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