Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-08T06:36:04.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Inhomogeneous random graphs, isolated vertices, and Poisson approximation

Part of: Limit theorems Graph theory

Published online by Cambridge University Press:  28 March 2018

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
University of Bath
*
* Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Inhomogeneous random graphrandom connection modelstochastic block modellatent variable modelrandom geometric graphPoisson approximationStein's methodU-statistic

MSC classification

Primary: 05C80: Random graphs
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 112 - 136
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation (Oxford Studies Prob. 2). Oxford University Press. CrossRefGoogle Scholar
[2]Bhamidi, S., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2012). Novel scaling limits for critical inhomogeneous random graphs. Ann. Prob. 40, 2299–2361. Google Scholar
[3]Bollobás, B. (2001). Random Graphs (Camb. Studies Adv. Math. 73), 2nd edn. Cambridge University Press. Google Scholar
[4]Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122. CrossRefGoogle Scholar
[5]Decreusefond, L., Schulte, M. and Thäle, C. (2016). Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry. Ann. Prob. 44, 2147–2197. CrossRefGoogle Scholar
[6]Devroye, L. and Fraiman, N. (2014). Connectivity of inhomogeneous random graphs. Random Structures Algorithms 45, 408–420. Google Scholar
[7]Díaz, J., Petit, J. and Serna, M. (2000). Faulty random geometric networks. Parallel Process. Lett. 10, 343–357. Google Scholar
[8]Franceschetti, M. and Meester, R. (2007). Random Networks for Communication. Cambridge University Press. Google Scholar
[9]Gupta, B. and Iyer, S. K. (2010). Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs. Adv. Appl. Prob. 42, 631–658. Google Scholar
[10]Gupta, P. and Kumar, P. R. (1999). Critical power for asymptotic connectivity in wireless networks. In Stochastic Analysis, Control, Optimization and Applications, Birkhäuser, Boston, MA, pp. 547–566. Google Scholar
[11]Hoff, P. D., Raftery, A. E. and Handcock, M. S. (2002). Latent space approaches to social network analysis. J. Amer. Statist. Assoc. 97, 1090–1098. Google Scholar
[12]Hsing, T. and Rootzén, H. (2005). Extremes on trees. Ann. Prob. 33, 413–444. Google Scholar
[13]Iyer, S. K. and Thacker, D. (2012). Nonuniform random geometric graphs with location-dependent radii. Ann. Appl. Prob. 22, 2048–2066. Google Scholar
[14]Kingman, J. F. C. (1993). Poisson Processes (Oxford Studies Prob. 3). Oxford University Press. Google Scholar
[15]Last, G. and Penrose, M. (2018). Lectures on the Poisson Process (IMS Textbooks 7). Cambridge University Press. Google Scholar
[16]Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York. Google Scholar
[17]Mao, G. and Anderson, B. D. O. (2012). Towards a better understanding of large-scale network models. IEEE/ACM Trans. Networking 20, 408–421. Google Scholar
[18]Meester, R. and Roy, R. (1996). Continuum Percolation (Camb. Tracts Math. 119). Cambridge University Press. CrossRefGoogle Scholar
[19]Penrose, M. D. (1998). Extremes for the minimal spanning tree on normally distributed points. Adv. Appl. Prob. 30, 628–639. Google Scholar
[20]Penrose, M. D. (1999). On k-connectivity for a geometric random graph. Random Structures Algorithms 15, 145–164. Google Scholar
[21]Penrose, M. (2003). Random Geometric Graphs (Oxford Studies Prob. 5). Oxford University Press. Google Scholar
[22]Penrose, M. D. (2016). Connectivity of soft random geometric graphs. Ann. Appl. Prob. 26, 986–1028. Google Scholar
[23]Rastelli, R., Friel, N. and Raftery, A. E. (2016). Properties of latent variable network models. Network Sci. 4, 407–432. Google Scholar
[24]Snijders, T. A. B. and Nowicki, K. (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classification 14, 75–100. Google Scholar
[25]Söderberg, B. (2002). General formalism for inhomogeneous random graphs. Phys Rev. E 66, 066121. CrossRefGoogle ScholarPubMed
[26]Yi, C.-W., Wan, P.-J., Lin, K.-W. and Huang, C.-H. (2006). Asymptotic distribution of the number of isolated nodes in wireless ad hoc networks with unreliable nodes and links. In Proc. Global Telecommun. Conf., 2006, IEEE. Google Scholar