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

Christopher A. Najim () and Ralph P. Russo ()
Additional contact information
Christopher A. Najim: University of Iowa
Ralph P. Russo: University of Iowa

Methodology and Computing in Applied Probability, 2003, vol. 5, issue 1, 23-33

Abstract: 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.

Keywords: random graphs and subgraphs; uniform strong law of large numbers; central limit theorem; law of the iterated logarithm (search for similar items in EconPapers)
Date: 2003
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1023/A:1024173204072 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:metcap:v:5:y:2003:i:1:d:10.1023_a:1024173204072

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/11009

DOI: 10.1023/A:1024173204072

Access Statistics for this article

Methodology and Computing in Applied Probability is currently edited by Joseph Glaz

More articles in Methodology and Computing in Applied Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().