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Giant components in three-parameter random directed graphs

Part of: Graph theory

Published online by Cambridge University Press:  01 July 2016

Tomasz Łuczak  and
Joel E. Cohen
Tomasz Łuczak*
Affiliation:
Rockefeller University
Joel E. Cohen*
Affiliation:
Rockefeller University
*
∗Permanent postal address: Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, 60–769 Poznan, Poland.
∗∗Postal address: The Rockefeller University, 1230 York Avenue, Box 20, New York, NY 10021, USA.
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Abstract

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertex i to vertex j where i < j, the probability of ‘down arrows' from i to j where i ≥ j, and the probability of bidirectional arrows between i and j. In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.

Keywords

PHASE TRANSITIONACYCLIC RANDOM DIGRAPHSRANDOM GRAPHSFOOD WEBECOLOGY

MSC classification

Primary: 05C80: Random graphs
Secondary: 92D40: Ecology
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 4 , December 1992 , pp. 845 - 857
Copyright
Copyright © Applied Probability Trust 1992 

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