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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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