Complex networks: A mixture of power-law and Weibull distributions

TL;DR

This paper reveals that in complex networks, the total degree distribution can be better modeled as a mixture of power-law and Weibull distributions, with the Weibull often providing a superior fit, due to combined preferential and random attachment mechanisms.

Contribution

It introduces a classification of network edges into p2c and p2p types and demonstrates that their degree distributions follow power-law and Weibull distributions respectively, revealing a new perspective on network degree modeling.

Findings

p2c degree distribution follows a power law more strictly than total degree.

p2p degree distribution fits Weibull distribution well.

Total degree distribution is often better described by Weibull than power law.

Abstract

Complex networks have recently aroused a lot of interest. However, network edges are considered to be the same in almost all these studies. In this paper, we present a simple classification method, which divides the edges of undirected, unweighted networks into two types: p2c and p2p. The p2c edge represents a hierarchical relationship between two nodes, while the p2p edge represents an equal relationship between two nodes. It is surprising and unexpected that for many real-world networks from a wide variety of domains (including computer science, transportation, biology, engineering and social science etc), the p2c degree distribution follows a power law more strictly than the total degree distribution, while the p2p degree distribution follows the Weibull distribution very well. Thus, the total degree distribution can be seen as a mixture of power-law and Weibull distributions. More surprisingly, it is found that in many cases, the total degree distribution can be better described by the Weibull distribution, rather than a power law as previously suggested. By comparing two topology models, we think that the origin of the Weibull distribution in complex networks might be a mixture of both preferential and random attachments when networks evolve.