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Advances in Applied Probability Advances in Applied Probability

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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
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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 TRANSITION ACYCLIC RANDOM DIGRAPHS RANDOM GRAPHS FOOD WEB ECOLOGY

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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References

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