A simple model of bipartite cooperation for ecological and organizational networks.

Abstract

In theoretical ecology, simple stochastic models that satisfy two basic conditions about the distribution of niche values and feeding ranges have proved successful in reproducing the overall structural properties of real food webs, using species richness and connectance as the only input parameters. Recently, more detailed models have incorporated higher levels of constraint in order to reproduce the actual links observed in real food webs. Here, building on previous stochastic models of consumer-resource interactions between species, we propose a highly parsimonious model that can reproduce the overall bipartite structure of cooperative partner-partner interactions, as exemplified by plant-animal mutualistic networks. Our stochastic model of bipartite cooperation uses simple specialization and interaction rules, and only requires three empirical input parameters. We test the bipartite cooperation model on ten large pollination data sets that have been compiled in the literature, and find that it successfully replicates the degree distribution, nestedness and modularity of the empirical networks. These properties are regarded as key to understanding cooperation in mutualistic networks. We also apply our model to an extensive data set of two classes of company engaged in joint production in the garment industry. Using the same metrics, we find that the network of manufacturer-contractor interactions exhibits similar structural patterns to plant-animal pollination networks. This surprising correspondence between ecological and organizational networks suggests that the simple rules of cooperation that generate bipartite networks may be generic, and could prove relevant in many different domains, ranging from biological systems to human society.

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LETTERS
Asimplemodelofbipartitecooperationforecological
and organizational networks
Serguei Saavedra1,2,3, Felix Reed-Tsochas2,4& Brian Uzzi5,6
Intheoreticalecology,simplestochasticmodelsthatsatisfytwobasic
conditionsaboutthedistributionofnichevaluesandfeedingranges
have proved successful in reproducing the overall structural
propertiesofrealfoodwebs,usingspeciesrichnessandconnectance
astheonlyinputparameters1–4.Recently,moredetailedmodelshave
incorporated higher levels of constraint in order to reproduce the
actuallinksobservedinrealfoodwebs5,6.Here,buildingonprevious
stochastic models of consumer–resource interactions between
species1–3, we propose a highly parsimonious model that can repro-
duce the overall bipartite structure of cooperative partner–partner
interactions, as exemplified by plant–animal mutualistic networks7.
Our stochastic model of bipartite cooperation uses simple speciali-
zationandinteractionrules,andonlyrequiresthreeempiricalinput
parameters. We test the bipartite cooperation model on ten large
pollination data sets that have been compiled in the literature, and
findthatitsuccessfullyreplicatesthedegreedistribution,nestedness
and modularity of the empirical networks. These properties are
regarded as key to understanding cooperation in mutualistic
networks8–10. We also apply our model to an extensive data set of
two classes of company engaged in joint production in the garment
industry. Using the same metrics, we find that the network of
manufacturer–contractor interactions exhibits similar structural
patterns to plant–animal pollination networks. This surprising
correspondence between ecological and organizational networks
suggests that the simple rules of cooperation that generate bipartite
networksmaybegeneric,andcouldproverelevantinmanydifferent
domains, ranging from biological systems to human society11–14.
Inecology,thecollectionandanalysisofempiricaldataonmutualistic
networks7can help identify the most significant features that have
evolved in bipartite cooperative networks. In these networks, species in
one class (class A, for animals) cooperate with species in a second class
(classP,forplants)tomutualadvantage.SpeciesinclassPofferrewards
withcertaincharacteristicsdeterminedbyrewardtraits,whichmayalso
have evolved to reduce exploitation and favour mutualism15. Species in
class A foraging for resources can benefit from the rewards offered by a
given species in class P if the respective foraging traits (for example
efficiency, morphology and behaviour) and reward traits (for example
quantity,qualityandavailability)arecomplementary8,16.Externalfactors
suchastheenvironmentalcontext(forexamplepopulationdensityand
geographic and temporal variation) attenuate or amplify the value of
rewardandforagingtraits,andhaveanimpactonthenumberofpoten-
tial partners with which a given species cooperates17–19. Furthermore,
mutualistic networks exhibit constraints introduced by phylogenetic
relationshipsbetweenspeciesinthesameclass,whichimpactmutualistic
interaction patterns by favouring ecological similarity20.
Recently, mutualistic models21,22have been proposed which incor-
porate the effectsofcomplementarity and exploitation or growth bar-
riersoninteractionsbetweenspeciesinclassesAandP(Supplementary
Information). Although a mixed model21and a differential limiting
size model22have proved successful at generating structural patterns
similartothoseobservedinrealmutualisticnetworks,wefindthatthey
are able to reproduce less than 25% of the total number of observed
metrics in different ecological networks (Supplementary Tables 1–4).
In organizational networks, cooperation between two different
classes of company (manufacturers and contractors) is subject to
equivalent structural constraints, which depend on the traits of the
companiesandthecomplementaritybetweentraitsofpotentialpart-
ners, as well as hierarchical relationships between companies in the
same class23. Companies are characterized by a set of organizational
traits24(for example company size, competitive niche space and
brand positioning), and face interaction barriers generated by differ-
ences in status25, which limit the number and range of potential
partners. Again, the values associated with trait complementarity
and interaction barriers are not absolute, but are modulated by the
specific market context in which the companies operate24,25.
Inspiredbypreviousfood-webmodels1–3,wedevelopanewmodel
of bipartite cooperation that can reproduce more than 70% of the
total number of observed metrics in both ecological and organiza-
tional networks (Table 1, Supplementary Tables 3, 4). In the corres-
ponding ecological (species-based) and organizational (company-
based) contexts, plants and manufacturers are treated as members
of class P, and animals and contractors as members of class A. The
threeinputsforthemodelarethesizeofclassA,thesizeofclassPand
the total number of links, L, all of which are given directly by the
empirical data(Table1).Themodelconsists oftwomechanisms (see
Methods for a description of the model).
The first mechanism is specialization. The specialization rule deter-
mineshowmanypartners,lp,eachmemberpgPwillcooperatewith.
ThisnumberisdeterminedbytherewardvalueassociatedwithpgP,
which is given by the reward trait, tRp, attenuated or amplified by an
externalfactorlpthataccountsforeffectssuchasgeographicvariation
andpopulationdiversity.Higherrewardvaluesincreasethenumberof
potential interactions. Reward traits tRpare the result of a hierarchical
process, which in our model corresponds to the generation of an
ordered sequence in trait space, so tRphas a role equivalent to that of
the niche value in the niche model1or nested-hierarchy model2. Note
that the specialization rule is only associated with class P. This is con-
sistent with previous findings which show that external factors affect
the level of specialization more strongly in plants than in animals18.
The second mechanism is interaction. The interaction rule deter-
mines which members agA cooperate with each member pgP.
Here interactions are limited by the complementarity between the
rewardtraits,tRp,forpgPandforagingtraits(organizationaltraits),
tFa, for agA. The foraging trait tFalimits the range of possible
partners for each member of class A, but again external factors llp
such as temporal variation and population density can modify these
1Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK.2CABDyN Complexity Centre,3Corporate Reputation Centre,4James Martin Institute, Saı ¨d Business
School, University of Oxford, Oxford OX1 1HP, UK.5Kellogg School of Management and Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois
60208, USA.6Haas School of Business, University of California, Berkeley, California 94720, USA.
doi:10.1038/nature07532
1
©2008 Macmillan Publishers Limited. All rights reserved

Page 2

interaction barriers. In both the specialization and interaction rules,
we assume that traits are normally distributed and that the effect of
external factors onmembers isinhomogeneous and afunction ofthe
diversity of members and the density of interactions in the network.
Wetestourmodelusingdatafromreal-worldecologicalandorgani-
zational networks. For the ecological networks, we use a diverse set of
ten extensive plant–animal pollination networks compiled in the lite-
rature (Supplementary Information), which can clearly be distin-
guished from random assemblages9. For organizational networks, we
use a data set of approximately 700,000 yearly bilateral production
transactionsbetweenmorethan10,000manufacturersandcontractors
in the New York garment industry (NYGI) from January 1985 to
December 2003 (Supplementary Information)23. The NYGI exhibits
a significant turnover of companies each year with different declining
trajectoriesforeachclass(SupplementaryFig.1),sothenetworkstruc-
tureatonetimedoesnottriviallymapintothestructureatothertimes.
For each network, we investigate three key features of bipartite
cooperation8–10: (1) degree distribution26, (2) nestedness27,28and (3)
modularity29.
First we consider the degree distribution. In Fig. 1a and b we
respectivelyshowthecumulativedistributionsformembersofclassP
Table 1 | Empirical values and model statistical significance
Data set/environmentL
|P||A|
KSP–KSA
NQ
Marsh (Japan)
Grassland (Cass, New Zealand)
Subalpine forest/meadow (Japan)
Subalpine (Arthur’s Pass, New Zealand)
Subalpine (Craigieburn, New Zealand)
Tundra (Canada)
Scrub/snow gum forest (Australia)
Deciduous forest (USA)
Arctic tundra (Greenland)
Subarctic rock slope (Sweden)
New York garment industry, 1985
New York garment industry, 1991
New York garment industry, 1997
New York garment industry, 2003
430
374
865
120
346
179
252
65
453
242
7,250
3,981
1,450
228
64
41
90
18
49
29
36
7
31
24
823
325
148
62
187
139
354
60
118
81
81
33
75
118
2,562
1,590
700
128
0.326*–0.438*
0.633*–0.385*
0.552*–0.0011
0.108{–0.999*
0.0021–0.0011
0.097{–0.989*
0.608*–0.076{
0.911*–0.642*
0.038{–0.118{
0.223{–0.005{
0.061{–0.115{
0.101{–0.531i
0.003{–0.264{
0.370{–0.002{
0.976* (0.969)
0.957* (0.960)
0.985{ (0.976)
0.8581 (0.936)
0.961* (0.955)
0.971{ (0.950)
0.935* (0.949)
0.953{ (0.930)
0.7931 (0.914)
0.927{ (0.952)
0.997{ (0.996)
0.994* (0.993)
0.990{ (0.988)
0.976{ (0.969)
0.551* (0.553)
0.474{ (0.465)
0.545{ (0.532)
0.553{ (0.527)
0.480{ (0.468)
NM
NM
NM
NM
NM
0.5981 (0.502)
0.6011 (0.529)
0.653{ (0.625)
0.711{ (0.700)
For each pollination data set and the four organizational networks used in this paper, the table presents its environment/location; the total number of links, L; the respective numbers of nodes, |P|
and |A|, in classes P and A; the combined Kolmogorov–Smirnov (KS) probabilities, KSP–KSA, calculated for the degree distributions using the two-group equivalence KS test between the empirical
and model-generated distributions for classes P and A, respectively; the observed nestedness, N; and the observed mean modularity value, Q. Note that all networks have a ratio |P|/|A| ,0.5,
whichhasbeenfoundtobeanimportantfactorlimitingtheappearanceofscale-freedistributions22.Themodel-generatedmeanvaluesforNandQareshowninsideparentheses.Fiveoftheobserved
pollination networks have already been found to be non-modular (NM)10. All comparisons are based on 1,000 model simulations. Note that the model reproduces more than 70% of the overall
number of observed metrics with a good or excellent fit (27 out of 35 and 11 out of 16 for the ecological and organizational networks, respectively).
*KS$0.30, normalized errors less than one model s.d. (excellent fit). {KS,0.30, normalized errors between one and two model s.d. (good fit). {KS,0.05, normalized errors between two and
three model s.d. (poor fit). 1KS,0.01, normalized errors greater than three model s.d. (bad fit).
02468 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k/zP
k/zA
Pcum(k)
Pcum(k)
a
0246810
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
b
Figure 1 | Scaled degree distribution. a, The cumulative degree
distribution, Pcum(k), for members of class P (plants, manufacturers);
b, Pcum(k) for members of class A (animals, contractors). The number of
partners, k, is scaled by a multiplicative factor of 1/zPfor members of P and
1/zAfor members of A, where zP5L/P and zA5L/A. Filled symbols
correspond to pollination networks and open symbols to organizational
networks. Note that all distributions collapse into a single curve. The solid
line corresponds to the model-generated distributions averaged over 1,000
simulations (Supplementary Fig. 2).
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Predicted
a
0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
Observed
1
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Predicted
b
1
Figure 2 | Nestedness. a,b,Comparisonsoftherandomlygenerated(green
crosses) and model-generated (orange circles) nestedness values
(mean62s.d.) with the observed nestedness values for all the pollination
(a) and the NYGI (b) networks. The dashed lines correspond to there being
perfect agreement between predicted and observed values. Note that a
matrix with a nestedness value of one is perfectly nested.
LETTERS
NATURE
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©2008 Macmillan Publishers Limited. All rights reserved

Page 3

and members of class A. Note that pollination networks (filled sym-
bols) and organizational networks (open symbols) exhibit the same
patternsinbothdegreedistributions.Thesolidlinecorrespondstothe
model-generateddegreedistributions(seealsoSupplementaryFig.2).
Table 1 provides statistical evidence for the correspondence between
the empirical and model-generated distributions for both ecological
and organizational networks.
To calculate the nestedness level, N, of the empirical and model-
generated networks, we use the BINMATNEST program28. Here
nestedness is defined in the interval [0,1], where 1 corresponds to
a perfectly nested network9. Figure 2 shows the nestedness values for
empirical networks (dashed line), random assemblages generated
following the null model II proposed in ref. 9 (green crosses) and
model-generated networks (orange circles). Note that the model
always significantly outperforms random assemblages. Table 1 pre-
sentstestsofstatisticalsignificanceandshowsthatthereisahighlevel
of correspondence between the model and data for most of the net-
works (Supplementary Fig. 3).
Modularityvalues,Q,forthenetworksarecalculatedusingtheone-
modeoptimizationalgorithmproposedinref.30,becausewewantto
extract only cooperative units from the network. This algorithm has
shown that only five of the observed ecological networks are truly
modular10(Table 1). For these five networks, we find a strong corres-
pondence between the empirical and model-generated modularity
values (Table 1). For each organizational network, we find a modu-
larity valuethat is higherthan that generatedfrom thecorresponding
random assemblages30(P,1026), and this empirical behaviour is
replicated by the model (Table 1). Nodes have been shown to have
differentconnectivityroles(forexampleglobalorlocalhubs)depend-
ingonhowtheyareembeddedwithintheirmoduleandparticipatein
other modules30. Here we also test the ability of our model to accur-
atelyreproducetheempiricallyobservednumberofnodeswithineach
connectivityrole30(Fig.3).Todoso,wemeasurethePearsoncorrela-
tion coefficient, r, and the ratio of the connectivity role norms, d, for
the observed and model-generated networks (Methods). We consis-
tently find values aligned with the empirical measurements for the
ecological and organizational networks (r50.98, 0.9,d,1.1).
Our study identifies striking similarities in the general structural
characteristics of networks that are formed as a result of cooperative
mechanisms operating in radically different contexts, linking part-
ners in ecological and socio-economic systems, respectively. This
empirical finding motivates the proposed simple model for bipartite
cooperation, which captures the most important generic features of
mutualistic interaction patterns starting from a minimal set of input
parameters. At the level of partner–partner interactions, equivalent
behaviourindifferentsystemsappearstobedrivenbysimilartypesof
interaction constraints. These correspond to complementarity in
traitsorcharacteristics,ahierarchicalorganizationlimitingtherange
1234567
0
10
20
30
40
50
60
70
80
90
Connectivity roleConnectivity role
Population (%)
c
Marsh (Japan)
Subalpine (Japan)
Subalpine (New Zealand)
Grassland (New Zealand)
Craigieburn (New Zealand)
Model
1234567
0
10
20
30
40
50
60
70
80
90
d
0 0.20.4 0.60.81
−2
0
2
4
6
8
10
Participation coefficient, Pc
Participation coefficient, Pc
Within-module degree, z
R5
R6
R7
R3 R4
R2 R1
00.2 0.4 0.6 0.81
−2
0
2
4
6
8
10
R1 R2
R3
R4
R5
R6
R7
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
Model
ab
Figure 3 | Connectivity roles. a, b, Division of connectivity roles (R1–R7)
for members of the subalpine (Japan) network (a) and the 1997 NYGI
network (b). Plants and manufacturers are indicated with blue circles and
animalsandcontractorswithredtriangles.Nodeswithanormalizedwithin-
module degree (z score) z$2.5 have been heuristically defined as module
hubs,andnodeswithz,2.5asnon-hubnodes30.Inaddition,hubsandnon-
hub nodes are classified according to a participation coefficient, Pc, which
gives the level of interaction of a node with the rest of the modules. The
completeclassificationisasfollows.Fornon-hubnodes,R1comprisesnodes
with all their links connected within their own module (Pc#0.05), R2
comprisesnodeswithmostoftheirlinksconnectedwithintheirownmodule
(0.05,Pc#0.62), R3 comprises nodes with many links connected to other
modules (0.62,Pc#0.80) and R4 comprises nodes with their links
homogeneously connected to all other modules (Pc.0.80). For hub-nodes,
R5 comprises nodes with most of their links connected to their own module
(Pc#0.30), R6 comprises nodes with most of their links connected to other
modules (0.30,Pc#0.75) and R7 comprises nodes with their links
homogeneouslyconnectedtoallothermodules(Pc.0.75).c,d,Proportions
of nodes from the population within each connectivity role for the modular
pollination networks (c) and the NYGI networks (d). Points (various
symbols) correspond to the empirical values and bars (mean6s.d.)
correspond to the model-generated values. Note that the model accurately
replicates the proportion of nodes within each connectivity role for both
types of network.
NATURE
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Page 4

of potential partners, and the environmental context. The success of
thissimplestochasticmodelingeneratingtheoverallstructuralchar-
acteristics of mutualistic networks makes it a suitable starting point
for the development of more elaborate ecological models, with the
aim of addressing the important question of reproducing the actual
links observed in real mutualistic webs. Such an approach would
require more comprehensive statistical comparisons across different
network metrics using maximum-likelihood methods6. Beyond this
ecological context, and the specific organizational network that has
been considered, the generic nature of the bipartite cooperation
model and its underlying assembly rules suggests that it should be
relevant to many other cooperative networks in domains beyond
those considered here.
METHODS SUMMARY
Model.Forthespecializationrule,thenumberoflinks,lp,foreachnodepgPis
defined by lp~1zRound((L{jPj)tRplp=
isuniformlydrawnfrom[0,1],theexternalfactorlpisrandomlydrawnfroman
exponentialdistribution, j?j denotes setcardinalityandRound(?) is thenearest-
integer function.
For the interaction rule, nodes from class P are sorted according to their
reward trait tRpin ascending order. Nodes from class A are sorted in descending
order according to their foraging traits tFa, which are uniformly drawn from
[0,1]. Starting from the first node, pi, and continuing sequentially subject to
tRpi.llpi, each link lpiis connected to the first node a9gA9, where A9 is the
subsetofnodesinA thathavenotalreadybeenlinkedtoby anothernodep?pi.
Here llpiis an external factor drawn randomly from the same exponential dis-
tributionaslpi.IftRpi#llpithenthelinkisrandomlyconnectedtoanothernode
a99gA99, where A99 is the subset of nodes in A that have been linked in a
previous time step. If the supply of nodes in either A9 or A99 is exhausted before
all lpilinks have been allocated, then nodes in the other subset are linked to
instead.Themodelisinitializedbyconnectingthefirstnode,pi,tolpinodesinA9.
The exponential distribution used for lp and llp is of the form
p(x)5bexp(2bx), with b5jPj(jAj21)/(2(L2jPj))21. If we generalize our
formalism and use a beta distribution1,3, we do not find significantly different
results (Supplementary Table 5).
Ratio of connectivity role norms. The ratio of the norms, d, is defined by
d5jxj/jyj, where x j j~(Pm
the empirical and, respectively, model-generated networks. Ratios within
(0.9,1.1) are fractions with a norm comparable to the empirical data.
X
jtRjlj), where the reward trait tRp
i~1x2
i)1=2, y j j~(Pm
i~1y2
i)1=2, m is the number of
connectivity roles and xiand yiare the numbers of nodes within each role i for
Received 5 June; accepted 10 October 2008.
Published online 3 December 2008.
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Acknowledgements We thank J. Dunne, R. Guimera `, J. Kerte ´sz, M. Sales-Pardo,
D. Stouffer and R. Williams for comments and suggestions. F.R.-T. acknowledges
funding from the European Commission under the FP6 NEST Pathfinder Initiative
‘TacklingComplexityinScience’(MMCOMNETproject,contractno.012999).S.S.
held a Doctoral Research Studentship funded by MMCOMNET and CONACYT,
and currently is supported by a Postdoctoral Fellowship at the Oxford University
Corporate Reputation Centre in conjunction with the CABDyN Complexity Centre.
Author Contributions B.U. provided the NYGI data; F.R.-T. designed the research;
S.S., F.R.-T. and B.U. analysed the data; S.S. ran the simulations; S.S. and F.R.-T.
wrote the paper.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. Correspondence and requests for materials should be
addressed to F.R-T. (felix.reed-tsochas@sbs.ox.ac.uk).
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