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Another installation in the series (see previous posts on this page).

System complexity.– The complexity of a food web depends upon the taxon richness of the system, as well as the topology and dynamics of interspecific interactions. Although richness and topology are captured by graphic depictions, the utility of the depictions is often limited to impressing upon the viewer the overwhelming structural complexity of the systems. For example, here is a Greater Antillean coral reef food web comprising 265 trophic guilds and 4,656 interactions, currently one of the most detailed food web networks available. The system is definitely complicated, as expected of a coral reef community, but not much else can be concluded from the graph. In fact, it is more complicated than illustrated, being based on a dataset comprising 750 species and 34,465 interspecific interactions. Many of the species have been aggregated into sets termed trophic guilds, where members of a guild share prey drawn from the same guild(s), and likewise for predators. Species aggregation is a common way in which to reduce food web network complexity, but there are few formulaic methods for aggregation. The most common method is based on the concept of trophic species (trophospecies), where aggregated species are assumed to have exactly the same prey and predators. The trophic guild concept on the other hand was formulated specifically for fossil taxa and assumes uncertainty in species interactions. It is very important to understand the impacts of aggregation on network structure and dynamics, and the implications for species’ roles in the system. Whether different aggregation schemes yield similar insights into complex systems is currently poorly understood. I will return to this topic in a later post.

Connectance.– A number of measures and summary statistics are used to describe and compare food webs, perhaps the most common one being connectance. Food web connectance differs from the graph connectance defined earlier, because the networks are now directional. Each node may link to every other node including itself, but a directional link from species A to B is no longer equivalent to a link from B to A. The maximum number of links possible is therefore the square of the number of nodes. Using symbols common in the food web literature,
C = \frac{L}{S^{2}}
where L is the number of directional links in the network, and S is the number of nodes or species. Connectance values are generally well below one, reflecting the relative sparsity of links in food webs, but it is difficult to compare connectances among food webs that use different aggregation schemes. Perhaps given this difficulty, it is quite surprising that there is a regular relationship between L and S spanning a large number of food webs, compiled from a variety of sources, and using different aggregation methods (see also Ings et al.). The exponential nature of the relationship shows that link density, or connectance, increases with increasing node richness. It is possible that increasing taxon richness in a community demands greater connectivity in order to maintain efficient energy transfer and hence stability, or the relationship is simply spurious and any true relationship is obscured by the heterogeneity of food web metadata. This remains, in my opinion, an open problem in food web theory.

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