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Neither version of fractional trophic level described in the previous post can be adapted to trophic networks. There are two obstacles. First, bi-directional links between nodes, representing inter-node predation, introduces recursion that prohibits the calculation of a static value. Second, Pauly’s version, while a very appropriate measure of trophic level, cannot work unless one has knowledge of the fraction of a node’s prey represented by the prey node. Those data are generally unavailable for complex network implementations of food webs (or for most of the species anyway). Many other measures of trophic level in complex trophic networks boil down simply to semi-quantitative assignments based on whether nodes are primary producers, primary consumers, true omnivores, or secondary consumers. The very simple network presented in the previous post could lead to conflicting assignments unless some conventional standard was formulated. But there’s another way.

The network is repeated here, but now with trophic levels assigned to each node. Trophic level was computed as the mean shortest path length from a node’s prey to a primary producer node. Primary producer nodes are therefore assigned a path length of zero (e.g. node 1). The formula, in the spirit of fractional trophic level, is for a node j
$\mathrm{ftl_{j}} = 1 + \frac{1}{r_{j}}\sum_{i=1}^{r}l_{i}$
where r is the in-degree of node j, and l is the shortest path from prey node i to a primary producer node. The formula is generalized to an entire network as
$\mathrm{ftl_{j}} = 1 + \frac{1}{r_{j}}\sum_{i=1}^{\vert U\vert}a_{ji}l_{i}$
where, as in many previous posts, $\vert U\vert$ is the size of the network, i.e. the number of nodes, and $a_{ji}$ is the ji-th entry of a binary adjacency matrix indicating whether node i is prey to node j.

For example, the trophic level of node 2 is
$1 + \frac{1}{1}(0) = 1$
while that of node 3 is
$1 + \frac{1}{2}(2 + 0) = 2$