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Oriented graphs generated by random points on a circle

Part of: Combinatorial probability Geometric probability and stochastic geometry

Published online by Cambridge University Press:  14 July 2016

Yoshiaki Itoh ,
Hiroshi Maehara  and
Norihide Tokushige
Yoshiaki Itoh*
Affiliation:
The Institute of Statistical Mathematics and The Rockefeller University
Hiroshi Maehara*
Affiliation:
Ryukyu University
Norihide Tokushige*
Affiliation:
Ryukyu University
*
∗Postal address: The Institute of Statistical Mathematics, 4–6–7 Minami-Azabu Minato-ku Tokyo, 106-8569 Japan. Email address: itoh@ism.ac.jp
∗∗Postal address: College of Education, Ryukyu University, Nishihara Okinawa, 903–0213 Japan
∗∗Postal address: College of Education, Ryukyu University, Nishihara Okinawa, 903–0213 Japan
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Abstract

Extending the cascade model for food webs, we introduce a cyclic cascade model which is a random generation model of cyclic dominance relations. Put n species as n points Q1,Q2,…, Qn on a circle. If the counterclockwise way from Qi to Qj on the circle is shorter than the clockwise way, we say Qi dominates Qj. Consider a tournament whose dominance relations are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the probability of getting a regular tournament of order 2r+1 as the largest regular tournament is equal to (n/(2r+1))/2n-1. This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.

Keywords

Random regular tournamentdistribution of orderconfiguration of pointscyclic cascade modelLotka-Volterra model

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 534 - 539
Copyright
Copyright © by the Applied Probability Trust 2000 

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