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308J. J. LUCZKOVICH ET AL.without loss of relational information. If twocolors blue and red are mutually connected inthe image graph, then, in the original graph,every blue node has a connection to every rednode, and every red node has an incoming tiefromevery blue node.The image graph of a regular equivalence alsoprovides a reduced model of the network(s) as awhole. Unlike structural equivalence, however,with regular equivalence there is some loss ofrelational information, although the extent andtype of loss are strictly controlled. Figure 2(b)gives the image graph associated with the regularequivalence in Fig. 1(e). Given the rules of aregular equivalence, we are guaranteed that if, inthe image graph, there is a tie from a verticallinednode to a cross-hatched node, then in theoriginal graph, every cross-hatched node has anincoming tie from at least one vertical-linednode, and every vertical-lined node has anoutgoing tie to at least one cross-hatched node.Thus, the image graph from a regular equivalenceprovides a simplified graph of the patternof connections among individual species; it is adisciplined way of representing complex foodweb structure in which some detailed informationis lost, but the fundamental pattern orstructure of ties is preserved.A look at Fig. 2(c) suggests that in addition tocapturing the notion of species trophic role,regular equivalence has some relationship to thenotion of trophic position. Node 3, which iscross-hatched, is two links above a primaryproducer, and one link below a top predator. Allthe other cross-hatched nodes are positioned thesame. Note that nodes 7 and 8 are very much likethe cross-hatched nodes: they have an outgoingtie to the top predator, and an incoming tie froma vertical-lined prey. But unlike the crosshatchednodes, they also have a tie directly tothe primary producers (white nodes), i.e. they areclassic omnivores. As a result, they are placed ina different class, which corresponds to a differenttrophic position. Similarly, node 6, which also isprey for the top predators, is in a different classfromthe vertical-lined nodes, the cross-hatchednodes and the gray nodes because it feedsdirectly and exclusively on a primary producer.Note that this means that node 11, the primaryproducer connected to node 6, cannot be in thesame class as the other producers 12 and 13. Inthis sense, regular equivalence makes finerdistinctions than do most other measures oftrophic position, because the producers will begrouped into separate classes with differingrelations to the herbivore and omnivore classes.Consequently, we can think of regular equivalenceas defining classes of isotrophic species,rather than trophic levels of species.The relationship between regular equivalenceand trophic position is clearest when the foodweb contains no cycles. In such webs, two speciesthat are colored the same are equally distantfromcorresponding top predators and producers.For example, if a b, then if a is positionedtwo links above a given producer species c, thenb is also two links above either c or some otherproducer d that is regularly equivalent to c.Thus, under such circumstances, nodes in thesame class are necessarily at the same trophicposition. In food webs with cycles, the correspondencenoted above can break down. However,there is a simple solution. In graph theory,the length of the shortest path between twonodes is known as the geodesic distance betweenthem. By calculating the geodesic distancebetween all pairs of nodes, we can construct anode-by-node geodesic distance matrix. Applyingregular equivalence to this matrix instead ofthe raw adjacency matrix generates a regularequivalence that necessarily preserves trophiclevels, as two nodes that are colored the samewill be the same geodesic distance from equivalentothers. This approach can also be applied toflow data by applying regular equivalencesimultaneously to both the flow matrix and thegeodesic distance matrix, but the details of thistechnique are beyond the scope of this paper.While regular equivalence classes are homogeneouswith respect to trophic position, asexplained above, they produce classes that differfromthe groupings that result using the classicdefinition of trophic level. One reason is thattrophic-level concept considers only distancefromthe producers, and does not include bothdownward and upward trophic paths. Forexample, if two nodes a and b are both twolinks above the producer, then they are notalways regularly equivalent, because they maydiffer in the number of predator links above
DEFINITION OF TROPHIC ROLE SIMILARITY 309themin the web. In addition, some approaches,such as Adams et al. (1983), are sensitive to therelative number of prey at different levels, whichis not the case with regular equivalence ingeneral, although it is true of a subset of regularcolorations known as exact regular colorations(Everett & Borgatti, 1996). For example, inFig. 2(c), nodes 1 and 2 are placed at the samelevel by a regular coloration, even though node 1preys on two cross-hatched species while node 2preys on only one.A distinctive feature of the definition ofregular equivalence is its recursive or implicitquality. That is, to determine whether two nodesare equivalent, one needs to know the colors ofall the other nodes, but to determine their colorsone needs to know the colors of all nodesincluding the two nodes one started with, and soon. This may appear to be computationallyintractable, but in fact is not. There are severalefficient algorithms available for regular equivalence(Borgatti et al., 1999), which vary in thetype of output they produce (e.g. pairwisecoefficients giving the degree of equivalence vs.approximate discrete classes), type of data theycan handle (e.g. valued or binary), and othervariables.that are attracted to a Malaysian pitcher plantand become drowned or are preyed upon byother insects that live in the plant. In a sense, it ismerely a sub-web of a larger food web, becausethe insects are consuming energy elsewhere andbringing it to the pitcher plant. Thus, the role of‘‘producers’’ is filled by the drowned insects andlive insects (ants) that visit the pitcher plant,which are really consumers in a larger food web.In the original presentation (Beaver, 1985), sometaxa were grouped into ‘‘trophic types’’, which infact were structurally equivalent, because theyhad the exact same links to other taxa in thefood web. Stiling (2002) disaggregated thesetaxa, and we did as well, so as to start with acompletely disaggregated food web at the specieslevel (Fig. 3).3.3. ST. MARKS, FLORIDA SEAGRASS CARBONFLOW WEBThe food web data were obtained fromdirectsampling and literature surveys of the St. Marksseagrass ecosystem(Baird et al., 1998; Christian& Luczkovich, 1999, Luczkovich et al., in press).3. Methods3.1. EMPIRICAL APPLICATIONSIn the last section we laid out the mathematicaland computational underpinnings of the regularequivalence model. Now we illustrate the empiricalapplication of these concepts using twoempirical food web datasets, namely the Malaysianpitcher plant insect web (Beaver, 1985) asreproduced and discussed in Stiling (2002), andthe St. Marks, Florida seagrass food web (Bairdet al., 1998; Christian & Luczkovich, 1999;Luczkovich et al., in press). The insect datasetis a topological web consisting of predation links,while the St. Marks, Florida seagrass ecosystemis a dynamic web, with data consisting ofestimated carbon flows between compartments.3.2. MALAYSIAN PITCHER PLANT WEBThis is one of several food webs that have beendescribed by Beaver (1985), involving insectsFig. 3. A simple food web diagram of the insects in thepitcher plant Nepenthes albomarginata in West Malaysia[modified from the diagram presented by Stiling (2001),based on orginal data fromBeaver, 1985]. Appendix A liststhe species’ numerical codes for each node. Predators arepositioned higher than their prey, and lines representtrophic linkages.
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