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Open AcceMethodology articleMeasuring specialization in species interaction networksNico Blüthgen*1, Florian Menzel1 and Nils Blüthgen2

Address: 1Department of Animal Ecology and Tropical Biology, University of Würzburg, Biozentrum, Am Hubland, 97074 Würzburg, Germany and 2Institute of Theoretical Biology, Humboldt University, 10115 Berlin and Institute of Molecular Neurobiology, Free University of Berlin, 14195 Berlin, Germany

Email: Nico Blüthgen* - bluethgen@biozentrum.uni-wuerzburg.de; Florian Menzel - menzel@biozentrum.uni-wuerzburg.de; Nils Blüthgen - nils.bluethgen@itb.biologie.hu-berlin.de

* Corresponding author

AbstractBackground: Network analyses of plant-animal interactions hold valuable biological information.They are often used to quantify the degree of specialization between partners, but usually based onqualitative indices such as 'connectance' or number of links. These measures ignore interactionfrequencies or sampling intensity, and strongly depend on network size.

Results: Here we introduce two quantitative indices using interaction frequencies to describe thedegree of specialization, based on information theory. The first measure (d') describes the degreeof interaction specialization at the species level, while the second measure (H2') characterizes thedegree of specialization or partitioning among two parties in the entire network. Both indices aremathematically related and derived from Shannon entropy. The species-level index d' can be usedto analyze variation within networks, while H2' as a network-level index is useful for comparisonsacross different interaction webs. Analyses of two published pollinator networks identifieddifferences and features that have not been detected with previous approaches. For instance, plantsand pollinators within a network differed in their average degree of specialization (weighted meand'), and the correlation between specialization of pollinators and their relative abundance alsodiffered between the webs. Rarefied sampling effort in both networks and null model simulationssuggest that H2' is not affected by network size or sampling intensity.

Conclusion: Quantitative analyses reflect properties of interaction networks more appropriatelythan previous qualitative attempts, and are robust against variation in sampling intensity, networksize and symmetry. These measures will improve our understanding of patterns of specializationwithin and across networks from a broad spectrum of biological interactions.

BackgroundThe degree of specialization of plants or animals has beenstudied and debated extensively, and a continuum fromcomplete specialization to full generalization can befound in various systems [1-6]. In general, two levels ofspecialization measures may be distinguished: first, thecharacterization of focal species and, second, the degree of

specialization of an entire interaction network, represent-ing an assemblage of species and their interaction partners(e.g. food webs, mutualistic networks, predator-prey rela-tionships). When interactions are considered as ecologicalniche, the first level describes the niche breadth of a spe-cies and the second level the degree of niche partitioningacross species. While the species level is more straightfor-

Published: 14 August 2006

BMC Ecology 2006, 6:9 doi:10.1186/1472-6785-6-9

Received: 04 May 2006Accepted: 14 August 2006

This article is available from: http://www.biomedcentral.com/1472-6785/6/9

© 2006 Blüthgen et al; licensee BioMed Central Ltd.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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ward in its biological interpretation, analyses at the net-work level can be useful for comparisons across differenttypes of networks. Such analyses have been performed tocompare plant-pollinator webs versus plant-seed dis-perser webs [4,5], different plant-pollinator networksalong geographic gradients [1,7,8], or food webs of varia-ble size [9,10]. Entire network analyses are also used tostudy patterns on a community level such as coevolution-ary adaptations [3], ecosystem stability or resilience [11-14].

Quantifying specialization at the species levelSpecialization or generalization of interactions are mostcommonly characterized as the number of partners (or'links'), e.g. the number of pollinator species visiting aflowering plant species or the number of food plant fam-ilies a herbivore feeds upon. In this qualitative approach,interactions between a consumer and a resource speciesare only scored in a binary way as 'present' or 'absent',ignoring any distinction between strong interactions andweak or occasional ones. For example, binary representa-tion of interactions do not distinguish a scenario where99% of the individuals of a herbivore species feed on asingle plant species only, but occasionally an individual isfound on another plant, from a different scenario where aherbivore regularly feeds on both food plants. The prob-lem is analogous to the measurement of biodiversityeither as a crude species richness versus as a more elabo-rate diversity index including relative abundances [15].Several approaches have thus been used to directlyinclude variation in interaction frequencies (i.e., theirevenness) in characterizing the diversity of partners, e.g.Simpson's diversity index for pollinators [16,17] orLloyd's index for host specificity [18]. Alternatively, otherstudies indirectly controlled for abundance or samplingintensity using rarefaction methods [13,19]. Correspond-ingly, Bersier and coworkers [20] have suggested to quan-tify the diversity of biomass flows in food webs using aShannon diversity measure. Niche breadth theory pro-vides several additional indices that include some meas-ure of resource frequency or resource use intensity [21],which can be viewed in analogy to 'partner diversity' inthe context of association networks. However, Hurlbert[22] emphasized that not only proportional utilization,but also the proportional availability of each niche shouldbe taken into account. A species that uses all niches in thesame proportion as their availability in the environmentshould be considered more opportunistic than a speciesthat uses rare resources disproportionately more. If varia-tion in resource availability is large, diversity-based meas-ures that ignore this availability may be highly misleading[22,23]. Several niche breadth measures thus combineproportional resource utilization with proportionalresource availability [22-24]. These concepts have beenrarely applied in the context of species interaction net-

works, e.g. plant-pollinator webs where binary data aremore common than quantitative webs.

Quantifying specialization at the community level

The measurement used most commonly to characterizecommunity-wide specialization is the 'connectance' index(C) [1,4,8-10, 25-27]. C is defined as the proportion of theactually observed interactions to all possible interactions.Consider a contingency table showing the associationbetween two parties, with r rows (e.g., plant species) andc columns (e.g., pollinators). Connectance is defined as C= I/(r·c), with I being the total number of non-zero ele-ments in the matrix. Therefore, like the number of part-ners or links (L) described above, C uses only binary infor-mation and ignores interaction strength. C is directly

related to the mean number of links ( ) of plant species

or pollinator species as C = plants/c = poll/r.

This measure, , has also been used to compare networks

[1,3,7,8,28]. Recently, it has been suggested to use instead of C to characterize networks [29]. However, notethat comparisons across networks of different size

(number of species) are problematic, since , unlike C, isnot scaled according to the number of available partners

(see also [2,10]). in a small network may represent alarger proportion of available partners compared to the

same value of in a large network.

Analyses based on binary data – both at the species andthe community level – have obvious shortcomings, sincethey are highly dependent on sampling effort, decisionswhich species to include or not, and the size of investi-gated networks. Several authors thus emphasized the needto move beyond binary representations of interactions toquantitative measures involving some measure of interac-tion strength [4,20,27,29-32]. A way to at least partly over-come these deficiencies is to cut off all rare species or weakinteractions below a frequency threshold [3,9,33,34] or tocontrol for sampling effort in null models[7,8,13,19,25,35]. However, for interaction webs where amore detailed information is available, simplification to

binary data as in C or remains unsatisfactory. Conven-iently, the observed interaction frequency may represent ameaningful surrogate for interaction strength, at least inpollination and seed-dispersal systems as shown byVázquez et al. [30] (see also [16]). Incorporating interac-tion frequency or even a direct measure of interaction

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strength in a network measure of specialization wouldthus provide an important progress frequently called for.

A severe additional problem of connectance is that itslower and upper constraints are not scale-invariant [25],which limits its use for comparisons across networks. Theminimum possible value (Cmin) to maintain at least onelink per species declines in a hyperbolic function with thenumber of interacting species, since Cmin = max(r, c)/(r·c),and an upper limit (Cmax) may be constrained by, or afunction of, total sampling effort. Across networks, Cdecays strongly with network size, which has beendebated in detail in the context of food web analysis[9,10,26,27,36,37]. The strong relationship between Cand network size generates a problem for disentanglingany biologically meaningful effect from this mathemati-cally inherent scale dependence. For instance, networkcomparisons may focus on residual variation in C after anaverage effect of network size has been controlled for[1,4], or C could be rescaled to account for this size effect(see [25,36]). For natural networks of similar size, therange of actual C values is typically very narrow [4], thusother structural forces may be poorly detectable.

The objective of this paper is to develop and discuss spe-cialization measures that are based on frequency data andthus account for sampling intensity, and that overcomethe problem of scale dependence. We then test theseapproaches by evaluating the effect of sampling effort andscale dependence on a published natural pollination net-work, and on randomly generated associations as a nullmodel. We differentiate between species-level measures ofspecialization, useful to investigate variability among spe-cies within a web, and a single network-wide measure thatcan be used for comparisons across networks.

ResultsPatterns in two pollinator networksTwo selected plant-pollinator networks (British meadowsstudied by Memmott [32], Argentinean forests studied byVázquez and Simberloff [33]) differ markedly in theirdegree of specialization when quantitative analyses areapplied. The qualitative network index, connectance, issimilar in both interaction webs (British web: C = 0.15,Argentinean web: C = 0.13). However, frequencies of pol-linator visits are much more evenly distributed in the Brit-ish community than in the Argentinean example. In theBritish web, the interaction between a dipteran speciesand Leontodon hispidus was the most frequent one, repre-senting 6% of the total 2183 interactions observed. In theArgentinean network, visits of Aristotelia chilensis by a col-letid bee species represented 20% of the 5285 interactionsalone. Interactions between the top five plant and top fivepollinator species made up 44% of the interactions in theBritish web, but 74% in the Argentinean web. This differ-

ence in the heterogeneity of interaction frequencies is notevident in measures based on binary information such asnumber of links (L) or connectance (C). In contrast, thedegree of specialization shown by the frequency-basedindex H2' (standardized two-dimensional Shannonentropy, see Methods: Network-level index) is much lower inthe British community (H2' = 0.24) compared to theArgentinean community (H2' = 0.63).

The variation of species-level specialization measures(standardized Kullback-Leibler distance, d') holds valua-ble information for the structural properties of a network(see Methods: Species-level index). The British pollinationweb is dominated by highly generalized pollinators (lowd', both in terms of individuals as well as species), whileputative specialists are represented by very few individualsand species (Fig. 1A). In contrast, most pollinators in theArgentinean web are moderately generalized to special-ized, with the second highest level of specialization foundin the most common species (Fig. 1B). Consequently, theweighted mean degree of specialization is much lower inthe former web (<d'poll> = 0.16) than in the latter (<d'poll>

= 0.54). The relationship between specialization of spe-cies i (d'i) and its interaction frequency (Ai) across the pol-

linator species differs between the two webs. In the Britishweb, d'i and Ai were not correlated significantly (Spear-

man's rs = -0.08, p = 0.46), while a highly positive correla-

tion was found in the Argentinean web (rs = 0.65, p <

0.0001). Note that designation of any specialization indexto a species i that is only represented by a single individualmay be critical. However, significances in the above corre-lations remain unaffected when pollinators with one sin-gle interaction are excluded. From the plants' point ofview, the species in Memmott's web are also more gener-alized in terms of their pollinator spectrum (Fig. 1C) thanthe plants studied by Vázquez and Simberloff (Fig. 1D).The respective weighted means are <d'plants> = 0.27 and

<d'plants> = 0.53. No significant correlation was found

between the plants' frequency and specialization in either

web (both p ≥ 0.16). Interestingly, plants were on averagemore specialized than pollinators in the British web(<d'plants> > <d'poll.>), but not in the Argentinean web. This

distinction is not found when only the weighted meannumber of links (L) are examined, since <Lplants> is much

greater than <Lpoll.> in both networks. The difference in

<L> may be driven by the highly asymmetrical matrixarchitecture in both webs, where the number of pollinatorspecies greatly exceeds the number of plant species. The

unweighted mean is even directly linked to the matrixL

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architecture (i.e., number of rows and columns, r and c)

by a constant (connectance C), since r = c·C and c =

r·C. In contrast, the matrix asymmetry does not affect d'(see also below, Null model patterns).

Simulation of sampling effortIn order to test whether specialization estimates aredependent on sampling and scale effects, we simulated adecreased sampling intensity in both networks using rare-faction (see Methods: Simulation of sampling effort andmatrix architecture). In both networks, H2' is robust andalready very well estimated by a small fraction of the inter-actions sampled (Fig. 2). The coefficient of variance of H2'remains below 5% from about half of the total number ofvisits onwards in the British web and even at one-tenth ofthe total sampling effort of the Argentinean web. The esti-mation of connectance (C) is also relatively stable at leastin the Argentinean web, although it shows a positive trendacross sampling effort in the British web (Fig. 2). Thesefindings suggest that network-wide measures of speciali-

zation, particularly H2', do not necessarily require a verylarge or even complete association matrix, but can also bevery well estimated from a smaller representative subset aslong as there is no systematic sampling bias.

Null model patternsThe degree of specialization can be further characterizedby comparison with a null model. The null model usedhere is that each species has a fixed total number of inter-actions (given by the observed association matrix), butinteractions are assigned randomly. In the above pollina-tor networks, random associations yield a specializationindex H2' that remains close to zero for almost the entirerange of sampling intensity, while connectance (C) showsa positive trend over the total number of interactions (m)(Fig. 2). Therefore, H2' derived from real networks maytypically be clearly distinguished from this null model,while the comparison of C is complicated by scaledependence and the relatively large values yielded by thenull model.

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Patterns within pollinator networksFigure 1Patterns within pollinator networks. Frequency distribution of the species-level specialization index (d') for pollinators and plants from two published networks, one from Britain [32] and one from Argentina [33]. Bars show the number of individ-uals in each category (label '0' defines 0.00 ≤ d' < 0.05, etc.). Bars are separated for different species, and total number of spe-cies in each category is given on top. Arrows indicate cases where bars are invisible due to low numbers of individuals.

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(C) Britain: plants (D) Argentina: plants

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Simulations of artificially generated random associations(see Methods: Simulation of sampling effort and matrix archi-tecture) confirm that the network-level specializationindex H2' is largely unaffected by network size (Fig. 3A),

network architecture (Fig. 3B) or total number of interac-tions (m) for a fixed matrix size (Fig. 3C). For randomassociations as shown here, H2' is usually close to zero.

Connectance values (C) of random matrices show theknown hyperbolic function over the number of associatedspecies (Fig. 3A), changes with matrix asymmetry (Fig.3B) and increase strongly with increasing m (Fig. 3C). Forspecialization measures at the species level, the average

number of links per species ( ) increases strongly withnetwork size, number of available partners, and m (Fig. 3).

While other niche breadth measures may also show somevariation across different network scales (not shown), theweighted mean Kullback-Leibler distance <d'> is poorlyaffected by network size, network asymmetry, andnumber of interactions (Fig. 3). Both H2' and d' may thus

be appropriate for comparisons across matrices of differ-ent scale.

DiscussionProperties of specialization measuresThe suggested indices, d' and H2', quantify the degree ofspecialisation of elements within an interaction networkand of the entire network, respectively. While the numberof links (L) and connectance (C) represent species-leveland community-level measures of interactions based on

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Sampling effect in pollinator networksFigure 2Sampling effect in pollinator networks. Rarefaction of sampling effort in a British and an Argentinean pollination web [32,33]. Two network-level measures of specialization – the frequency-based specialization index (H2') and the 'connectance' index (C) – are shown for networks in which the total number of interactions (m) has been reduced by randomly deleting interactions. Black dots show the effect of sampling effort for the original association matrix, gray dots the effect for a null model, i.e. five networks in which partners were randomly associated (same row and column totals as in the original matrix).

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binary data, respectively, d' and H2' represent correspond-ing measures for frequency-based data. The need toinclude information on interaction strength or interactionfrequency into network analyses has been announced byvarious authors [4,20,27,30,31,38]. Parallel to earlieradvances in diversity measures compared to species rich-ness, quantitative network measures account for the heter-

ogeneity in link strength rather than assigning equalweights to every link. Moreover, we have shown that d'and H2' are largely robust against variation in matrix size,shape, and sampling effort. In several cases, C may bestrongly affected by sampling effort [25,27], while H2'remained largely unchanged in simulations of randomassociations over a range of network sizes, variable net-

Simulated random networksFigure 3Simulated random networks. Behavior of specialization measures in simulated random networks. Each point represents one matrix with random associations, based on specific row and column totals that follow a lognormal distribution. The size of squared matrices in (A) increased from 2 × 2 to 200 × 200. In (B), only the number of rows changed, while the number of col-umns was fixed at 20, rectangular matrices thus increased from 2 × 20 to 200 × 20. In (C), the network size was fixed at 20 × 20. The total number of interactions (m) increased with matrix size in (A), where each species had on average 20 individuals. In (B), m was fixed at 4000, resulting in a reduced interaction density for larger matrices. In (C), m increased from 20 to 4000. The index H2' and connectance C are specialization measures of the whole matrix and thus reciprocal, while the average

number of links ( ), and weighted mean standardized Kullback-Leibler distance (<d'>) are given for all columns (rows give a similar pattern).

Matrix size (number of rows)

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work asymmetries, and number of interactions. This scaleinvariance suggests that both d' and H2' can be useddirectly for comparisons across different networks, whilecomparisons of L and C are more problematic [1,35].

Qualitative methods like the indices suggested here alsoallow a more detailed analysis of interaction patternswithin and across networks. Fruitful areas include com-parisons of networks across different interaction types [4],biogeographical gradients [1], biodiversity and land usegradients [13], robustness of networks against extinctionrisks [39], asymmetries between plants and animals [38],and relationships between specialisation and abundance[35]. While a comparison of the average number of part-ners between plants versus animals is solely dependent onthe matrix architecture (i.e., the number of rows r versus

columns c, since plants = c·C and poll = r·C), this limita-

tion does not apply to d'. In the two selected pollinatorwebs, plants are either similarly or more specialised thanpollinators in regard to weighted mean d'. This allows anscale-independent evaluation of asymmetries in thedegree of specialization between partners (see also [38]).Moreover, Vázquez and Aizen [35] noted that the numberof links of a species (Li) is strongly positively correlated

with its overall frequency (Ai) in five pollination networks

including the datasets analyzed above. They argued thatthis apparent higher generalization of common plantsand common pollinators may be largely explained by nullmodels, calling for an improved measurement of special-ization. Our results for the correlation between d'i and Ai

in two pollinator webs suggest that the relationshipbetween specialization and abundance may be more vari-able, and even positive as in the Argentinean network.

CaveatsSome problems apply to any measure of network analysesincluding the proposed indices. Measures of specializa-tion mostly ignore phylogenetic relationships or ecologi-cal similarity within an association matrix. For example, aplant species that is pollinated by multiple moth speciesmay be unsuitably regarded as more generalized than aplant pollinated by few insect species comprising severaldifferent orders [40]. In addition, the fact that herbivoresare commonly specialized on host plant families ratherthan species may skew network patterns if not carefullyaccounted for. A first approach to investigate such effectsmay be to compare the level of specialization after a step-wise reduction of the matrix by pooling species to highertaxonomic units, such as genera, families, and orders. Forknown phylogenies, more advanced techniques for analy-ses with a particular evolutionary focus are available [41-

43]. Another deficiency may be that species or their part-ners are all given the same individual 'weight' in the anal-yses, whether they may be small bees or large bats visitinga small herb with little nectar or a mass flowering tree.Null models as in the calculation for both C and H2.'imply that all individuals can be shifted around betweenresources in the same way, irrespective of their size ornon-fitting parameters. The role of 'forbidden links' asconstraints to network analyses has been discussed else-where [44,45]. Similarly, calculations of d' or other nichebreadth measures are based on the implicit assumptionthat each species adjusts its interactions according to theavailability of partners (niches), irrespective of morpho-logical or behavioral constraints. Moreover, if data are col-lected from a large heterogeneous habitat or over aprolonged time period, calculations of the degree of spe-cialization may be severely constrained by the spatiotem-poral overlap or non-overlap between partners for otherreasons than resource preferences, e.g. when not all spe-cies are able to reach all sites in the same way, or whensome resources and consumers have asynchronous phe-nologies. Consequently, network analyses as suggestedhere will be most useful to study resource-consumer par-titioning within a short time frame and limited spatialscale.

For both indices d' and H2', we proposed above to use thetotal number of interactions for each species as a measureof partner availability (qj) and as constraint for standardi-zation (fixed row and column totals). It may be debatedwhether independent measures of plant and animal abun-dances could be more appropriate than using interactionfrequency data as such. However, despite the fact that suchabundance data barely exist for most networks, note thatthe actual number of interactions often more suitablyreflects resource availability and consumer activity thanan independent measure of species abundance. Forinstance, a flower of one species may have a much highernectar production than another and consequently receivea higher number of visitors, while the local abundance ofthe plant species does not reflect such differences inresource quality and/or quantity. Both d' and H2' thusfocus on the actual partitioning between the interactingspecies. In studies where detailed knowledge or theoreti-cal assumptions about resources (availability and quality)or consumers (activity density and consumption rate) areavailable or under experimental control, such data may beincorporated into the analysis (defining qj and con-straints) instead of interaction frequencies. The constraintof fixed row and column totals has been debated else-where in the context of species co-occurrence patterns,where it was found to be most appropriate in null modelcomparisons, although critics have argued earlier thatthese marginals themselves may already reflect competi-tive interactions ([46] and references therein). Any

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approach to compare networks based on fixed marginalsfor standardization will fail to detect potentially meaning-ful patterns displayed by these architectural features,namely the number of resource and consumer species andthe heterogeneity of total interaction frequencies. Thisnetwork architecture may already be shaped by past com-petitive interactions or indicate fundamental constraints,a largely unexplored hypothesis that merits additionalinvestigations.

It should also be emphasized that analyses of frequencydata may be susceptible for pseudoreplication of repeatedassociations of the same individuals or close associationsderived from a single dispersal event (e.g. a social insectcolony, aggregating individuals, multiple offspring from asingle egg cluster, or monospecific plant clusters). Thesemay lead to an overestimation of specialization. To bemore meaningful on a population level, frequency analy-ses should thus be based on spatially independent associ-ation replicates. Note that all species-wise specializationmeasures such as d' are sensitive to the behavior of theother species. Any systematic sampling bias (e.g. a taxo-nomic focus within a guild) will therefore affect the con-clusions of comparisons within or across networks.

ConclusionIn accordance with previous calls [4,20,27,30,31,38], wesuggest that the explicit inclusion of frequency datareflects an important step forward in network analyses, astoo many assumptions are implicit in any measure basedon binary representation. Most notably, connectance and'number of partners' imply an equal availability of allpartners – an unlikely scenario. Qualitative indices are notrobust against sampling effort. On the contrary, the pro-posed quantitative measures based on interaction fre-quencies explicitly account for this source of variation.Our study suggests that d' and H2' represent scale-inde-pendent and meaningful indices to characterize speciali-zation on the level of single species and the entirenetwork, respectively. These novel indices allow us toinvestigate patterns within and across networks that havenot been detected with qualitative measures such as corre-lations with species frequencies, network size and asym-metries in specialization between partners. Recently,Bascompte et al. [38] showed that the incorporation offrequency data may unveil pervasive asymmetries withinnetworks. Particularly since Vázquez et al. [30] demon-strated that interaction frequencies in plant-pollinatorand plant-seed disperser systems often correlate with themagnitude of mutualistic services for the plant (althoughvariation in pollinator effectiveness can be important, see[47]), an increased collection of frequency data andappropriate quantitative analyses would greatly benefitfuture network studies.

MethodsSpecies-level indexAs species-level measure of 'partner diversity', we proposethe Kullback-Leibler distance (or Kullback-Leibler diver-gence, relative entropy) in a standardized form (d'). Com-ing from information theory, this index quantifies thedifference between two probability distributions [48].While the standardized Hurlbert's and Smith's measure ofniche breadth could be used alternatively [21,22,24], d'has some advantages in the context of networks. While allthree indices regard an exclusive pairing between two spe-cies as high degree of specialization as long as interactionsbetween the two partners are infrequent, Hurlbert's andSmith's indices show a undesired trend towards full gen-eralization when the number of interactions between thetwo partners increase, although this should be considereda stronger indication of specialization (see below, Proper-ties of alternative niche breadth measures). The interactionbetween two parties is commonly displayed in a r × c con-tingency table, with r rows representing one party such asflowering plant species, and c columns representing theother party such as pollinator species. In each cell, the fre-quency of interaction between plant species i and pollina-tor species j (or another useful measure of interactionstrength) is given as aij, (Table 1).

Instead of frequencies (aij), each interaction can beassigned a proportion of the total (m) as

, where .

Let p'ij be the proportion of the number of interactions(aij) in relation to the respective row total (Ai), and qj theproportion of all interactions by partner j in relation tothe total number of interactions (m). Thus,

, , , and .

To quantify the specialization of a species i, the followingindex di is suggested. This di is related to Shannon diver-sity, similar to an index recently suggested to characterizebiomass flow diversity in food webs [20]. However, anappropriate index in this context should not only considerthe diversity of partners, but also their respective availabil-ity (see [22]). Consequently, the following index com-pares the distribution of the interactions with each partner(p'j) to the overall partner availability (qj). The Kullback-Leibler distance for species i is denoted as

p a mij ij= / pijj

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which can be normalized as

The theoretical maximum is given by dmax = ln (m/Ai), andthe theoretical minimum (dmin) is zero for the special casewhere all p'ij = qj. However, a realistic dmin may be con-strained at some value above zero given that p'ij and qj arecalculated from discrete integer values (aij). To take thisinto account, dmin is more suitably computed algorithmi-cally as in a program available from the authors andonline [49], providing all d' for a given matrix. This stand-ardized Kullback-Leibler distance (d') ranges from 0 forthe most generalized to 1.0 for the most specialized case.Thus, d' can be interpreted as deviation of the actual inter-action frequencies from a null model which assumes thatall partners are used in proportion to their availability. Anaverage degree of specialization among the species of aparty can be presented as a weighted mean of the stand-ardized index, e.g. <d'i> for pollinators as

While <d'i> usually differs from <d'j>, the weighted meansof the non-standardized Kullback-Leibler distances are thesame for both parties, hence <di> = <dj>.

Network-level indexThe following network-wide measure is based on thebipartite representation of a two mode network of interac-tions such as plant-animal or other resource-consumerinteractions where members of each party interact withmembers of the other party but not among themselves(unlike many food webs). The two-dimensional Shannonentropy (termed H2 in order to avoid confusion with thecommon one-dimensional H) is obtained as

H2 decreases with higher specialization. This measure isclosely related to the weighted mean of the non-standard-ized Kullback-Leibler distance of all species, since

<di> = <dj> = H2max - H2

(see below, Relationship between di and H2). H2 can bestandardized between 0 and 1.0 for extreme specializationversus extreme generalization, respectively, when its min-imum and maximum values (H2min and H2max) areknown. H2min and H2max can be calculated for given con-straints. The constraints used here are the maintenance ofthe total number of interactions of each species, thus allrow and column totals, Ai and Aj, being fixed (see also[46]). Alternative constraints may be defined dependingon the knowledge of the system studied.

H2 reaches its theoretical maximum where each pij equalsits expected value from a random interaction matrix(qi·qj), such that

d pp

qi ijij

jj

c= ′ ⋅

′⎛

⎝⎜⎜

⎞

⎠⎟⎟

=∑ ln ,

1

′ =−

−d

d d

d dii min

max min.

⟨ ′⟩ = ′ ⋅( ) = ′ ⋅( )= =∑ ∑d

md A d qi i i

i

r

i ii

r1

1 1

H p pij ijj

c

i

r

211

= − ⋅( )==∑∑ ln .

Table 1: Elements in a species association matrix. Interaction frequencies (aij) between c animal and r plant species and their respective totals (rows:Ai, columns: Aj, total elements: m).

Animal sp.1 sp. 2 ... sp. c Total

Plant sp. 1 a11 a12 ... a1c

sp. 2 a21 a22 ... a2c

... ... ... ... ... ...sp. r ar1 ar2 ... arc

Total ...

A ai jj

c

==

= ∑1 11

A ai jj

c

==

= ∑2 21

A ai r rjj

c

==

= ∑1

A aj ii

r

==

= ∑1 11

A aj ii

r

==

= ∑2 21

A aj c ici

r

==

= ∑1

m aijj

c

i

r=

==∑∑

11

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while its theoretical minimum (H2min) may be close tozero depending on the matrix architecture. Like for dminabove, H2max and H2min are constrained by the fact thatthey are derived from integer values. A program imple-menting a heuristic solution to obtain H2max and H2min,and to perform the entire analysis is available from theauthors or online [49].

The degree of specialization is obtained as a standardizedentropy on a scale between H2min and H2max as

Consequently, H2' ranges between 0 and 1.0 for extremegeneralization and specialization, respectively.

Comparison with random associationsH2 can be tested against a null model of random associa-tions (H2ran). A number of random permutations of thematrix can be performed using a r × c randomization algo-rithm (also available at [49]). The probability (p-value)that the observed H2 is more specialized than expected byrandom associations is simply given as the proportion ofvalues obtained for H2ran that are equal or larger than H2,a common procedure in randomization statistics [25,50].H2ran is usually only slightly larger than H2min.Previously,permutations of r × c contingency tables often used a dif-ferent test statistics instead of H2 [25,51,52]:

The relationship between T and H2 is described by a con-stant, the total number of interactions (m), as T = m·ln m- m·H2. Consequently, both methods yield exactly thesame p-values.

Relationship between di and H2In the following we derive the relationship between theindividual levels of specialization (di) and the communitylevel (H2). The non-standardized Kullback-Leibler dis-tance for row i can be rewritten as

because .

The weighted mean of di for all i rows (each row weightedby qi) yields

since and

While the first summand in the final equation for <di>equals -H2, the remaining two summands correspond tothe maximum entropy H2max, because

Therefore,

<di> = H2max -H2.

The same calculation applies for <dj>, thus <di> = <dj>.Consequently, the degree of specialization of the entirenetwork (corresponding to the deviation of the network-wide entropy from its maximum value) equals theweighted sum of the specialization of its elements (spe-cies).

Properties of alternative niche breadth measuresThe standardized Hurlbert's (B') and Smith's (FT) meas-ure can be applied widely for niche breadth analysis[21,22,24]. In this context, the Kullback-Leibler distance(d) can be viewed as a modified Shannon-Wiener measureof niche breadth that accounts for niche availabilities.Like the Kullback-Leibler distance, both B' and FT com-pare the proportional distribution of individuals (p) tothe proportional resource availability (q) (here: partneravailability). For a certain species i, the two measures arein our notation:

Each p'ij is the proportion of the number of interactions inrelation to the respective row total, and qj is the propor-

H q q q qi j i jj

c

i

r

211

max ln ,= − ⋅( )==∑∑

′ =−

−H

H H

H H22 2

2 2

max

max min.

T a aij ijj

c

i

r= − ( )

==∑∑ .ln .

11

d pp

q

p

q

p

q qi ijij

jj

c ij

i

ij

i jj

c= ′ ⋅

′⎛

⎝⎜⎜

⎞

⎠⎟⎟ = ⋅

⋅⎛

⎝⎜⎜

⎞

⎠⎟⎟

= =∑ ln ln

1 1∑∑ ,

pa

A

A

mp qij

ij

i

iij i= ⋅ = ′ .

⟨ ⟩ = ⋅( ) = ⋅⋅

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= ⋅

= ==∑ ∑∑d d q p

p

q q

p p

i i ii

r

ijij

i jj

c

i

r

ij

1 11

ln

ln iijj

c

i

r

ij ij

c

i

r

ij jj

c

i

rp q p q( ) − ⋅( ) − ⋅( )

=

== == ==∑∑ ∑∑ ∑∑

11 11 11

ln ln

pp p q q q qij ijj

c

i ii

r

j jj

c

i

r⋅( ) − ⋅( ) − ⋅( )

= = ==∑ ∑ ∑∑ ln ln ln ,

1 1 11

q pi ijj

c=

=∑

1

q pj iji

r=

=∑

1

H q q q q q q q q qi j i jj

c

i

r

i j ij

c

i

r

i j211 11

max ln ln= − ⋅( ) = − ⋅( ) −== ==∑∑ ∑∑ ⋅⋅( )

= − ⋅( ) − ⋅( )==

= =

∑∑

∑ ∑

ln

ln ln .

q

q q q q

jj

c

i

r

i ii

r

j jj

c

11

1 1

′ =−

−=

( )= ⋅(

=∑

BB q

qB

p q

FT p qii

i

ij jj

c i ij jmin

min, ,

11

2

1

with and ))=∑j

c

1

.

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tion of all interactions by partner j in relation to the totalnumber of interactions. Thus,

Both the standardized Hurlbert's (B') and Smith's (FT)measure range between 0 for the most specialized case to1.0 for extreme generalization (broadest niche). In thecontext of niche breadth, it has been shown that the Shan-non-Wiener measure is most sensitive, while Hurlbert'sand particularly Smith's measure are less sensitive for theselection of rare resources [21] (see also [20]).

For the application in network analyses, however, both B'and FT may show some undesired properties. Generally,B', FT and d' are reasonably well correlated with eachother across the species within a network (e.g., rs = -0.49between d' and B', and rs = -0.36 between d' and FT for the90 pollinators in the network of Vázquez and Simberloff[33], both p < 0.001). However, differences with d' aresubstantial when a highly specialized species interactslargely exclusively with a specialized partner, e.g. a spe-cialized pollinator with a plant that is almost exclusivelypollinated by this one. Imagine a scenario where oneexclusive interaction occurs between a plant species and apollinator species in a 3 × 3 matrix (Table 2). If the inter-action between pollinator sp. 3 and plant sp. 3 is onlyinfrequent (e.g. a33 = 1), all indices show a high degree ofspecialization (d' = 1.0, B' = 0, FT = 0.14) for both part-ners. However, as the number of exclusive interactions(a33) increases, the values for both B' and FT of pollinatorsp. 3 and plant sp. 3 show a highly undesired changetowards generalization, although a higher a33 is intuitivelyconsidered as extreme specialization (e.g., for a33 = 50 thevalues for pollinator sp. 3 are B' = 0.31 and FT = 0.70),while only d' remains unaffected (d' = 1.0). FT is alwayslarger than zero, and B' becomes larger than zero when thespecialists interact more frequently than one of the otherpartners, thus when qj > min(q1, q2, ... qc). Both FT and B'approach a value of 1.0 (maximum generalization) forvery large a33. This undesired effect of FT and B' is notrestricted to completely exclusive interactions betweentwo partners.

Simulation of sampling effort and matrix architectureTwo published plant-pollinator networks were selected toinvestigate the behavior of different specialization meas-ures [32,33]. Both articles use their observed interactionmatrices as a model to discuss network properties basedon the number of links per pollinator or plant species,allowing a comparison of conclusions drawn. Both net-works may be compared as they comprise relatively largedatasets from temperate ecosystems, reporting interactionfrequencies between plants and their floral visitors: theBritish meadow community studied by Memmott [32]involved 79 pollinator and 25 plant species (2183 polli-nator visits observed), the forests in Argentina studied byVázquez and Simberloff [33] involved 90 pollinator and14 plant species (5285 visits). The datasets can beobtained from the Interaction Web Database [53]. Wesimulated a decreased sampling intensity in both net-works using a rarefaction method in order to investigatehow sampling effort affects the estimation of specializa-tion indices. Real association matrices were reduced byrandomly extracting interactions, e.g. from the total of m= 2183 visits in Memmott's web down to m = 5 visits (insteps of five, repeated ten times for each m).

In order to compare the null model characteristics of thespecialization measures, we simulated artificial matriceswith randomly associated partners and plotted the indicesagainst an increasing number of partners and/or totalnumber of interactions. We assumed that the total fre-quency of participating species approximates a lognormaldistribution, which is typical for biological communities[21,22,24]. All row and column totals were randomly gen-erated from a lognormal distribution (μ = 50, ∑= 1) thatwas scaled to the desired total number of interactions. Tendifferent combinations of row and column totals wereobtained for each matrix size and taken as template to ran-domly associate the partners five times, thus each matrixsize was represented by 50 random associations.

Authors' contributionsNB1 conceived of the study and all authors (NB1, FM,NB2) were involved in designing the methods, analyses,interpretation and drafting the manuscript.

AcknowledgementsWe thank Diego Vázquez, Pedro Jordano, Thomas Hovestadt, and Michel Loreau for helpful comments and valuable discussion on earlier versions of this manuscript and the Interaction Web Database [53] for providing the datasets used here.

References1. Olesen JM, Jordano P: Geographic patterns in plant-pollinator

mutualistic networks. Ecology 2002, 83:2416-2424.2. Novotny V, Basset Y: Host specificity of insect herbivores in

tropical forests. Proc R Soc London Ser B 2005, 272:1083-1090.3. Waser NM, Chittka L, Price MV, Williams NM, Ollerton J: General-

ization in pollination systems, and why it matters. Ecology1996, 77:1043-1060.

′ = = = ( )= =∑ ∑p q q qijj

c

jj

c

j1 1

1 1, , min .min and

Table 2: Association matrix example. Fictive association matrix between three pollinator species and three plant species. Numbers in each cell are counts of interaction frequencies.

Pollinator sp. 1 Pollinator sp. 2 Pollinator sp. 3Plant sp. 1 21 5 0Plant sp. 2 23 4 0Plant sp. 3 0 0 a33

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4. Jordano P: Patterns of mutualistic interactions in pollinationand seed dispersal: connectance, dependence asymmetries,and coevolution. Am Nat 1987, 129:657-677.

5. Bascompte J, Jordano P, Melian CJ, Olesen JM: The nested assem-bly of plant-animal mutualistic networks. Proc Natl Acad Sci USA2003, 100:9383-9387.

6. Waser NM, Ollerton J, Eds: Plant-pollinator interactions: fromspecialization to generalization. Chicago: University of ChicagoPress; 2006.

7. Ollerton J, Cranmer L: Latitudinal trends in plant-pollinatorinteractions: are tropical plants more specialised? Oikos 2002,98:340-350.

8. Devoto M, Medan D, Montaldo NH: Patterns of interactionbetween plants and pollinators along an environmental gra-dient. Oikos 2005, 109:461-472.

9. Winemiller KO: Must connectance decrease with species rich-ness? Am Nat 1989, 134:960-968.

10. Martinez ND: Constant connectance in community food webs.Am Nat 1992, 139:1208-1218.

11. May RM: Will a large complex system be stable? Nature 1972,238:413-414.

12. Rejmánek M, Starý P: Connectance in real biotic communitiesand critical values for stability of model ecosystems. Nature1979, 280:311-313.

13. Vázquez DP, Simberloff D: Ecological specialization and suscep-tibility to disturbance: conjectures and refutations. Am Nat2002, 159:606-623.

14. Dunne JA, Williams RJ, Martinez ND: Network structure and bio-diversity loss in food webs: robustness increases with con-nectance. Ecol Lett 2002, 5:558-567.

15. Magurran AE: Ecological diversity and its measurement Princeton: Princ-eton University Press; 1988.

16. Sahli HF, Conner JK: Characterizing ecological generalizationin plant-pollination systems. Oecologia 2006, 148:365-372.

17. Parrish JAD, Bazzaz FA: Difference in pollination niche relation-ships in early and late successional plant communities. Ecol-ogy 1979, 60:597-610.

18. Basset Y: Diversity and abundance of insect herbivores forag-ing on seedlings in a rainforest in Guyana. Ecol Entomol 1999,24:245-259.

19. Herrera CM: Plant generalization on pollinators: species prop-erty or local phenomenon? Am J Bot 2005, 92:13-20.

20. Bersier LF, Banasek-Richter C, Cattin MF: Quantitative descrip-tors of food-web matrices. Ecology 2002, 83:2394-2407.

21. Krebs CJ: Ecological Methodology Menlo Park: Benjamin Cummings;1999.

22. Hurlbert SH: Measurement of niche overlap and some rela-tives. Ecology 1978, 59:67-77.

23. Feinsinger P, Spears EE, Poole RW: A simple measure of nichebreadth. Ecology 1981, 61:27-32.

24. Smith EP: Niche breadth, resource availability, and inference.Ecology 1982, 63:1675-1681.

25. Fonseca CR, Ganade G: Asymmetries, compartments and nullinteractions in an Amazonian ant-plant community. J AnimEcol 1996, 65:339-347.

26. Kenny D, Loehle C: Are food webs randomly connected? Ecol-ogy 1991, 72:1794-1799.

27. Goldwasser L, Roughgarden J: Sampling effects and the estima-tion of food-web properties. Ecology 1997, 78:41-54.

28. Vázquez DP, Aizen MA: Asymmetric specialization: a pervasivefeature of plant-pollinator interactions. Ecology 2004,85:1251-1257.

29. Kay KM, Schemske DW: Geographic patterns in plant-pollina-tor mutualistic networks: comment. Ecology 2004, 85:875-878.

30. Vázquez DP, Morris WF, Jordano P: Interaction frequency as asurrogate for the total effect of animal mutualists on plants.Ecol Lett 2005, 8:1088-1094.

31. Borer ET, Anderson K, Blanchette CA, Broitman B, Cooper SD, Halp-ern BS, Seabloom EW, Shurin JB: Topological approaches to foodweb analyses: a few modifications may improve our insights.Oikos 2002, 99:397-401.

32. Memmott J: The structure of a plant-pollinator food web. EcolLett 1999, 2:276-280.

33. Vázquez DP, Simberloff D: Changes in interaction biodiversityinduced by an introduced ungulate. Ecol Lett 2003, 6:1077-1083.

34. Dicks LV, Corbet SA, Pywell RF: Compartmentalization in plant-insect flower visitor webs. J Anim Ecol 2002, 71:32-43.

35. Vázquez DP, Aizen MA: Null model analyzes of specialization inplant-pollinator interactions. Ecology 2003, 84:2493-2501.

36. Auerbach MJ: Stability, probability, and the topology of foodwebs. In Ecological communities: conceptual issues and the evidenceEdited by: Strong DR, Simberloff D, Abele LG, Thistle AB. Princeton:Princeton University Press; 1984:413-436.

37. Gotelli NJ, Graves GR: Null models in ecology Washington: SmithsonianInstitution; 1996.

38. Bascompte J, Jordano P, Olesen JM: Asymmetric coevolutionarynetworks facilitate biodiversity maintenance. Science 2006,312:431-433.

39. Memmott J, Waser NM, Price MV: Tolerance of pollination net-works to species extinctions. Proc R Soc London Ser B 2004,271:2605-2611.

40. Johnson SD, Steiner KE: Generalization versus specialization inplant pollination systems. Trends Ecol Evol 2000, 15:140-143.

41. Symons FB, Beccaloni GW: Phylogenetic indices for measuringthe diet breadths of phytophagous insects. Oecologia 1999,119:427-434.

42. Webb CO, Ackerly DD, McPeek MA, Donoghue MJ: Phylogeniesand community ecology. Annu Rev Ecol Syst 2002, 33:475-505.

43. Novotny V, Basset Y, Miller SE, Weiblen GD, Bremer B, Cizek L,Drozd P: Low host specificity of herbivorous insects in a trop-ical forest. Nature 2002, 416:841-844.

44. Jordano P, Bascompte J, Olesen JM: Invariant properties in coev-olutionary networks of plant-animal interactions. Ecol Lett2003, 6:69-81.

45. Vázquez DP: Degree distribution in plant-animal mutualisticnetworks: forbidden links or random interactions? Oikos2005, 108:421-426.

46. Gotelli NJ: Null model analysis of species co-occurrence pat-terns. Ecology 2000, 81:2606-2621.

47. Fenster CB, Armbruster WS, Wilson P, Dudash MR, Thomson JD:Pollination syndromes and floral specialization. Annu Rev EcolEvol Syst 2004, 35:375-403.

48. Kullback S, Leibler RA: On information and sufficiency. Ann MathStat 1951, 22:79-86.

49. Montecarlo statistics on RxC matrices [http://itb.biologie.hu-berlin.de/~nils/stat/]

50. Manly B: Randomization bootstrap and Monte Carlo methods in biologyLondon: Chapman and Hall; 1997.

51. Blüthgen N, Verhaagh M, Goitía W, Blüthgen N: Ant nests in tankbromeliads – an example of non-specific interaction. InsectSoc 2000, 47:313-316.

52. Patefield WM: An efficient method of generating random RxCtables with given row and column totals. Appl Stat 1981,30:91-97.

53. Interaction web database [http://www.nceas.ucsb.edu/interactionweb/]

Page 12 of 12(page number not for citation purposes)