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May, 1993 Stratigraphy of a Random Acyclic Directed Graph: The Size of Trophic Levels in the Cascade Model
Tomasz Luczak, Joel E. Cohen
Ann. Appl. Probab. 3(2): 403-420 (May, 1993). DOI: 10.1214/aoap/1177005431

Abstract

When an ecological food web is described by an acyclic directed graph, the trophic level of a species of plant or animal may be described by the length of the shortest (or the longest) food chain from the species to a green plant or to a top predator. Here we analyze the number of vertices in different levels in a stochastic model of acyclic directed graphs called the cascade model. This model describes several features of real food webs. For an acyclic directed graph D, define the ith lower (upper) level as the set of all vertices ν of D such that the length of the shortest (longest) maximal path starting from ν equals i,i=0,1. In this article, we compute the sizes of the levels of a random digaph D(n,c) obtained from a random graph on the set {1,2,,n} of vertices in which each edge appears independently with probability c/n, by directing all edges from a larger vertex to a smaller one. The number of edges between any two levels of D(n,c) is also found.

Citation

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Tomasz Luczak. Joel E. Cohen. "Stratigraphy of a Random Acyclic Directed Graph: The Size of Trophic Levels in the Cascade Model." Ann. Appl. Probab. 3 (2) 403 - 420, May, 1993. https://doi.org/10.1214/aoap/1177005431

Information

Published: May, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0790.05076
MathSciNet: MR1221159
Digital Object Identifier: 10.1214/aoap/1177005431

Subjects:
Primary: 05C80
Secondary: 05C20‎ , 92D40

Keywords: acyclic , ‎digraph‎ , food web , random graph , trophic level

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.3 • No. 2 • May, 1993
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