# Maximum Likelihood Estimation of Power-law Degree Distributions via Friendship Paradox based Sampling

## TL;DR

This paper introduces a novel maximum likelihood estimation method for power-law degree distributions that leverages the friendship paradox to improve sampling efficiency, reducing bias and variance compared to traditional methods.

## Contribution

It presents a new sampling-based MLE approach exploiting the friendship paradox, enhancing estimation accuracy for degree distributions in networks.

## Key findings

- Proposed method achieves lower bias and variance than uniform sampling.
- Method extends to other distributions like exponential.
- Numerical and empirical results validate improved performance.

## Abstract

This paper considers the problem of estimating a power-law degree distribution of an undirected network using sampled data. Although power-law degree distributions are ubiquitous in nature, the widely used parametric methods for estimating them (e.g. linear regression on double-logarithmic axes, maximum likelihood estimation with uniformly sampled nodes) suffer from the large variance introduced by the lack of data-points from the tail portion of the power-law degree distribution. As a solution, we present a novel maximum likelihood estimation approach that exploits the friendship paradox to sample more efficiently from the tail of the degree distribution. We analytically show that the proposed method results in a smaller bias, variance and a Cramer-Rao lower bound compared to the vanilla maximum-likelihood estimate obtained with uniformly sampled nodes (which is the most commonly used method in literature). Detailed numerical and empirical results are presented to illustrate the performance of the proposed method under different conditions and how it compares with alternative methods. We also show that the proposed method and its desirable properties (i.e. smaller bias, variance and Cramer-Rao lower bound compared to vanilla method based on uniform samples) extend to parametric degree distributions other than the power-law such as exponential degree distributions as well. All the numerical and empirical results are reproducible and the code is publicly available on Github.