Common Organizing Mechanisms in Ecological and Socio-economic Networks
aa r X i v : . [ phy s i c s . s o c - ph ] O c t Common Organizing Mechanisms in Ecological andSocio-economic Networks
Serguei Saavedra , , , , Felix Reed-Tsochas , and Brian Uzzi , Kellogg School of Management, Management and Organizations Department,Northwestern University, Evanston, Illinois, USA, 60208 Northwestern Institute on Complex Systems, Northwestern University, Evanston,Illinois, USA, 60208 CABDyN Complexity Centre, Oxford University, OX1 1HP, UK Oxford University Centre for Corporate Reputation, Said Business School, Oxford,OX1 1HP, UK James Martin Institute, Said Business School, Oxford University, UK, OX1 1HPE-mail: [email protected],[email protected], [email protected]
Abstract.
Previous work has shown that species interacting in an ecosystem andactors transacting in an economic context may have notable similarities in behavior.However, the specific mechanism that may underlie similarities in nature and humansystems has not been analyzed. Building on stochastic food-web models, we proposea parsimonious bipartite-cooperation model that reproduces the key features ofmutualistic networks - degree distribution, nestedness and modularity – for bothecological networks and socio-economic networks. Our analysis uses two diversenetworks. Mutually-beneficial interactions between plants and their pollinators, andcooperative economic exchanges between designers and their contractors. We findthat these mutualistic networks share a key hierarchical ordering of their members,along with an exponential constraint in the number and type of partners they cancooperate with. We use our model to show that slight changes in the interactionconstraints can produce either extremely nested or random structures, revealing thatthese constraints play a key role in the evolution of mutualistic networks. This couldalso encourage a new systematic approach to study the functional and structuralproperties of networks. The surprising correspondence across mutualistic networkssuggests their broadly representativeness and their potential role in the productiveorganization of exchange systems, both ecological and social.
1. Introduction
The analogy between ecological and economic systems is not new. Biologists havealways being intrigued about the economic aspects of nature [1, 2], and economistsand sociologists have taken insight from biological systems to shed new light onthe factors shaping socio-economic systems. For instance, researchers have adaptedbiological models that focus on constructs such as niche width, resource partitioning, orspecialization and generalization to explain the birth and death rates of organizationalpopulations [3, 4]. More recently, multidisciplinary approaches have led to the discoveryof significant structural similarities across different network domains, including biologicaland socio-economic networks [5, 6, 7]. This has awakened an even more spirited searchfor common structural properties between ecological and economic networks [8, 9], andpointed to a greater need for work on mutually-beneficial interactions across realms.Cooperation [10, 11, 12] is a central concept in biological and social studies,and although the evolution of cooperation was initially modeled for homogeneouspopulations, subsequent work has also included spatial effects [13]. The recentdevelopment of simulation models for evolutionary games on graphs [14] andcollaborative social networks [15] provides a starting point for addressing the questionof how cooperative structures are assembled. In ecology, contemporary research onmutualistic networks provides us with an increasingly detailed picture of the complex setof cooperative interactions between different species in an ecosystem, and demonstratesthat purely local interactions can generate highly structured macroscopic patterns ofmutually beneficial exchanges [16].However, despite the increasing interest in cooperative systems, we currently lackmodels that allow us to connect the cooperative behavior at the level of individualswith emergent global network properties. Furthermore, although cooperation appearsas distinctive characteristic at different levels of organization ranging from groups ofanimals to human society [17], it has been difficult to find empirical evidence showingsimilar patterns of cooperation shared across these levels. In previous work [18], we haveproposed a bipartite-cooperation model (BC model) that can replicate key propertiesof mutualistic networks. To test the BC model across ecological and socio-economicnetworks, we have used ten large pollination datasets that have been compiled inthe literature, and a unique and extensive, economy -wide dataset of designers andcontractors engaged in joint production in the New York City garment industry. In thepresent paper, we attempt to go beyond showing associations in the assembling principlesof these ecological and socio-economic networks to show the effects of different organizingmechanisms on the hierarchical arrangement of these networks. In Sections 2 and 3 wediscuss some of the main features of mutually-beneficial interactions in ecological andsocio-economic networks respectively. Sections 4, 5 and 6 describe the BC model, theempirical data and the validation of the BC model respectively. Section 7 introduces afurther justification of the BC model using empirical data. In Section 8 we use the BCmodel to study the effects of different organizing mechanisms and interaction constraintson the hierarchical arrangement of empirical networks. Finally, Section 9 summarizesour conclusions and overall findings.
2. Ecological networks
In ecology, mutualistic networks are formed by the mutually-beneficial interactionsbetween populations of different species (e.g. P for plants and A for animals) [16].Species in class P offer rewards with certain characteristics to attract species in classA. These individual attributes, determined by their own reward traits, may also haveevolved to reduce exploitation and favor mutualism [19]. Species in class A foragingfor resources can benefit from the rewards offered by a given species in class P if therespective foraging traits (e.g. efficiency, morphology, behavior) and reward traits(e.g. quantity, quality, availability) are complementary [20, 21]. External factorssuch as the environmental context (e.g. population density, geographic variation)attenuate or amplify the value of reward and foraging traits, and impact the numberof potential partners that a given species cooperates with [22, 23, 24]. Furthermore,Rezende et al.[25] have shown that mutualistic networks exhibit hierarchical constraintsintroduced by phylogenetic relationships between species in the same class, which impactmutualistic interaction patterns by favoring ecological similarity.Recent work has found key structural properties in mutualistic networks. Mutually-beneficial interactions between species exhibit broad-scale distributions and a significantpresence of asymmetric interactions (i.e. links connecting high-degree to low-degreenodes), which can be the result of mechanisms such as aging, forbidden interactions,capacity constraints, etc. [20, 26, 27]. Mutualistic networks also display nested andmodular structures that differ from appropriate random assemblages, which play acrucial role in structural robustness and function of these networks [28, 23]. In addition,studies have found that the relationship between the number of species S and mutually-beneficial interactions L follows a power-law given by L = S α , where α = 1 .
13 [28].This suggests that mutualistic networks might display a universal mode of organizationthat appears to enhance the ability of species to respond to environmental changes andcompetition pressures [16, 29, 30].
3. Socio-economic networks
Exchanges and beneficial relationships between members in social and economic contextsare also known to give rise to large-scale cooperative networks through which resourcesand information can flow [31, 32, 33, 34]. In an organizational context, firms arecharacterized by a set of reward or organizational traits (e.g. firm size, competitive nichespace, brand positioning), which are modulated by hierarachical barriers generated bydifferences in status, which limit the number and range of potential partners [4, 3, 35, 36].Hence, cooperation between two different classes of firms (e.g designers and contractorsin a manufacturing industry) can also be subject to structural constraints equivalentto those found in ecological networks. These constraints depend on the traits of firmsand the complementarity between traits of potential partners, as well as hierarchicalrelationships between firms in the same class. Note that the values associated withreward traits and interaction barriers between firms are not absolute, but modulated bythe specific market context that the firms operate in [4, 35].Interestingly, in common with mutualistic networks, studies have found that thenumber of collaborative partners in different socio-economic networks follows a broad-scale distribution characterized by asymmetric interactions, which not only preventthe network from collapsing, but also enhance the efficiency of the network [34, 37].Previously, we found that although the total number of firms F in socio-economicnetwork varies from year to year, the relation between the total number of firms and links L also follows a constant relation [18]. Figure 1 shows that this relation is defined by apower-law L = F α with α = 1 .
22. In addition, structurally cohesive social networks doenhance a hierarchical nesting of groups [38]. These results suggest that mutualisticnetworks and cooperative socio-economic networks might have both structural andorganizing mechanism in common.
4. The bipartite-cooperation model
In theoretical ecology, stochastic models incorporating simple interaction rules haverecently proved remarkably successful at reproducing overall structural properties ofreal food webs [39, 40]. Food webs are formed by interactions between predatorsand preys that reflect all possible flows of energy and biomass between species inan ecosystem. It has been shown that these models need to satisfy only two basicconditions about the distribution of niche values and feeding ranges to reproduce manyaspects of the complex network of predator-prey interactions, using species richness S and connectance L/S as the only input parameters [41]. These two basic conditionsare: (1) an ordered set of species’ niche values; (2) an exponentially decaying probabilityof preying on species with lower niche values. Note that niche values can be betterexplained by a hierarchical ordering of body-sizes [42]. Recently, more detailed modelshave incorporated higher levels of constraints in order to reproduce the actual linksobserved in real food webs [43, 44, 45]. Hence, the notable success in modeling thecomplex set of predator-prey interactions between species has suggested that simplemodels may equally well account for the structure of networks generated by cooperativepartner-partner interactions between two distinct classes of species, as in the case ofplant-animal mutualistic networks [46, 47].Here, building on food web models [39, 40], we developed the BC model that applieswhen members in a network can be divided into two distinct classes (e.g. class P andclass A ). Hence, the inputs for the BC model are the size of class A , the size of class P and the total number of links L , all of which are given directly by the empirical data.The BC model consists of a specialization and an interaction mechanism, which aremotivated by the fact that specialists (members with low number of partners) will faceless competition and fewer interaction barriers by cooperating with generalists (memberswith high number of partners) [30].(i) Specialization . The specialization rule determines how many partners l p eachmember p ∈ P will cooperate with, and is calculated following the nested-hierarchymodel [40]. The number of partners l p is given by l p = 1 + Round ( ( L −| P | ) t Rp λ p ) P j t Rj λ j ),where | · | denotes set cardinality and Round ( · ) is the nearest integer function.Here, t Rp is the reward trait associated with member p , uniformly drawn from[0 , λ p , randomly drawn froman exponential distribution (described below), that accounts for effects such asgeographic variation and population diversity. Reward traits t Rp are the result ofa hierarchical process, which in the BC model corresponds to the generation of anordered sequence in trait space, so t Rp plays an equivalent role to the niche valuein the niche model [39].(ii) Interaction . The interaction rule determines which members a ∈ A cooperatewith each member p ∈ P . Mutualistic interactions are hierarchically limited bythe complementarity between reward traits t Rp for p ∈ P and foraging traits t F a for a ∈ A , which both are uniformly drawn from [0 , P are sorted according to their reward trait t Rp in ascending order given their abilityto attract partners; whereas members from class A are sorted according to theirforaging traits t F a in descending order given their ability to cooperate with partners.Starting from the first -specialist- member p i and continuing sequentially subject to t Rp i > λ l pi , each link l pi is connected to the first -generalist- member a ′ ∈ A ′ , where A ′ is the subset of members in A that have not already been linked to by anothermember p = p i . If t Rp i ≤ λ l pi , the link is randomly connected to another member a ′′ ∈ A ′′ that belongs to a previously randomly selected member p j with lower traitvalue, where A ′′ is the subset of members in A that have been allocated links in aprevious timestep. This is to say, for each individual link of the new member p i ,first we randomly select a different member p j with lower trait value, and secondwe randomly select a member a ′′ from the partners of p j . This process is repeatedindependently for each individual link. Note that λ l pi corresponds to an externalconstraint associated with each link l p i that accounts for interaction barriers suchas competition and population density. This is drawn randomly from the sameexponential distribution as λ p (described below). In applying the interaction rule,if the supply of partners in either subset A ′ or A ′′ is exhausted before all l pi linkshave been allocated, then partners in the other subset are chosen to instead. Theeffect of this additional rule is equivalent to imposing a size limit on the population[46]. The BC model is initialized by connecting the first member p i to l pi membersin A ′ .In line with prior food web models [41], and in order to account for inhomogeneouseffects in the population, we assume that external constraints λ p and λ lp are randomlydrawn from an exponential distribution given by p ( x ) = βexp ( − βx ), with β = | P | ( | A | − / (2( L − | P | )) −
1. Note that L/ ( A · P ) is equivalent to the connectance ofa bipartite network. As has been shown [41], this exponential distribution is equivalentto a beta distribution under low-connectance levels. We also found that the BC modelproduces similar results using a beta distribution of the form p ( x ) = β (1 − x ) β − (seesupplementary information in [18]).
5. Empirical data
In the present paper, we use a diverse set of ten extensive plant-animal pollinationnetworks compiled in the literature (see Table), which can clearly be distinguished fromrandom assemblages [28]. Mutually-beneficial interactions in a pollination network areformed when an animal such as a bee or wasp gets food in form of nectar in exchangeof transferring pollen from one plant to another of the same species. Hence, mutually-beneficial interactions are established by the transportation mechanism, which enablesfertilization and sexual reproduction for plants.Our observed socio-economic network uses information collected by the workersunion UNITE (Union of Needle Trades and Industrial and Textile Employees) on thedesigner-contractor network of the New York Garment Industry (NYGI). UNITE hasorganized around 90 percent of all firms in the NYGI, and has developed a reliable systemto ensure the validity of transaction data [32]. The dataset includes approximately700,000 designer-contractor bilateral exchanges from January 1985 to December 2003.We formed yearly network snapshots from 1985-2003 given the seasonality and volatilityof the NYGI industry [34]. A cooperative interaction exists between a designer andcontractor if they co-produce a garment in a particular year. For example, the typicalproduction process in a year begins with a designer that develops a line of clothing.Each garment in the line is made into a sample prototype, which is disassembled intoits component parts such as shelves, collars, waistbands, and so forth. The componentsof the sample are then sent by the designer to contractors that cut components fromfabric in lots large enough to be mass produced. The cut fabric is then sent by thedesigner to sewing contractors that sew the fabric together into the garments that aresold directly to consumers at retailers. All firms are free to make connections of theirown choice; there is no governing body that suggests or mandates connections (see [34]for more details on the data).
6. Model validation
To analyze the performance of the BC model, we measure the ability of the BC modelat reproducing the degree distribution, nestedness and modularity of both pollinationnetworks and the NYGI networks. These network metrics are key organizational featurespresent in networks formed by bipartite cooperation [20, 28, 23, 30]. To capture thestatistical relevance of our model-generated networks, we use a Kolmogorov-Smirnov(KS) comparison test and a z -score analysis (i.e. normalized errors) to test the overallgoodness of fit of the BC model to the empirical data. All comparisons are basedon 1000 model simulations for each network (for comparisons of the BC model toother mutualistic models see supplementary information in [18]). In the BC model,plants and designers are treated as members of class P , and animals and contractorsas members of class A . This also follows the rationale that animals and contractorsmight experience higher competitive pressures than animals and contractors, given thedifferences in population size. From the Table we can see that the ratio between plantsanimals is P/A < .
5, which also plays a significant role limiting the appearance ofscale-free distributions [46].
The degree distribution P ( k ) is a widely used statistical metric that measures theprobability that a node has up to k network connections [5, 6, 7]. This measureaffects network growth and decline in different complex networks [34, 48]. Figures 2Aand 2B show the scaled cumulative distribution for members of class P and membersof class A respectively. Note that pollination networks (solid symbols) and NYGInetworks (crosses) exhibit the same patterns in both degree distributions. The solidline corresponds to the model-generated degree distributions. The Table shows that theBC model reproduced most of the empirical distributions. Nestedness is a concept applied in ecology to metacommunity populations, where siteswith low biodiversity constitute proper subsets of sites with higher biodiversity [49]. Thishas been extended to ecological networks, where a network is said to be nested whenspecialist species interact with proper subsets of the ecological interactions of generalistspecies [27]. To calculate nestedness N , we use the BINMATNEST program [50]. Thisprogram uses a genetic algorithm to find the matrix configuration that minimizes thelevel of disorder, and calculates an isocline curve that is used to measure the distanceto the situation of perfect order or nestedness for each pair-wise interaction. Herenestedness is defined in the interval [0,1], where 1 corresponds to a perfectly nestednetwork. In addition, we use a null model (i.e. randomized version) to compare whetherthe empirical networks and model-generated networks are indeed different from randomassemblages. For this, we use the null model II proposed in [27], which has been shownto produce conservative results with respect to nestedness [51].Figure 3A and 3B displays the nestedness pattern for empirical networks (dashedline), random assemblages (red), and model-generated networks (blue) for pollinationnetworks and NYGI networks respectively. The dashed line would correspond to aperfect agreement with the observed values. Bars are equivalent to avg. ± s.d. values-a nestedness of 1 means a perfectly nested matrix-. Note that the BC model alwaysperforms significantly better than random assemblages. To capture the statisticalrelevance of our model-generated nestedness values, we use a z -score analysis givenby Z = ( N − N m ) /σ m where N is the nestedness value for the empirical network, and N m and σ m are the average nestedness value and the standard deviation for the model-generated network respectively. Our Table shows the high goodness of fit of the BCmodel at reproducing the empirical nestedness values for the observed ecological andNYGI networks. Different nestedness algorithms do not change the qualitative natureof our results [52, 53]. The third characteristic that we analyze is network modularity [54]. The modularityof a network depends on its number of groups or modules and on the deviations fromthe number of groups expected in a properly randomized network [54]. The modularityvalues Q for the networks were calculated using the one-mode optimization algorithm[55]. A good partition generates many within-community links and as few as possiblebetween-community links. This algorithm does not discriminate between plants andpollinators or designers and contractors, where communities comprise nodes from bothclasses. This is important because we want to extract only cooperative units from thenetwork, and using a two-module partition algorithm [56] (i.e. modules with only oneclass of nodes), we could erroneously form communities between competitive membersinstead.To test our fit, we are not looking for a specific partition of the network [57], buta standard and well defined benchmark of comparison. Recent work has shown thathalf of the pollination networks studied in the present paper are modular (see Table)[23]. Using again a z -score analysis, the Table shows a surprisingly high degree ofcorrespondence between the five empirical modularity values and the model-generatedones. For all the observed NYGI networks, we always found higher modularity valuescompared to properly randomized networks [55] ( p - value < − ), while the BC modelproduced good approximations to the empirical modularity (see Table 1). Shifting ourfocus to the individual level, nodes have different connectivity roles according to theirnumber and distribution of connections within and outside their own community [55].Guimer`a and Nunes Amaral [55] have heuristically established seven different roles fornodes: roles 1-4 define different non-hub nodes, roles 5-7 are assigned for different classesof hub nodes, and the higher the role the higher the connectivity of the node with othercommunities. This categorization scheme classifies nodes according to their normalizedwithin-module degree z i and a participation coefficient P C i with the rest of the modules.Figure 4A and 4B show the connectivity-role space generated for a pollination and aNYGI network respectively. Note the similar patterns between the two networks. Plantsand designers (black dots) are mainly the nodes acting as hubs (roles 5-6), while animalsand contractors (red crosses) show high connectivity among modules (roles 3-4).To test the ability of the BC model to accurately reproduce the same number ofnodes within each connectivity role observed in the empirical networks, we measured thePearson correlation and the ratio of the connectivity role norms d for the observed andmodel-generated networks. The ratio of the norms d is defined by d = | x | / | y | , where | x | = qP mi =1 x i ; | y | = qP mi =1 y i ; m is the number of connectivity roles; and x and y are the proportion of nodes within each role i for the empirical and model-generatednetworks respectively. Note that the ratio measures the relative length between twovectors in an m -dimensional space, and values within 0 . < d < . r = 0 . . < d < . avg. ± s.d. ) to reproduce the same number of nodes within each connectivity role(circles) for pollination networks and NYGI networks respectively.
7. Ecological niche markets
As previously mentioned, a key feature in food-web models is the existence of ahierarchical organization established by niche values. In the BC model we also assumedthat members in the population are classified according to a niche or hierarchicalvalue. In pollination networks, these niches or hierarchies can be the result of differentmorphological, temporal and geographical variables that constrain the amount andidentity of partners [23, 24, 25]. Similarly, firms face interaction barriers accordingto their niche or position in the market [4, 35]. This position can be assessed by thediversity of products and the hierarchical nesting of groups [38]. Here, to empiricallytest this organizational feature on the NYGI network, we analyze the extent to whichthe emerging communities in the network correspond to actual niche markets.To identify the communities or groups in the network, we use the same communitydetection algorithm that we used in Section 6.3 to validate the BC model [55]. To identifythe niche of a community, we use data on the different types of garment (i.e. men’s coats,women’s skirts, t-shirts, etc.) that a firm designs or manufactures. Thus, the diversityof products or garment could be taken as a proxy for the niche characteristics of aparticular community. This is to say, two different communities of firms would be inseparated niches if they produce different products. We measure the correlation betweencommunities and products using a principal components analysis. To do this we built amatrix P , where each element P ij corresponds to the percentage of firms in community i working on garment j . Using this matrix, we calculated the correlation of productsbetween communities given by a transformation of values P ij into principal components(i.e. eigenvectors). Using the two largest eigenvalues, which accounted for the 95% ofthe variability (see Figure 6A), we generated a two-dimensional projection of the values.Figure 6B shows the projection (dots) and in fact reveals different production trends(i.e different positions in the two-dimensional space) between communities.To test whether the niche or product differentiation is an artifact of the differentnumber of firms in a community, for each community i we measure its level of productiondiversification defined by d ( i ) = P j ( P ij ) . Here, d ( i ) = 0 if every firm in the communityworks on a different product, and d ( i ) = 1 if all firms work on a single type of product.0The correlation between the size of a community and its level of product diversificationwas r = 0 .
01, which confirms that the network’s organizational structure is confirmed bynon-structural niche barriers. Hence, we believe future studies should investigate howthe model-generated ordering map into empirical hierarchies at the level of individuals.
8. Organizing mechanisms
Social networks, metabolic networks, socio-economic networks, and ecological networks,among others display network structures that differ from random assemblages [5, 6, 7,34]. These comparisons are based on an appropriate randomization of the network(i.e. null model), which assigns a probability p to the existence of each pairwiseinteraction. These null models can range from models that only preserve the number ofconnections in the network (e.g. p = L/S ) to models that keep the actual distributionof connections intact and allow a further exploration of the structural correlations inthe network [58, 59]. However, null models based on organizing mechanisms ratherthan structural characteristics in the network have not been fully explored. Here, wemake use of the BC model to show the effects that different constraints acting overthe organizing mechanisms of mutualistic networks might produce on the hierarchicalor nested structure of ecological networks, which is a key feature in the robustness andbiodiversity of these systems [27, 30].In the BC model, the number and identity of partners are constrained by ahierarchical ordering, which are modulated by an external factor λ p and λ l pi actingover the specialization and interaction mechanisms respectively. In line with food webmodels [41], we have assumed random inhomogeneous values for λ p and λ l pi given byan exponential distribution. In food webs, this distribution imposes a specific rangeof possible interactions, which might correspond to the actual handling and foragingcapacities of species [43]. To study the effects of these constraints on the specializationand interaction mechanisms of the BC model, we replaced the exponential distributionby a randomly uniformed distribution. By modifying the specialization mechanismalone (i.e. λ p ), we found nestedness values below the ones displayed by the observedpollination networks. In line with standard null models, Figure 7 shows an under-representation of nestedness by using both the modified version of the BC model (redline) and the standard null model II (blue line) [27]. This suggests that the exponentialdistribution limiting the number of connections for plants might in fact be responsible forthe nested organization of pollination networks. However, if we modify the interactionmechanism (i.e. λ l pi ), we found the opposite behavior. Figure 7 (green line) shows anover-representation of nestedness compared to the observed networks. Figure 7 (blackline) confirms that this pattern is robust even if we modify both the specialization andthe interaction mechanisms simultaneously. These results suggest that the organizationof mutualistic networks is neither extremely nested nor random, which might emergeas a compromise between the structural robustness (specialization mechanism) and thefunctionality of the network (interaction mechanism). We believe this conjecture should1be tested in future work.
9. Conclusions
The study of direct member-to-member interactions have allowed us to find that thestructure of ecological and socio-economic networks generated by mutually-beneficialinteractions exhibits remarkably similar features. This empirical finding motivates theproposed model for bipartite cooperation, starting from a generalization of the nichemodel [39], which can successfully reproduce the overall structure of pollination andNYGI networks using the number of members and the total number of links as theonly input parameters. We have identified common organizing mechanisms operatingin these radically different networks, which are the result of a hierarchical ordering thatfavors the presence of asymmetric interactions between their members. These organizingmechanisms are exponentially constrained by external factors (such as environmentaland market pressures), which modulate the number and identity of potential partners.Using our model, we investigated the effects that different constraints acting on thespecialization and interaction mechanisms of mutualistic networks might produce on thehierarchical arrangement of these networks. We have found that real-world mutualisticnetworks display network organizations, which are between highly nested and randomstructures. This could enable a systematic approach to study the interplay betweenfunctionality and structural robustness in mutualistic networks. The success of thissimple stochastic model in generating the overall structural characteristics of mutualisticnetworks makes it a suitable starting point for more elaborate ecological and socio-economic models which seek to understand the effects of changes in population size,number of interactions, and harsher environmental or market conditions.
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L P A KS P − KS A N Q
Kato and Miura (1996) 430 64 187 0.326 †† − . †† †† (0.969) 0.551 †† (0.553)Primack (1983) 374 41 139 0.633 †† − . †† †† (0.96) 0.474 † (0.465)Kato et al. (1993) 865 90 354 0.552 †† -0.001* 0.985 † (0.976) 0.545 † (0.532)Primack (1983) 120 18 60 0.108 † − . †† ∗∗ (0.527)Primack (1983) 346 49 118 0.002*-0.001* 0.961 †† (0.955) 0.480 † (0.468)Hocking (1968) 179 28 81 0.097 † − . †† † (0.950) nmInouye and Pyke (1988) 252 36 81 0.608 †† − . † †† (0.949) nmSchemske et al. (1978) 65 7 33 0.911 †† − . †† † (0.930) nmElberling and Olsesen (1999) 453 31 75 0.038** − . † ) 0.793*(0.914) nmElberling and Olsesen (1999) 242 24 118 0.223 † -0.005**) 0.927 † (0.952) nmNYGI (1985) 7250 823 2562 0.061 † − . † † (0.996) 0.598*(0.502)NYGI (1991) 3981 325 1590 0.101 † − . †† †† (0.993) 0.601*(0.529)NYGI (1997) 1450 148 700 0.003** − . † † (0.988) 0.653**(0.625)NYGI (2003) 228 62 128 0.370 † − . † (0.969) 0.711 † (0.700) Table 1.
Model Validation. For ten pollination datasets and four NYGI networksused in this paper, the table presents its source; total number of links L , P and A are the number of members in class P and class A respectively. For thedegree distributions, ( KS P − KS A ) shows the combined Kolmogorov-Smirnov (KS)probability using the two-group equivalence KS test between the empirical and model-generated distributions for class P and class A respectively. N and Q correspondto the observed nestedness and mean modularity values respectively, along with thenormalized errors ( z -score) for the comparison between the empirical and model-generated values. The model-generated mean values for N and Q are shown insidethe parentheses. Five of the observed pollination networks have already been found tobe non-modular (nm) [28]. †† : KS > .
30 or normalized errors < † : KS < .
30 ornormalized errors between 1 and 2 model s.d. (good fit); **:
KS < .
05 or normalizederrors between 2 and 3 model s.d. (poor fit); *:
KS < .
01 or normalized errors > Firms L i n ks datafit Figure 1.
Scale invariability. The figure shows the relationship on a log-log scalebetween the total number of firms F and links L in the NYGI network from 1985 to2003 (dots). The solid line is the fit to the data defined by L = F α with α = 1 . ± . R = 0 . k / z P c u m ( k ) A P Links per plant/designer k / z P c u m ( k ) B A Links per animal/contractor
Figure 2.
Cumulative degree distribution. Panel A and Panel B show the scaledcumulative degree distribution P cum ( k ) for members of class P (plants, designers), andmembers of class A (animals, contractors) respectively. The number of partners k isscaled by a multiplicative factor of 1 /z P for members of P, and 1 /z A for members ofA, where z P = L/P and z A = L/A . Solid symbols correspond to pollination networksand crosses correspond to NYGI networks. Note that all distributions collapse into asingle curve. The solid line corresponds to the model-generated distributions averagedover 1000 simulations. Observed P r ed i c t ed A Pollination networks
Observed P r ed i c t ed B NYGI networks
Figure 3.
Nestedness. Panel A and Panel B compare the random-generated (red)and model-generated (blue) nestedness values to the observed nestedness values for thepollination and NYGI networks respectively. The dashed line would correspond to aperfect agreement with the observed values. Bars are equivalent to avg. ± s.d. values.Here, a nestedness of 1 means a perfectly nested matrix. Participation coefficient, PC W i t h i n − m odu l e deg r ee , z Pollination networkR5 R6 R7R3 R4R2R1 A Participation coefficient, PC W i t h i n − m odu l e deg r ee , z R1 R2 R3 R4R5 R6 R7 B NYGI network
Figure 4.
Empirical connectivity roles. Panel A and Panel B show the connectivity-role space for members of the Kato and Miura (1996) network and the 1997 NYGInetwork respectively. Plants and designers are shown in black dots, whereas animalsand contractors in red crosses. Note that the two networks present similar patterns.The classification is as follows [55]: Nodes with z ≥ . z < . P C . For non-hub nodes (R1) is for nodes with all their links connected within their own module P ≤ .
05, (R2) is for nodes with most of their links connected within their own module0 . < P ≤ .
62, (R3) is for nodes with many links connected to other modules0 . < P ≤ .
80 and (R4) is for nodes with their links homogeneously connected toall other modules
P > .
80. For hub-nodes, (R5) is for nodes with most of their linksconnected to their own module P ≤ .
30, (R6) is for nodes with most of their linksconnected to other modules 0 . < P ≤ .
75 and (R7) is for nodes with their linkshomogeneously connected to all other modules
P > . Connectivity Role % P opu l a t i on A Marsh, JapanSubalpine, JapanSubalpine, New ZealandGrassland, New ZealandCraigieburn, New Zealandmodel
Connectivity Role % P opu l a t i on B Figure 5.
Model validation for connectivity roles. Panel A and Panel B showthe proportion of members within each connectivity role for the modular pollinationnetworks and the NYGI networks respectively. The bars show the accuracy of the BCmodel to reproduce the same number of nodes within each connectivity role (circles) asthe ones observed in the empirical networks. Dots correspond to the empirical valuesand bars ( avg. ± s.d. ) correspond to the model-generated values.
1 2 30102030405060708090100
Principal Component V a r i an c e E x p l a i ned ( % )
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% A −0.5 0 0.5−0.5−0.4−0.3−0.2−0.100.10.20.3 p r i n c i pa l c o m ponen t B Figure 6.
Niche markets. Panel A shows the variability of each principal componentmeasured for the product correlation j between communities i in the 1990 P ij network.Note that we only need the first two components in order to account for the 95% ofthe variability. Panel B shows the transformation of the original P ij matrix by itsprojection into eigenvectors using the first two principal components. Note that eachcommunity i (dots) is defined by a different trend (i.e. position in the two-dimensionalspace). Different years produce similar results. Network z − sc o r e standardExp−UnifUnif−ExpUnif−Unifz=2z=−2 Figure 7.
An alternative null model. The figure shows the z -score values (y-axis) for each of the pollination networks (x-axis). The z -score values are given by Z = ( N − N m ) /σ m where N is the nestedness value for the empirical network, and N m and σ m are the average nestedness value and the standard deviation for the null-model-generated network respectively. The blue line and red line correspond to the standardnull model II [27] and the modified version of the BC model using a uniform distributionfor the specialization factor λ p respectively. Note that both null models produce amajority of Z < −
2, which indicates that the empirical values are significantly nestedcompared to these null models. In contrast, the green line and black line correspond tothe modified versions of the BC model using a uniform distribution for the interactionfactor λ lp alone and using a uniform distribution for both factors respectively. Notethat both null models produce a majority of Z >
2, which indicates that the empiricalvalues are significantly less nested compared to these null models. Dashed lines arejust eye guidelines for Z = 2 and Z = −−