The Coreness and H-Index of Random Geometric Graphs

TL;DR

This paper investigates the asymptotic behavior of node centrality measures, specifically coreness and H-index, in random geometric graphs, revealing their convergence to new continuum limits as the network grows.

Contribution

It introduces the concept of continuum coreness and demonstrates the convergence of coreness and H-index in large random geometric graphs.

Findings

Coreness converges to a new continuum object.

H-index and its iterates also converge to new limits.

Results apply as the number of nodes increases with proper connectivity radius.

Abstract

In network analysis, a measure of node centrality provides a scale indicating how central a node is within a network. The coreness is a popular notion of centrality that accounts for the maximal smallest degree of a subgraph containing a given node. In this paper, we study the coreness of random geometric graphs and show that, with an increasing number of nodes and properly chosen connectivity radius, the coreness converges to a new object, that we call the continuum coreness. In the process, we show that other popular notions of centrality measures, namely the H-index and its iterates, also converge under the same setting to new limiting objects.