Supernodes

TL;DR

This paper introduces vertex and edge participation concepts to analyze subgraph involvement, generalizing the rich-club idea, and demonstrates their application on various network types including brain networks.

Contribution

It presents novel participation-based measures for subgraph analysis and extends the rich-club concept to super rich-club and rich edge-club, with experimental validation.

Findings

Vertex participation generalizes the rich-club concept.

Experimental results on Erdős-Rényi and Watts-Strogatz networks.

Application to brain networks showing new insights.

Abstract

In this paper, 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 generalisation of the rich-club. We present experimental results on randomised Erd\"{o}s R\'{e}nyi 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.