Perhaps the most obvious structural elements of real food webs that distinguishes them from the graphs presented earlier is directionality of the links. Links are trophic interactions, that is, predator-prey relationships, and describe the passage of energy from prey species to predators. They can also be used to describe the impact of predation on a prey species, recognizing that the relationship is an asymmetrical one between nodes. The “traditional” manner in which to depict this graphically is with arrows between nodes (Fig. A). Whereas the graphs illustrated so far have been undirected graphs, a food web is defined properly as a directed graph, or digraph. The asymmetry is also reflected by the adjacency matrix, which is no longer symmetric about the diagonal.
The most straightforward applications of Graph Theory to food web biology are analyses of the structure or topology of digraphs. Digraphs are often referred to as networks in modern usage, and the study of digraphs, especially those describing real-world networks such as the Internet or social networks, is described as Network Theory. The reader should be aware, however, that networks are technically graphs that are digraphs having weighted or parameterized links. A network therefore depicts a food web when it contains species interactions, the direction of those interactions, and some measure of the interactions, such as interaction strength. A digraph without measures or weights on the links is in reality a special case of a food web digraph, one in which all links are considered equivalent.
A very simple three species food web is illustrated in Fig. A. Species 1 (S1) is prey only (perhaps a primary producer), S2 is both a predator or consumer of S1 while being prey to S3, and S3 is the top consumer in the network. Alternative arrangements for three species are illustrated in Fig. B-D, including a simple food chain (Fig. B), a web where the top consumer is also cannibalistic (Fig. C), and a cycle among the three species (Fig. D). These networks bear only information about the existence and direction of interactions among species, but this information is important because structure always affects function (Strogatz, 2001). The basic network approach has proven useful as a means of capturing the complexity of food webs, deriving basic comparative properties such as connectance and link distributions, and assessing one type of robustness against perturbation.