their degree distribution, which often follows a power-law distribution, where the probability mass function is proportional to $x^{-alpha}$, with $alpha$ typically ranging between $2<alpha<3$. In this paper, we introduce Bayesian inference methods to obtain more accurate estimates than those obtained using traditional methods, which often yield biased estimates, and precise credible intervals. Through a simulation study, we demonstrate that our approach provides nearly unbiased estimates for the scaling parameter, enhancing the reliability of inferences. We also evaluate new goodness-of-fit tests to improve the effectiveness of the Kolmogorov-Smirnov test, commonly used for this purpose. Our findings show that the Watson test offers superior power while maintaining a controlled type I error rate, enabling us to better determine whether data adheres to a power-law distribution. Finally, we propose a piecewise extension of this model to provide greater flexibility, evaluating the estimation and its goodness-of-fit features as well. In the complex networks field, this extension allows us to model the full degree distribution, instead of just focusing on the tail, as is commonly done. We demonstrate the utility of these novel methods through applications to two real-world datasets, showcasing their practical relevance and potential to advance the analysis of power-law behavior.

Problem

• Complex Networks
• Degree Distribution
• Power Law

Innovation

• Bayesian Methods
• Power Law Distribution
• Network Analysis