Metanetwork state space

Each metanetwork is a higher (than species) level representation of ecological/functional diversity within the community. Species sharing the same potential prey and predators are grouped into guilds. Trophic connections among guilds are certain. From each such metanetwork $U$ are derived species-level trophic networks, where links or interactions among individual species are specified. The CEG model derives these lower level networks stochastically, reflecting both uncertainty in the specific topology of the food web, as well as the fact that this topology is expected to vary on ecological spatial scales, and on evolutionary temporal scales. Stochastic derivation is controlled by drawing the number of in-links of any given consumer from a trophic link distribution assigned to that species guild. The link distributions are decay distributions (generally exponential, power law, or a mixed exponential-power law), reflecting the relative proportions and ranging of species within a guild from extreme specialists to generalists. The distributions are also truncated, with an upper limit set, of course, by the maximum number of prey species available to a guild.

Truncation of the link distributions therefore means that while derivation of a species-level network from $U$ is stochastic, the number of such networks is finite, defining a state space for $U$ . The size of the state space is calculated by first determining the in-degrees of all consumer species. Let the in-degree of species $x_{i} \in G_{i}$ ( $x$ is a member of guild $G_{i}$ ) range $\left[ 1, \sum a_{ij}| G_{j}| \right]$ , where $a_{ij}$ is the $ij^{th}$ element of $U$ ‘s binary adjacency matrix. The sum is therefore the total species richness of all guilds that are prey to guild $G_{i}$ . Given a guild trophic link distribution $P(r_{i})$ , and an in-degree of $x_{i}$ equal to $r_{x}$ , the total number of configurations of $x_{i}$ ‘s in-links is

$S(x_{i}) = \left( \begin{array}{c}\sum a_{ij}| G_{i}| \\ r_{x}\end{array} \right)$

Within a guild, the number of species of a particular in-degree $r$ is estimated discretely as

$E(|x_{i}|) = \frac{\int_{r-1}^{r} P(r_{i})}{\int_{0}^{b_{i}} P(r_{i})}$

where $b_{i} = \sum a_{ij}| G_{j}|$ is, again, the species richness of all $G_{i}$ ‘s prey guilds. The total number of species-level topologies in $U$ is therefore the product of all in-link configurations possible for all species, and where $x_{i}$ represents a type or category of species within a guild $G_{i}$ , this total is