Two Cumulative Distributions For Scale-freeness of Dynamic Networks

It is well-known that the scale-free networks are ubiquitous in nature and society and have been one of the hotspot topic in complex networks. Recently, scholars presented a large quantity of scale-free networks by calculating cumulative distribution. The purpose of this paper is to discuss the relationship between two cumulative distributions, namely, cumulative distribution, edge-cumulative distribution. Here, firstly, we introduce a relationship between degree distribution and cumulative distribution. Secondly, we introduce the definition of cumulative distribution and edge-cumulative distribution, and compare the relationship between them. Thirdly, we apply algorithmic techniques to construct three deterministic networks, calculate their cumulative distribution and edge-cumulative distribution, and analyze the relationship between cumulative distribution and edge-cumulative distribution. Finally, we offer some open problems for future research in order to understand the interplay between the degree distribution, cumulative distribution and edge-cumulative distribution.