Rich-club and page-club coefficients for directed graphs

TL;DR

This paper introduces rich-club and page-club coefficients for directed graphs, analyzes their null models, and applies them to various real-world networks to reveal patterns of elite node interactions.

Contribution

It develops new metrics for directed networks and demonstrates their effectiveness through analysis of multiple real-world network datasets.

Findings

Rich-club and page-club ordering observed in citation and patent networks.

Web network shows limited rich-club and page-club effects, indicating selective linking.

E-mail network lacks both rich-club and page-club ordering.

Abstract

Rich-club and page-club coefficients and their null models are introduced for directed graphs. Null models allow for a quantitative discussion of the rich-club and page-club phenomena. These coefficients are computed for four directed real-world networks: Arxiv High Energy Physics paper citation network, Web network (released from Google), Citation network among US Patents, and Email network from a EU research institution. The results show a high correlation between rich-club and page-club ordering. For journal paper citation network, we identify both rich-club and page-club ordering, showing that {}"elite" papers are cited by other {}"elite" papers. Google web network shows partial rich-club and page-club ordering up to some point and then a narrow declining of the corresponding normalized coefficients, indicating the lack of rich-club ordering and the lack of page-club ordering, i.e. high in-degree (PageRank) pages purposely avoid sharing links with other high in-degree (PageRank) pages. For UC patents citation network, we identify page-club and rich-club ordering providing a conclusion that {}"elite" patents are cited by other {}"elite" patents. Finally, for e-mail communication network we show lack of both rich-club and page-club ordering. We construct an example of synthetic network showing page-club ordering and the lack of rich-club ordering.