Analysis of a Model for Generating Weakly Scale-free Networks

TL;DR

This paper introduces a new network model that combines preferential attachment with uniform edge addition, producing weakly scale-free networks that better match real-world degree distributions across all degrees.

Contribution

The paper proposes a novel two-step network growth model that captures the entire degree distribution, including the uniform region below the scale-free tail.

Findings

The model generates networks with degree distributions similar to real-world networks.

Mathematical analysis confirms the model's ability to produce weakly scale-free structures.

Comparison shows improved fit over traditional models for the entire degree range.

Abstract

It is commonly believed that real networks are scale-free and fraction of nodes $P(k)$ with degree $k$ satisfies the power law $P(k) \propto k^{-\gamma} \text{ for } k > k_{min} > 0$. Preferential attachment is the mechanism that has been considered responsible for such organization of these networks. In many real networks, degree distribution before the $k_{min}$ varies very slowly to the extent of being uniform as compared with the degree distribution for $k > k_{min}$ . In this paper, we proposed a model that describe this particular degree distribution for the whole range of $k>0$. We adopt a two step approach. In the first step, at every time stamp we add a new node to the network and attach it with an existing node using preferential attachment method. In the second step, we add edges between existing pairs of nodes with the node selection based on the uniform probability distribution. Our approach generates weakly scale-free networks that closely follow the degree distribution of real-world networks. We perform comprehensive mathematical analysis of the model in the discrete domain and compare the degree distribution generated by these models with that of real-world networks.