tailestim 0.0.4

A Python package for estimating tail parameters of heavy-tailed distributions, which is useful for analyzing power-law behavior in complex networks.

tailestim

A Python package for estimating tail parameters of heavy-tailed distributions, which is useful for analyzing power-law behavior in complex networks.

[!NOTE] The original estimation implementations are from ivanvoitalov/tail-estimation. This is a wrapper package that provides a more convenient/modern interface and logging, that can be installed using pip and conda.

Features

• Multiple estimation methods including Hill, Moments, Kernel, Pickands, and Smooth Hill estimators
• Double-bootstrap procedure for optimal threshold selection
• Built-in dataset loader for example networks
• Support for custom network data analysis
• Comprehensive parameter estimation and diagnostics

Installation

pip install tailestim

Quick Start

Using Built-in Datasets

from tailestim.datasets import TailData
from tailestim.estimator import TailEstimator

# Load a sample dataset
data = TailData(name='CAIDA_KONECT').data

# Initialize and fit the estimator
estimator = TailEstimator(method='hill')
estimator.fit(data)

# Get the estimated parameters
result = estimator.get_parameters()
gamma = result['gamma']

# Print full results
print(estimator)

Using degree sequence from networkx graphs

import networkx as nx
from tailestim.estimator import TailEstimator

# Create or load your network
G = nx.barabasi_albert_graph(10000, 2)
degree = list(dict(G.degree()).values()) # Degree sequence

# Initialize and fit the estimator
estimator = TailEstimator(method='hill')
estimator.fit(degree)

# Get the estimated parameters
result = estimator.get_parameters()
gamma = result['gamma']

# Print full results
print(estimator)

Available Methods

The package provides several methods for tail estimation. For details on parameters that can be specified to each methods, please refer to the original repository ivanvoitalov/tail-estimation, original paper, or the actual code.

1. Hill Estimator (method='hill')
   • Classical Hill estimator with double-bootstrap for optimal threshold selection
   • Default method, generally recommended for power law analysis
2. Moments Estimator (method='moments')
   • Moments-based estimation with double-bootstrap
   • More robust to certain types of deviations from pure power law
3. Kernel-type Estimator (method='kernel')
   • Kernel-based estimation with double-bootstrap and bandwidth selection
   • Additional parameters: hsteps (int, default=200), alpha (float, default=0.6)
4. Pickands Estimator (method='pickands')
   • Pickands-based estimation (no bootstrap)
   • Provides arrays of estimates across different thresholds
5. Smooth Hill Estimator (method='smooth_hill')
   • Smoothed version of the Hill estimator (no bootstrap)
   • Additional parameter: r_smooth (int, default=2)

Results

The results can be obtained by estimator.get_parameters(), which returns a dictionary. This includes:

• gamma: Power law exponent (γ = 1 + 1/ξ)
• xi_star: Tail index (ξ)
• k_star: Optimal order statistic
• Bootstrap results (when applicable):
  • First and second bootstrap AMSE values
  • Optimal bandwidths or minimum AMSE fractions

Example Output

When you print(estimator) after fitting, you will get the following output.

==================================================
Tail Estimation Results (Hill Method)
==================================================

Parameters:
--------------------
Optimal order statistic (k*): 6873
Tail index (ξ): 0.6191
Gamma (powerlaw exponent) (γ): 2.6151

Bootstrap Results:
--------------------
First bootstrap minimum AMSE fraction: 0.6899
Second bootstrap minimum AMSE fraction: 0.6901

Built-in Datasets

The package includes several example datasets:

• CAIDA_KONECT
• Libimseti_in_KONECT
• Pareto

Load any example dataset using:

from tailestim.datasets import TailData
data = TailData(name='dataset_name').data

References

• I. Voitalov, P. van der Hoorn, R. van der Hofstad, and D. Krioukov. Scale-free networks well done. Phys. Rev. Res., Oct. 2019, doi: 10.1103/PhysRevResearch.1.033034.
• I. Voitalov. ivanvoitalov/tail-estimation, GitHub. Mar. 2018. https://github.com/ivanvoitalov/tail-estimation.

License

tailestim is distributed under the terms of the MIT license.