Supernodes: a generalization of the rich-club

In this article, we present two new concepts related to subgraph counting where the focus is not on the number of subgraphs that are isomorphic to some fixed graph
$H$
, but on the frequency with which a vertex or an edge belongs to such subgraphs. In particular, we are interested in the case where
$H$
is a complete graph. These new concepts are termed vertex participation and edge participation, respectively. We combine these concepts with that of the rich-club to identify what we call a Super rich-club and rich edge-club. We show that the concept of vertex participation is a generalization of the rich-club. We present experimental results on randomized Erdös–Rényi and Watts–Strogatz small-world networks. We further demonstrate both concepts on a complex brain network and compare our results to the rich-club of the brain.