Inhomogeneous random graphs, isolated vertices, and Poisson approximation

Research output: Contribution to journalArticle

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Abstract

Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability φ(x, y). Suppose the number of points is large but the mean number of isolated points is O(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

Original languageEnglish
Pages (from-to)112-136
Number of pages25
JournalJournal of Applied Probability
Volume55
Issue number1
DOIs
Publication statusPublished - 1 Mar 2018

Fingerprint

Poisson Approximation
Random Graphs
Statistics
Stein's Method
Poisson Point Process
U-statistics
Number of Components
Poisson process
Set of points
Siméon Denis Poisson
Arbitrary
Graph in graph theory

Keywords

  • Inhomogeneous random graph
  • Poisson approximation
  • Stein's method
  • U-statistic
  • latent variable model
  • random connection model
  • random geometric graph
  • stochastic block model

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

Cite this

Inhomogeneous random graphs, isolated vertices, and Poisson approximation. / Penrose, Mathew D.

In: Journal of Applied Probability, Vol. 55, No. 1, 01.03.2018, p. 112-136.

Research output: Contribution to journalArticle

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