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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


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 $i$th lower (upper) level as the set of all vertices $\nu$ of $D$ such that the length of the shortest (longest) maximal path starting from $\nu$ equals $i, i = 0, 1\cdots$. 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 $\{1, 2,\cdots,n\}$ 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.


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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.


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

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

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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