Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Coreness and H-Index of Random Geometric Graphs

Eddie Aamari, Ery Arias-Castro, Clément Berenfeld

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.

The Coreness and H-Index of Random Geometric Graphs

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.

Paper Structure

This paper contains 18 sections, 18 theorems, 84 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Setting
  4. Outline
  5. A preliminary result of stochastic convergence
  6. H-Index
  7. Continuum H-indices: $r > 0$ fixed
  8. Continuum H-indices: $r\to 0$
  9. Coreness
  10. Continuum coreness: $r>0$ fixed
  11. Continuum coreness: $r\to 0$
  12. Numerical Simulations
  13. Illustrative Examples
  14. Convergence Rates
  15. Iterations of the H-Index
  16. ...and 3 more sections

Key Result

Lemma 1.1

Let $(\mathcal{S}_r)_{r>0}$ be a family of classes of subsets of $\mathbb{R}^d$ such that: Then, for any sequence $r = r_n$ such that $n r^d \gg \log n$, we have where we recall that $P_n$ is the empirical distribution of a $n$-sample drawn from $P$.

Figures (7)

  • Figure 1: These are the main relationships that we establish. In this diagram, ${\rm H}^k_r f_r(x)$ is defined in \ref{['def:hr']} and arises as the large-$n$ limit of $\frac{1}{N} {\rm H}^k_r(x,\mathcal{X}_n)$; ${\rm C}_r(x,f)$ is defined as the large-$k$ limit of ${\rm H}^k_r(x,f)$, and is shown to be the large-$n$ limit of $\frac{1}{N} {\rm C}_r(x,\mathcal{X}_n)$; and ${\rm C}_0(x,f)$ is defined as the small-$r$ limit of ${\rm C}_r(x,f)$.
  • Figure 2: A density $f$, a function $\phi$, and its transform ${\rm H}_r \phi$ for $r = 0.1$. Both $f$ and $\phi$ are smooth. ${\rm H}_r \phi$ does not appear to be continuously differentiable everywhere but is nonetheless Lipschitz, with Lipschitz constant no bigger than that of $f$ and $\phi$ (see Lemma \ref{['lem:monotonous']}).
  • Figure 3: The successive iterations of ${\rm H}_r^k f_r$ (solid) for a given density $f$ (dashed), for $k$ ranging from $0$ to $100$ with $r=0.1$. The hundredth iteration is very close to its limit ${\rm C}_r(x,f)$.
  • Figure 4: An illustration of $f$ (blue), $f/2$ (red) and ${\rm C}_0(\cdot,f)$ (black) for a mixture of $6$ Gaussians in dimension $d=1$. In the zones where ${\rm C}_0(\cdot,f)$ does not coincide with $f/2$, it exhibits plateaus over intervals $[x_{\min},x_{\max}]$. For $x \in (x_{\min},x_{\max})$, the supremum of Lemma \ref{['lem:c0-variational']} is attained for $S = (x_{\min},x_{\max})$. Otherwise, this supremum is asymptotically attained for $S = \{x\}$.
  • Figure 5: Illustrative examples in dimension $d=1$ and in dimension $d=2$.
  • ...and 2 more figures

Theorems & Definitions (37)

  • Lemma 1.1
  • proof
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 27 more