likelihood « Roopnarine’s Food Weblog

Trophic network probability

February 7, 2009 proopnarine Leave a comment

Given any species level topology $(E,V)\text{,}$ where $E$ represents edges or links, and $V$ represents vertices or species, the probability of that specific topology is the probability of the set of interspecific links specified or composing the topology. Therefore, the probability of the network, given metanetwork $U\text{,}$ is

$p(E,V) = \prod_{x=1}^{\sum \vert G_{u}\vert} \prod _{u=1}^{\vert U \vert}p(r_{x}^{u} \mid r_{x})$
where the right product is the probability of any species $x$ having a particular pattern or topology of links to other species, and the left product is the product of those probabilities for all species in the community. We can see immediately how the probability of any particular species-level food web is built from the probabilities of individual species networks.

The number of species-level networks that can be derived from a metanetwork depends upon permutations of all possible combinations of link topologies of species in the community, and is generally an astronomical number for even a modest number of species and guilds. The network with the greatest probability, or maximum likelihood of occurrence, is one where the probabilities in the formula above are maximized. This can be approximated if one considers that the probability of $x_{i}$ being linked to any $x_{j}$ is equal, regardless of the in-degree of $x_{j}\text{.}$ One can therefore consider the probability of linking to any $x_{j}$ to simply be the proportion of the predator’s set of prey species that is represented by $\vert G_{j}\vert\text{.}$ These are simply maximum likelihood estimates of the metanetwork link probabilities.

The in-links of each species are assigned randomly to species in other guilds for which a metanetwork in-link exists, i.e. those other guilds comprise potential prey of the species, and $a_{iu}=1\text{,}$ where the species belongs to guild $i.$ The resulting network may be represented as a $N\times N$ adjacency matrix, $A_{N}\text{,}$ where $N$ is the total number of species in the community. Binary entries $n_{xy}$ indicate whether species $y$ is prey to species $x\text{.}$ Furthermore, row sums, $\sum n_{xy}$ equal the in-degree of species $x.$

The probability of a species $a\text{'s}$ link topology, $p(a_{i})\text{,}$ or the binary pattern of the row $A_{a}\text{,}$ is the product of the probabilities of each link. This can be calculated efficiently as the multinomial probability of a pattern of links spread among the guilds. Say that the probability of a link between $a$ and a species in guild $G_{j}$ is equal to the fraction of the diversity of $a\text{'s}$ prey that is represented by the species richness of $G_{j}\text{,}$ then the probability of $a$ is
$p(a_{i}) = \frac{r_{a}!}{r_{a}^{1}!r_{a}^{2}!,\ldots,r_{a}^{n}!} \prod_{x=1}^{n}\left( \frac{\vert G_{x}\vert}{\vert R_{i}\vert}\right) ^{r_{a}^{x}} \Rightarrow \frac{r_{a}!}{r_{a}^{1}!r_{a}^{2}!,\ldots,r_{a}^{n}!} \left( \frac{1}{\vert R_{i}\vert}\right)^{r_{a}} \prod_{x=1}^{n}\left( \vert G_{x}\vert\right) ^{r_{a}^{x}}$
where $r_{a}^{x}$ is the number of $a\text{'s}$ links to species in guild $G_{x}\text{,}$ $\vert G_{x}\vert$ is the species richness of $G_{x}\text{,}$ and $\vert R_{i}\vert$ is the total number of prey potentially available to $a$ according to metanetwork $U$ , where $a\in G_{i}\text.$ It is important to note that each term in the formula exists iff $a_{ix}=1\text{,}$ i.e. a metanetwork link exists $G_{i}\leftarrow G_{x}.$ This prevents the inclusion of zero probabilities. Finally, the overall probability of the network is calculated as $p(E,V)=\prod_{a=1}^{a=N}p(a)\text{,}$ the product of all species probabilities, which in turn are the products of all link probabilities.