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

Nixon Jerez-Lillo, Francisco A. Rodrigues, Paulo H. Ferreira, Pedro L. Ramos

TL;DR

This paper introduces Bayesian inference methods to obtain more accurate estimates than those obtained using traditional methods, which often yield biased estimates, and precise credible intervals, and proposes a piecewise extension of this model to provide greater flexibility.

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 $x^{-α}$, with $α$ typically ranging between $2 < α< 3$. 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.

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

TL;DR

This paper introduces Bayesian inference methods to obtain more accurate estimates than those obtained using traditional methods, which often yield biased estimates, and precise credible intervals, and proposes a piecewise extension of this model to provide greater flexibility.

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 , with typically ranging between . 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.
Paper Structure (29 sections, 4 theorems, 40 equations, 11 figures, 7 tables, 1 algorithm)

This paper contains 29 sections, 4 theorems, 40 equations, 11 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Bayesian estimators
  3. Testing power-law distribution
  4. Kolmogorov-Smirnov test
  5. Alternative goodness-of-fit tests
  6. Discrete piecewise power-law model
  7. Simulation study
  8. Study 1
  9. Study 2
  10. Size of the test.
  11. Power of the test.
  12. Conclusion.
  13. Study 3
  14. Study 4
  15. Size of the test.
  16. ...and 14 more sections

Key Result

Proposition 2.3

Let $g:(a, b) \rightarrow \mathbb{R}^{+}$ and $h:(a, b) \rightarrow \mathbb{R}^{+}$ be continuous functions on $(a, b) \subset \mathbb{R}$, where $a, b \in \overline{\mathbb{R}}$. Then, $g(x) \propto g(x)$ if and only if $\lim\limits_{x \to a} {g(x)}/{h(x)} < +\infty$ and $\lim\limits_{x \to b} {g(x

Figures (11)

  • Figure 1: Jeffreys priors for the discrete and continuous power-law model taking $x_{\min}=1$.
  • Figure 2: Performance of the estimators in terms of bias (left), MSE (middle), and CP (right) across different cases using random samples of sizes $n=10, 20, \ldots, 100$ generated from a power-law distribution.
  • Figure 3: Empirical sizes of the GOF tests based on the $K$, $W^2$, $U^2$, and $A^2$ statistics at a 5% significance level. The dashed line represents the limits of $0.05 \pm 1.96 \sqrt{0.05(1-0.05)/\text{1,000}}$, derived from the normal approximation.
  • Figure 4: Empirical power of GOF tests based on the $K$, $W^2$, $U^2$, and $A^2$ statistics at a 5% significance level.
  • Figure 5: Histogram of the estimated scaling parameter, derived from equation \ref{['eqn:map1']}, for power-law degree sequences under the Watson test, highlighting the range where the strong scale-free evidence taxonomy holds.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 4.1
  • proof