Defining and Measuring Trophic Role Similarity in Food Webs Using ...

DEFINITION OF TROPHIC ROLE SIMILARITY 307graph may have more than one regular equivalence.Since our goal is usually to seek thesimplest model, our discussion will center onthe maximal non-trivial regular equivalence (i.e.the one with the fewest equivalence classes).Consequently, when we write ‘‘regular equivalence’’it is understood that this means the maximalregular equivalence unless otherwise stated.An example of a regular equivalence is shownin Fig. 1(c), again using same color and fill toindicate equivalence. To determine whether itis regular, we check each pair of nodes that iscolored the same against the definition. Forexample, nodes 2 and 3 are colored the same,indicating they are equivalent. According to thedefinition, their out-neighborhoods (and their inneighborhoods)must contain equivalent nodes(i.e. the same colors). Node 2 has incoming tiesfromcross-hatched nodes and vertical-linednodes and no other colors. Node 3 is the same.Node 2 has outgoing ties to stippled nodes andcross-hatched nodes. Node 3 is again the same(recall that the definition says nothing about thenumber of nodes of each type that a node maybe connected to). Hence this pair satisfies thedefinition. If all pairs do, the equivalence isregular. In contrast, Fig. 1(d) gives an exampleof a non-regular equivalence. One reason it isnot regular is that node 4 is colored the same asnode 2, yet node 4 has no tie to a vertical-linednode, while node 2 does.It is useful to note three desirable features ofregular equivalence. The first is that cycles, suchas the one connecting nodes 2, 3 and 11 inFig. 1(c), pose no difficulty. The second is thatomnivory also provides no particular problem,as nodes 10 and 1 feed at the same two levels ofthe food chain (and are equivalent) whereasnodes 4 and 2 feed at different levels (and are notequivalent). The third is that two species can beregularly equivalent without having any prey orpredators in common, as is the case with nodes 5and 6. Thus, regular equivalence captures theidea that species can occupy analogous positionsin the food web, without the limitation found inthe Yodzis–Winemiller approach that requiresspecies assigned to the same class to interacttrophically with the very same species. Animplication of the last feature is that regularequivalence can be used to class together speciesthat are members of different food webs, yetoccupy analogous positions. An example is givenin Fig. 1(e) in which nodes 12 and 2 are regularlyequivalent, yet obviously belong to separatewebs. In this way, regular equivalence may beused to improve upon the practice of comparingwebs by counting numbers of species by basal-,intermediate- and top-level species (Cohen &Newman, 1985).2.4. IMAGE GRAPHSThe image graph induced by an equivalencerelation R on G is defined as the digraphG 0 (C(V),E 0 ) in which the nodes are the equivalenceclasses of R, and two nodes (i.e. classes) xand y are adjacent in G 0 if there exist adjacentnodes u and v in G such that C(u) ¼ x and C(v)¼ y [recall that C(u) denotes the equivalenceclass of node u]. In this sense, an equivalencerelation induces a mapping, known as a graphhomomorphism, from the original network to areduced model of the network. Figure 2(a) showsthe image graph for the structural equivalence inFig. 1(a). It should be noted that in a structuralequivalence, the image graph is a completedescription of the entire systemof relations,Fig. 2. (a) Image graph for the structural coloration inFig. 1(a); (b) image graph for the regular coloration inFig. 1(c); (c) a graph of a hypothetical food web showingthe regular equivalence of feeding relations. Directed tiesare depicted by arrows that show the predator’s consumptionof a prey. Color codes are the same as in Fig. 1.