From Densification Power Law to Degree of Separation: A Case Study

Abstract

The densification power law is a concept in the realm of temporal graph evolution. The number of edges grows in a power law over the number of nodes over time, replacing the pre-2005 general assumption of a linear trend. The densification power law has been verified by several real networks over a long period of time. In this work, one such graph, the arXiv citation network is investigated to examine how the densification power law is working ten years after its publication. The network is evaluated and compared with the discussion in a previous work. It is observed that the graph densification continues over time, but instead of maintaining a constant densification power exponent, as suggested by previous work, the exponent is actually dropping over time, which suggests the densification power law is now fading away. Here, this fading effect is literature analysed, and it is suggested that node capability is the major obstacle to the continuation of the original trend. To fully compare with the previous work on graph evolution, the change of the average path length over time is also investigated on our and other's results. The results imply the decreasing of the average path length in the temporal evolution is very slow, which suggests that there exists a new universal degree of separation in social networks of around three.