# Generalised thresholding of hidden variable network models with scale-free property

## TL;DR

This paper generalizes and relaxes the thresholding mechanism in hidden variable network models, enabling the generation of diverse scale-free networks with adjustable degree distribution exponents.

## Contribution

It introduces a flexible framework for hidden variable models, allowing for a broader range of degree distributions and sparse scale-free networks with tunable exponents.

## Key findings

- Extended connection probabilities produce varied degree distributions.
- Relaxed thresholds enable sparse networks with arbitrary power-law exponents.
- Analytical conditions for scale-free behavior are derived.

## Abstract

The hidden variable formalism (based on the assumption of some intrinsic node parameters) turned out to be a remarkably efficient and powerful approach in describing and analyzing the topology of complex networks. Owing to one of its most advantageous property - namely proven to be able to reproduce a wide range of different degree distribution forms - it has become a standard tool for generating networks having the scale-free property. One of the most intensively studied version of this model is based on a thresholding mechanism of the exponentially distributed hidden variables associated to the nodes (intrinsic vertex weights), which give rise to the emergence of a scale-free network where the degree distribution $p(k)\sim k^{-\gamma}$ is decaying with an exponent of $\gamma =2$. Here we propose a generalization and modification of this model by extending the set of connection probabilities and hidden variable distributions that lead to the aforementioned degree distribution, and analyze the conditions leading to the above behavior analytically. In addition, we propose a relaxation of the hard threshold in the connection probabilities, which opens up the possibility for obtaining sparse scale free networks with arbitrary scaling exponent.