The sizes of components in random circle graphs

• Autorzy: Ramin Imany-Nabiyyi
• Języki publikacji: EN

Abstrakty

EN

We study random circle graphs which are generated by throwing n points (vertices) on the circle of unit circumference at random and joining them by an edge if the length of shorter arc between them is less than or equal to a given parameter d. We derive here some exact and asymptotic results on sizes (the numbers of vertices) of "typical" connected components for different ways of sampling them. By studying the joint distribution of the sizes of two components, we "go into" the structure of random circle graphs more deeply. As a corollary of one of our results we get the exact, closed formula for the expected value of the total length of all components of the random circle graph. Although the asymptotic distribution for this random characteristic is well known (see e.g. T. Huillet [4]), this surprisingly simple formula seems to be a new one.

Słowa kluczowe

EN

random interval graphs; geometric graphs; geometric probability

Kategorie tematyczne

• 60D05: Geometric probability and stochastic geometry
• 05C80: Random graphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2008
• Tom: 28
• Numer: 3
• Strony: 511-533

Daty

wydano

2008

otrzymano

2008-02-18

poprawiono

2008-04-04

zaakceptowano

2008-04-07

Twórcy

autor

Ramin Imany-Nabiyyi

• Shahid Beheshti University, Department of Statistics, Evin, Tehran, Iran

Bibliografia

• [1] L. Fatto and A.G. Konheim, The random division of an interval and the random covering of a circle, SIAM Review 4 (1962) 211-222, doi: 10.1137/1004058.
• [2] E. Godehardt and J. Jaworski, On the connectivity of a random interval graphs, Random Structures and Algorithms 9 (1996) 137-161, doi: 10.1002/(SICI)1098-2418(199608/09)9:1/2<137::AID-RSA9>3.0.CO;2-Y
• [3] L. Holst and J. Husler, On the random coverage of the circle, J. Appl. Prob. 21 (1984) 558-566, doi: 10.2307/3213617.
• [4] T. Huillet, Random covering of the circle: the size of the connected components, Adv. in Appl. Probab. 35 (2003) 563-582, doi: 10.1239/aap/1059486818.
• [5] H. Maehara, On the intersection graph of random arcs on a circle, Random Graphs' 87 (1990) 159-173.
• [6] M. Penrose, Random Geometric Graphs (Oxford Studies in Probability, 2003), doi: 10.1093/acprof:oso/9780198506263.001.0001.
• [7] A.F. Siegel, Random arcs on the circle, J. Appl. Prob. 15 (1978) 774-789, doi: 10.2307/3213433.
• [8] H. Solomon, Geometric Probability (Society for Industrial and Applied Mathematics, Philadelphia, 1976).
• [9] F.W. Steutel, Random division of an interval, Statistica Neerlandica 21 (1967) 231-244, doi: 10.1111/j.1467-9574.1967.tb00561.x.
• [10] W.L. Stevens, Solution to a geometrical problem in probability, Ann. Eugenics 9 (1939) 315-320, doi: 10.1111/j.1469-1809.1939.tb02216.x.

Identyfikatory

DOI

10.7151/dmgt.1424