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The topological extinction of a species x_{i} 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 x_{i} and a particular class of prey, y_{j} \in G_{j}, is

E(|x_{i}\leftarrow y_{j}|) = \frac{|y_{j}|}{b_{i}}r_{x}

(we’ll ignore our integer links for now). The probability of topological secondary extinction of x_{i} 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

\mathrm{pr}(e,x_{i}|\omega) = \prod_{j=1}^{|U|}\prod_{r_{y}=1}^{b_{j}}\mathrm{pr}(e,y_{j}|r_{y})^{\frac{|y_{j}|r_{x}a_{ij}}{b_{i}}}

where |U| is the number of guilds in metanetwork U, b_{j} is the number of potential prey species (in-links) of y_{j}, and a_{ij} is the ij^{th} element of U‘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.

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