The topological extinction of a species requires extinction of all its prey resources, or incoming links. Since the probability of extinction is a function of in-degree, it is helpful to distinguish among prey of different in-degrees. Also, prey guild membership is also a necessary parameter, as the manner in which a perturbation propagates through the network is a function of metanetwork topology and hence guild linkages.
The expected number of links between and a particular class of prey, , is
(we’ll ignore our integer links for now). The probability of topological secondary extinction of is then a function of the probabilities of extinction of all its prey, those prey being distinguished by guild membership and in-degree. This may be written as
where is the number of guilds in metanetwork , is the number of potential prey species (in-links) of , and is the element of ‘s adjacency matrix. The use of the adjacency matrix allows us to generalize the formula to all guilds and species in the community, regardless of the metanetwork’s topology.