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What I mean here, when I say “analytical approach”, is basically a non-simulation approach to the problem. I adopt an ensemble approach to estimate the level of topological secondary extinction, where an ensemble consists of all species $x_{i}$ of in-degree $r_{x}$, where $x_{i} \in G_{i} \forall x_{i}$. That is, all species, of a particular in-degree or dietary breadth, belonging to a specific guild. Let $|G_{i}|$ be the species richness of that guild. Then, on the basis of the guild’s trophic link distribution $P(r,i)$, the number of species of in-degree $r_{x}$, or $|x_{i}|$, is estimated as

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

The fraction expresses the relative frequency of species of degree $r_{x}$ in the guild, and hence the function is an estimate of the number of such species, given the guild species richness. The numerator integral is taken over an integer interval of course since species interactions only occur as whole numbers :-). The range of dietary breadth, $r$ is $0 \rightarrow b_{i}$. The minimum recognizes that a species must have at least one other species which it consumes, but no more than the maximum number of species with which its guild-dictated ecology can interact given guild species richnesses.

The probability of topological extinction of a species $x_{i}$ is equal to the probability that, given a perturbation to the community (network), all species (nodes) to which it is linked are lost. The basic formula for this, ignoring considerations of multiple metanetwork connections, is given by a hypergeometric probability. Stated simply, given $b_{i}$ prey, an extinction or perturbation of magnitude $\omega$, what is the probability that $n_{x}$ out of $r_{x}$ links will be lost?

$\mathrm{pr}(n_{x}\mid \omega) = \left( \begin{array}{c}r_{x}\\n_{x}\end{array}\right) \left( \begin{array}{c}b_{i}-r_{x}\\\omega -n_{x}\end{array}\right) \left( \begin{array}{c}b_{i}\\ \omega \end{array}\right)^{-1}$

Topological secondary extinction occurs when $n_{x}=r_{x}$, and the above formula then yields the probability of secondary extinction as

$\mathrm{pr}(e,x_{i}\mid \omega) = \left( \begin{array}{c}b_{i}-r_{x}\\ \omega -r_{x}\end{array}\right) \left( \begin{array}{c} b_{i}\\ \omega \end{array}\right)^{-1} = \frac{\omega !(b_{i}-r_{x})!}{b_{i}!(\omega -r_{x})!}$

This formulation must be elaborated to account for links to multiple different guilds, and consumed/prey species of different in-degrees, and hence their own varying probabilities of secondary extinction.