### 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 language | English |
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Pages (from-to) | 112-136 |

Number of pages | 25 |

Journal | Journal of Applied Probability |

Volume | 55 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Mar 2018 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 55, no. 1, pp. 112-136. https://doi.org/10.1017/jpr.2018.9

}

TY - JOUR

T1 - Inhomogeneous random graphs, isolated vertices, and Poisson approximation

AU - Penrose, Mathew D.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

AB - 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.

KW - Inhomogeneous random graph

KW - Poisson approximation

KW - Stein's method

KW - U-statistic

KW - latent variable model

KW - random connection model

KW - random geometric graph

KW - stochastic block model

UR - http://www.scopus.com/inward/record.url?scp=85044568966&partnerID=8YFLogxK

U2 - 10.1017/jpr.2018.9

DO - 10.1017/jpr.2018.9

M3 - Article

VL - 55

SP - 112

EP - 136

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 1

ER -