Stratigraphy of a Random Acyclic Directed Graph: The Size of Trophic Levels in the Cascade Model

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 $i$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.