IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v5y2003i1d10.1023_a1024173204072.html
   My bibliography  Save this article

On the Number of Subgraphs of a Specified Form Embedded in a Random Graph

Author

Listed:
  • Christopher A. Najim

    (University of Iowa)

  • Ralph P. Russo

    (University of Iowa)

Abstract

Let U 1, U 2,... be a sequence of i.i.d. random elements in Rd. For x>0, a graph G n (x) may be formed by connecting with an edge each pair of points in $$\left\{ {U_i :1 \leqslant i \leqslant n} \right\}$$ that are separated by a distance no greater than x. The points of G n (x) could represent the stations in a telecommunications network and the edge set the lines of communication that exist among them. Let $$A$$ be a collection of graphs on m≤n points having a specified “form” or structure, and let $${\varepsilon }_n \left( {x,A} \right)$$ denote the number of subgraphs embedded in G n (x) and contained in $$A$$ . It is shown that a SLLN, CLT and LIL for $${\varepsilon }_n \left( {x,A} \right)$$ follow easily from the theory of U-statistics. In addition, a uniform (in x) SLLN is proved for collections $$A$$ that satisfy a certain monotonicity condition. Some applications are mentioned and the results of some simulations presented. The scaling constants appearing in the CLT are usually hard to obtain. These are worked out for some special cases.

Suggested Citation

  • Christopher A. Najim & Ralph P. Russo, 2003. "On the Number of Subgraphs of a Specified Form Embedded in a Random Graph," Methodology and Computing in Applied Probability, Springer, vol. 5(1), pages 23-33, March.
    • Handle: RePEc:spr:metcap:v:5:y:2003:i:1:d:10.1023_a:1024173204072
    DOI: 10.1023/A:1024173204072
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1024173204072
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1024173204072?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Yusuke Ide & Norio Konno & Naoki Masuda, 2010. "Statistical Properties of a Generalized Threshold Network Model," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 361-377, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metcap:v:5:y:2003:i:1:d:10.1023_a:1024173204072. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.