# Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

## TL;DR

This paper investigates a modified network growth model combining preferential and uniform attachment rules, demonstrating that it maintains a power-law degree distribution and analyzing how the uniform rule affects the emergence of scale-free behavior.

## Contribution

It introduces and analyzes a new network model with mixed attachment rules, showing robustness of scale-free properties and effects of recent node selection on degree distribution.

## Key findings

- The model exhibits asymptotic power-law degree distribution.
- Uniform attachment delays the onset of the scale-free regime.
- Analytical expressions for degree distribution and mean node degree.

## Abstract

Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale free behavior of the degree distribution. We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1-p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion (\alpha n) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold. The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.