The metanetwork is a higher guild-level representation of the community food web or trophic network, incorporating uncertainty and variability of the species-level links. The number of species-level networks that can be derived from the network is finite, but large. This species-level network space encompasses all the possible configurations and behaviours that can be exhibited by the community. An interesting excercise is to calculate the size of the space under particular parameterizations of the metanetwork. Let’s do that for the Late Miocene community whose metanetwork is described here. An example of one of the species-level food webs is illustrated in this figure. Okay, here we go…
A metanetwork U is a set of guilds, . represents the power of U, or the number of guilds, and is the species-richness or number of species in guild . The number of potential prey species for each species in is the sum of the species-richnesses of all guilds designated by the topology of U as prey to . This number is calculated as
where is the number of potential prey species and is the element of U’s binary adjacency or connectivity matrix. The order ij designates i as a predator of j, and 1 if this is true, and zero otherwise.
If for simplicity we assume that all species in the community are extreme specialists, that is, they consume specifically a single resource, then the number of different species link configurations possible in is . Therefore in U there are exactly
possible species-level networks. Given that links within each guild are described, however, by a trophic link distribution, and that species within a guild are expected to differ in their dietary breadths, the expected (average) number of species-level networks is found by first calculating the number of possible ways in which the links of a single species may be arranged among potential prey nodes, and the applying that to the above product.
where is the number of species in of in-degree (with number of prey) . (The formula for calculating was given in a previous post.)