Rich-clubness test: how to determine whether a complex network has or doesn't have a rich-club?

TL;DR

This paper introduces a novel, unified method to statistically determine the presence of rich-club organization in complex networks, addressing limitations of previous approaches by providing a single significance measure.

Contribution

The study proposes a new null-model, normalization strategy, and statistical test to quantify rich-clubness with a unique significance value, improving analysis of network organization.

Findings

Introduces a null-model with lower rich-club coefficient

Develops a normalization strategy for rich-clubness

Provides a statistical test with a single p-value for rich-club presence

Abstract

The rich-club concept has been introduced in order to characterize the presence of a cohort of nodes with a large number of links (rich nodes) that tend to be well connected between each other, creating a tight group (club). Rich-clubness defines the extent to which a network displays a topological organization characterized by the presence of a node rich-club. It is crucial for the investigation of internal organization and function of networks arising in systems of disparate fields such as transportation, social, communication and neuroscience. Different methods have been proposed for assessing the rich-clubness and various null-models have been adopted for performing statistical tests. However, a procedure that assigns a unique value of rich-clubness significance to a given network is still missing. Our solution to this problem grows on the basis of three new pillars. We introduce: i) a null-model characterized by a lower rich-club coefficient; ii) a fair strategy to normalize the level of rich-clubness of a network in respect to the null-model; iii) a statistical test that, exploiting the maximum deviation of the normalized rich-club coefficient attributes a unique p-value of rich-clubness to a given network. In conclusion, this study proposes the first attempt to quantify, using a unique measure, whether a network presents a significant rich-club topological organization. The general impact of our study on engineering and science is that simulations investigating how the functional performance of a network is changing in relation to rich-clubness might be more easily tuned controlling one unique value: the proposed rich-clubness measure.