Arbitrary degree distribution networks with perturbations

Xiaomin Wang,Bing Yao,Fei Ma

doi:10.1063/5.0031500

Abstract

Complex networks have played an important role in the field of natural science and social science, attracting considerable attention of more and more scholars. Currently, scholars have proposed numbers of complex networks, in which some show a required degree distribution and others follow arbitrary degree distribution. The goal of this paper is to discuss the impact of perturbations on degree distribution. To this end, we first introduce two types of perturbations, i.e., edge perturbations and vertex perturbations, and investigate networks whose structure can be determined by tuning perturbation rules. Next, we calculate the degree distribution using two popularly utilized mathematical methods, namely, rate equation and generating function. Afterward, we analyze several networks with different degree distributions, for example, Poisson distribution, stretched exponential distribution, and power-law distribution; there are, in practice, some pronounced differences among three cases. Therefore, to a certain extent, the above three cases can serve as the measures for degree distribution to help us clearly distinguish among different degree distributions.

Highlights

The investigation of complex networks has received extensive attention in many fields in recent decades, on the one hand because of the application of complex networks to complex and dynamical systems and on the other hand because of the remarkable successes of mathematical analysis and computations in shedding light on network phenomena.[1–5] Due to the randomness and variability of real networks, real networks become very complicated
Many efforts have been put into constructing some networks with a desired degree distribution. These constructed networks usually have similar dynamic properties to real networks, so the investigation of topological properties in these networks can help people understand the properties of real networks more intuitively and deeply, for example, the Internet, WWW, social relationship, annotated network, musical solos network, protein network, information network, biological neural networks, blog network, and Chinese railway network.[6–18]
III, we demonstrate three networks with different degree distributions, i.e., Poisson degree distribution, stretched exponential degree distribution, and power-law degree distribution

Summary

INTRODUCTION

The investigation of complex networks has received extensive attention in many fields in recent decades, on the one hand because of the application of complex networks to complex and dynamical systems and on the other hand because of the remarkable successes of mathematical analysis and computations in shedding light on network phenomena.[1–5] Due to the randomness and variability of real networks, real networks become very complicated. In 1999, Barabási and Albert proposed another intensive article on the BA-model (Barabási and Albert) with scale-free property, which characterizes a class of networks whose degree distribution displays a power-law form.[7]. These two pioneering articles revealed the most fundamental characteristic of various complex networks around us. We investigate the network perturbations in three cases, i.e., rewiring edge, adding vertex, and removing vertex, and analyze their effect on degree distribution.

NETWORKS WITH SPECIFIC PROPERTIES

Network perturbations

Rate equation

Example

EXAMPLES AND APPLICATIONS

CONCLUSION AND DISCUSSION