Evolutionary dynamics of group interactions on structured populations: A review

2.1 MB·English

by  Matjaz Perc

Evolutionary dynamics ofgroup interactions onstructured populations: A review
Matjazˇ Perc,1,∗ Jesu´sGo´mez-Garden˜es,2,3 Attila Szolnoki,4 LuisM.Flor´ıa,2,3 andYamirMoreno3,5,6
1FacultyofNaturalSciencesandMathematics,UniversityofMaribor,Korosˇkacesta160,SI-2000Maribor,Slovenia
2Department ofCondensed MatterPhysics, Universityof Zaragoza, E-50009 Zaragoza, Spain
3InstituteforBiocomputationandPhysicsofComplexSystems(BIFI),UniversityofZaragoza,E-50018Zaragoza,Spain
4Instituteof Technical Physics andMaterials Science, Research Centre forNatural Sciences,
Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
5Department of Theoretical Physics, University of Zaragoza, E-50009 Zaragoza, Spain
6ComplexNetworksandSystemsLagrangeLab,InstituteforScientificInterchange,VialeS.Severo65,I-10133Torino,Italy
3 Interactions among livingorganisms, frombacteriacolonies tohuman societies, areinherently morecom-
1 plexthaninteractionsamongparticlesandnonlivingmatter.Groupinteractionsareaparticularlyimportantand
0 widespreadclass,representativeofwhichisthepublicgoodsgame. Inaddition,methodsofstatisticalphysics
2 haveprovenvaluableforstudyingpatternformation,equilibriumselection,andself-organisationinevolution-
n ary games. Here we review recent advances in the study of evolutionary dynamics of group interactions on
a structured populations, including lattices, complex networks and coevolutionary models. We also compare
J theseresultswiththoseobtainedonwell-mixedpopulations. Thereviewparticularlyhighlightsthatthestudy
0 ofthedynamicsofgroupinteractions,likeseveralotherimportantequilibriumandnon-equilibriumdynamical
1 processesinbiological,economicalandsocialsciences,benefitsfromthesynergybetweenstatisticalphysics,
networkscienceandevolutionarygametheory.

]
h
PACSnumbers:02.50.-r,87.23.-n,89.65.-s,89.75.-k
p Keywords:evolution,cooperation,publicgoods,phasetransitions,patternformation,cyclicalinteractions,self-organization,
- lattices,complexnetworks,coevolution,statisticalphysics
c
o
s
. 1.INTRODUCTION 1.1.Motivationandbasicconcepts
s
c
i
s Wepresentareviewofrecentadvancesontheevolutionary Given that fundamental interactions of matter are of pair-
y
dynamicsofspatialgamesthataregovernedbygroupinterac- wise nature, the consideration of N-particle interactions in
h
tions. Thefocusisonthepublicgoodsgame,ormoregener- traditional physical systems is relatively rare. In computa-
p
[ allyN-playergames,whicharerepresentativeforthistypeof tionalapproachesaimed at modelingsocial, economicaland
interaction. Althoughrelevantaspectsof 2-playergamesare biological systems, however, where the constituents are nei-
1 surveyedaswell,wereferto[1]foramorethoroughexposi- therpointmassparticlesnormagneticmoments,N-playerin-
v
tion. Anotherimportantaspect of this review is its focuson teractionsarealmostasfundamentalas2-playerinteractions.
7
structured populations. In the continuation of this introduc- Mostimportantly,groupinteractionsingeneralcannotbere-
4
2 torysectionwe willalsosummarisebasicresultsconcerning ducedtothecorrespondingsumofpairwiseinteractions.
2 thepublicgoodsgameonwell-mixedpopulations,butwere- Asimplemodelinspiredbyexperiments[4–10]canbein-
1. ferthereaderto[2,3]fordetails. vokedforbothmotivatingtheusageofthepublicgoodsgame
0 Themethodologicalperspectivethatpermeatesthroughout as well as forintroducingbasic notation. Consider a colony
3 the review is that of statistical physics. The advances re- of N microbial agents where a fraction of them (producers
1 viewedthereforeoughttobeofinteresttophysiciststhatare orcooperators)pouramountsofafastdiffusivechemicalinto
:
v involvedintheinterdisciplinaryresearchofcomplexsystems, theenvironment. Thelatterhasthestatusofapublicgoodas
i buthopefullyalsotoexpertsongametheory,sociology,com- itisbeneficialalsoforthosethatdonotproduceit(free-riders
X
puterscience, ecology,as wellas evolutionand modelingof or defectors). For N = 3 such a setup is depictedschemat-
r socio-technicalsystemsingeneral. Groupinteractionsarein- ically in Fig. 1. The metabolic expensesstemming from the
a
deedinseparablylinkedwithourincreasinglyinterconnected productioncostofthepublicgoodaregivenbythecostfunc-
world,andthuslieattheinterfaceofmanydifferentfieldsof tion α(ρC), while the individual benefit for each of the N
research. We note that there are many studies which are not microbes is β(ρC), where 0 ≤ ρC ≤ 1 is the fraction of
coveredinthisreview. However,wehavetriedtomakeitas producers. Eachnon-producing(D-phenotype)microbethus
comprehensiveaspossibletofacilitatefurtherresearch. receives the payoff PD = β(ρC), while each microbe that
doesproduce(C-phenotype)bearstheadditionalcost,sothat
Wedescribeourmotivation,notation,andotherelementary
itsnetbenefitisP =β(ρ )−α(ρ ).
conceptsinsubsection1.1,followedby“inanutshell”survey C C C
ofresultsonwell-mixedpopulationsinsubsection1.2,andan ForN =2andthesimplechoiceofα(ρC)=c/(2ρC)and
overviewoftheorganisationofthereviewinsubsection1.3. β(ρC)=bθ(ρC)[whereθ(x)isthestepfunction],werecover
twowell-knowngamesthataregovernedbypairwiseinterac-
tions.Namelytheprisoner’sdilemmafor2b>c>b>0and
thesnowdriftgameforb > c > 0,assummarizedinTableI.
∗Electronicaddress:[email protected] When N ≥ 3, however, the problembecomes that of group
2
C D
C R=b−(c/2) S =b−c
C D T =b P =0
C
D TABLE I: Payoff matrix of 2-player games if α = c/(2ρC) and
β = θ(ρC)b. For2b >c>b>0wehavetheprisoner’sdilemma,
andforb>c>0thesnowdriftgame.
FIG.1: SchematicconfigurationofN = 3microbes,whereafrac-
tionρC = 2/3arecooperators(producers) and ρD = 1/3arede- producersρC. Thisistherealmofevolutionarygamedynam-
fectors(free-riders). Forthemostpopularchoiceofbenefitandcost ics,whichimplementsDarwiniannaturalselectionofpheno-
functions, β(ρC) = rρC (r > 1)andα(ρC) = 1respectively,in- types in populationsunder frequency-dependentfitness con-
dividual payoffsarePC(2/3) = 2r/3−1andPD(2/3) = 2r/3. ditions[1,2,13,14],aswellasinrelatedthoughnon-genetic
AnexplicitcomputationofPC(ρC)(1/3 ≤ ρC ≤ 1)andPD(ρC) socialand economicsystems. In the latter, “sociallearning”
(0 ≤ ρC ≤ 2/3)revealsthattheycannotbegeneratedbymeansof assumptionsmayleadtoaverysimilarevolutionarydynamics
pairwise interactions, thus illustrating the inherent irreducibility of
providedsimple assumptionsconcerningthe cognitivecapa-
groupinteractions.
bilitiesofagentsareaccepted.
A calculation that invokes a standard well-mixed popula-
tionsetting(seebelowand[3,15,16])leadstothedifferential
interactions. We see that, under the sensible assumption of equation for the expected value x = hρCi of the fraction of
producers
additivityofindividuallyobtainedpayoffs,thedefinedpayoff
structurecannotbereproducedbymeansofpairwiseinterac-
x˙ =x(1−x)[W (x)−W (x)] , (1)
tions (see caption of Fig. 1 for details). This example also C D
suggeststhat,providedthebenefitandcostfunctionscouldbe
whereW (x)istheaveragepayoffpereitheracooperative
C,D
inferredfromexperiments,theexperimentercouldpotentially
ordefectiveindividual. Thisisthereplicatorequation,which
determinewhetheracolonyisgovernedbypairorgroupinter-
isnonlinearalreadyforlinearpayoffs. Dependingfurther on
actions. Indeed,itwasrecentlynoted[3]thattheoversimpli-
the additionalproperties of α(x) and β(x), its analysis may
fyingrestrictionofpairwisesocialinteractionshasdominated
thusbeallbutstraightforward.
theinterpretationofmanybiologicaldatathatwouldlikelybe
Theorems relate the asymptotic states of the replicator
muchbetterinterpretedintermsofgroupinteractions.
equation with Nash’s stability criteria, and Motro’s [11] re-
ThepayoffsPD = β(ρC)andPC =β(ρC)−α(ρC)have sultsonthepublicgoodsgameinturntranslateintothechar-
ageneralpublicgoodsgamestructureinthatcooperatorsbare acterizationoftheevolutionarystablestatesforourmicrobial
anadditionalcostbesidesthebenefitsthatarecommontoboth
population. In particular, for constant α and concave bene-
strategies. Theanalysisofdecisionmakingbya“rationalmi- fitfunctionsβ(x), awell-mixedcolonyofmixedphenotypic
crobe”thusfallswithintherealmofclassicalgametheory. In
composition is evolutionary stable. Notably, in addition to
this framework, for a constant individual production cost α the replicator dynamics, best-response and related learning
andan arbitraryconcavebenefitfunction β(ρC), Motro[11] dynamicscan also be formulatedfor the evolutionof x, and
showed that even values from within the 0 < ρC < 1 in- indeedtheycan be ofmuchrelevancein specific contextsof
tervalarestableNashequilibria. Undercertainconditionsto agentbasedmodeling.
be met by the benefit and cost functions (β and α), there is Atthispoint,itisinformativetospellouttheoperationalas-
thus no “tragedy of the commons” [12]. This may be wel- sumptionsthattraditionallyunderliethewell-mixedapproxi-
comed news for the liberal (“invisible hand”) supporters of
mation[3]. Inparticular,itisassumedthattheN−1individ-
publicgoodssystems: The tragedyof the commonsis ratio- ualsthatinteractwiththefocalplayerarerandomlysampled
nally avoidable even without the “cognitive” or “normative”
from an infinite population of cooperators and defectors, so
capacities required for the existence of additional strategies. thattheprobabilityofinteractingwith j cooperatorsisgiven
Nevertheless,the“tragedyofthecommons”doesoccurinthe by f (x) = CN−1xj(1−x)N−1−j, where x (1−x) is the
j j
majorityofothercases(e.g.,linearbenefitfunctionβ),where average fraction of cooperators (defectors). Other formally
noproductionofthepublicgoodistheonlyrationalindividual
moresophisticatedsettingscanalsobeofinterest.Wereferto
choice. [16]foronethatallowstoconsideracontinuumofstrategies
parameterizedbytheamountofpublicgoodproducedperin-
dividual,andto[17]fora“grandcanonical”treatmentwhere
1.2.Evolutionarygamedynamics thegroupsizeN isconsideredarandomvariable.
Note that the payoffs of the focal individual are collected
Turningback to microbialpopulations,underthe assump- from group configurations that are statistically uncorrelated.
tion that the reproductive power of each microbe is propor- Moreover,inordertoimplementtheassumptionthattheindi-
tional to the net metabolic benefit enjoyed, one arrives at a vidualreproductivepoweris proportionalto the netbenefits,
formal description for the time evolution of the fraction of whileoperationallykeepingaconstantpopulationsizeN,one
3
can(amongotheroptions,likeusingthestochasticbirth/death 1.3.Organisationofthereview
Moranprocess)useareplicator-likeruleinwhich,inthenext
time step, the focal player imitates the current strategy of a The remainder of this review is organised as follows. In
randomlychosenagentfromthegroup,withaprobabilityde- Sec.2wesurveytheimplementationofthepublicgoodsgame
pendingonthepayoffdifference.Itisworthemphasisingthat on lattices. We focus on recent studies investigating the ef-
thebasicunderlyingassumptionhereishomogeneity,sothat fectsoflatticestructureontheemergenceofcooperation. In
individualsdo notdifferentiateorassort, asboth(i) the pay- addition,wereviewboththeeffectsofheterogeneityinthedy-
off’sarecollectedfrom,and(ii)thecompetitivereproduction namicalingredientsofthepublicgoodsgameaswellastheef-
isagainstconfigurationssampledfromanunbiased(uncorre- fectsofstrategiccomplexityontheevolutionofcooperation.
lated)strategicdistributionfj(x). InSec. 3we focusonstructuresthataremorerepresentative
If departing from the assumptions of well-mixed popula- forhumansocieties. Inthisframeworkwewillrevisitthefor-
tions,however,severalissuesopenup. Tobeginwith: mulationofthepublicgoodsgameoncomplexnetworksand
showhowsocialdiversitypromotescooperation. Inaddition,
• Which criterion determines how group configurations we will surveyhow publicgoodsgameson networkscan be
are sampled to provide instantaneous payoff to focal formulatedbymeansofabipartiterepresentation. Thelatter
players? Is the group size N also a random variable includesbothsocial as wellas groupstructure, thusopening
inthatsampling? thepathtowardsamoreaccuratestudyofgroupinteractions
inlargesocialsystems. WeconcludeSec.3byreviewingdif-
• Whatkindofpopulationsamplingisusedtoimplement ferent networked structures in which the public goods game
replicating competition among strategies? In other has been implemented, most notably modularand multiplex
words, who imitates whom? Should members of all networks,aswellas populationsofmobileagents. InSec. 4
groupsbe potential imitators (or should potentially be wereviewadvancesonstructuredpopulationswherethecon-
imitated)? Orshouldjustafractionofthem(forexam- nectionscoevolvewiththeevolutionarydynamics,andwhere
plethoseinasmallerspatialneighborhoodofthefocal thus the topology of interactions changes depending on the
player)qualifyassuch? payoffs and strategies in the population. We round off the
review by discussing the main perspectives, challenges and
Thereareseveralpossibleanswerstobothgroupsofques- openquestionsinSec.5,andbysummarisingtheconclusions
tions, and they depend significantly on the particular prob- inSec.6.
lem one wishes to address. For example, for a quantitative
modeling of a yeast colony of invertase-producersand non-
producing cells, the answers should be based on considera- 2.LATTICES
tionsinvolvingcharacteristictimescalesofmanybiochemical
processesandthespatialmicrobialarrangementsthataretyp- Beyond patch-structured populations where under certain
ical amongst the measured samples of the microbial colony, updatingrulesthespatialstructurehasnoeffectontheevolu-
to namebuta fewpotentiallyimportantissues. Onthe other tionofaltruism[21–23],latticesrepresentverysimpletopolo-
hand, in systems where best-response or other non-imitative gies, which enjoy remarkable popularityin game theoretical
evolutionaryrulesareconsidered,onlythefirstgroupofques- models[18, 24, 25]. Despite theirdissimilarity to actual so-
tionswouldlikelybeofrelevance. cialnetworks[26],theyprovideaveryusefulentrypointfor
Recent research concerning public goods games on struc- exploring the consequences of structure on the evolution of
tured populations is in general very indirectly, if at all, re- cooperation. Moreover,therearealsorealisticsystems,espe-
lated to a particular experimental setup. Instead, it is of an ciallyinbiologyandecology,wherethecompetitionbetween
exploratorynatureoverdifferentpotentiallyrelevanttheoret- the speciescanbe representedadequatelybymeansofa lat-
icalissuesthatcanbeeitherformulatedorunderstoodaslat- tice [27, 28]. In general, lattices can be regardedas an even
tice or network effects. A quite commonground motivation fieldforallcompetingstrategieswherethepossibilityofnet-
is the search for analogs of network reciprocity [18]. Is the workreciprocityisgiven[18].Furthermore,astherearemany
resilience of cooperative clusters against invading defectors differenttypesoflattices(seeFig.2fordetails),wecanfocus
on networks and lattices enough to effectively work against onveryspecificpropertiesofgroupinteractionsandtestwhat
the mean-field tendencies [19]? More generally, what are istheirroleintheevolutionaryprocess.
the effects of structure in a population when confluent with The basic setup fora public goodsgamewith cooperators
knownsourcesofpublicgoodssustainability,suchaspunish- anddefectorsasthetwocompetingstrategiesonalatticecan
mentorreward? Are these synergisticconfluences? Indeed, be described as follows. Initially, N ∝ L2 players are ar-
theinterestofreviewedresearchgoesfarbeyonditsrelevance rangedintooverlappinggroupsofsizeGsuchthateveryoneis
to a specific experiment. Evolutionarygame dynamics is of surroundedbyitsk =G−1neighborsandbelongstog =G
fundamentalinterest to the makingof interdisciplinarycom- differentgroups,where L is the linearsystem size and k the
plex systems science, encompassing biological, economical degree (or coordination number) of the lattice. Each player
as well as social sciences, and from this wider perspective, on site i is designated either as a cooperator (C) s = 1 or
i
the universalfeaturesof dynamicalprocessesofgroupinter- defector(D)s = 0withequalprobability. Cooperatorscon-
i
actionsarestillratherunexplored. tributeafixedamounta, normallyconsideredbeingequalto
4
groups. These elementary steps are repeated consecutively,
(a) (b) whereby each full Monte Carlo step (MCS) gives a chance
foreveryplayertoenforceitsstrategyontooneoftheneigh-
bors once on average. Alternatively, synchronous updating
can also be applied so that all the players play and update
theirstrategiessimultaneously,butthelattercanleadtospuri-
ousresults,especiallyinthedeterministicK → 0limit[29].
Likewise,asanticipatedabove,thereareseveralwaysofhow
to determinewhen a strategytransferoughtto occur, yetfor
latticestheFermifunctioncanbeconsideredstandardasitcan
(c) (d) easily recover both the deterministic as well as the stochas-
tic limit. The average fraction of cooperators ρ and defec-
C
torsρ inthepopulationmustbedeterminedinthestationary
D
state. Dependingon the actual conditions, such as the prox-
imity to extinction points and the typical size of the emerg-
ingspatialpatterns, the linearsystem size hasto bebetween
L = 200 and 1600 in order to avoid accidental extinction,
andtherelaxationtimehastoexceedanywherebetween 104
and106MCStoensurethestationarystateisreached.Excep-
tionstothisbasicrequirementsarenotuncommon,especially
FIG. 2: Schematic presentation of different types of lattices. On
whenconsideringmorethantwocompetingstrategies,aswe
thesquarelattice(a)eachplayerhasfourimmediateneighbors,thus
willemphasiseattheendofthissection.
forminggroupsofsizeG=5,whileonthehoneycomblattice(b)it
hasthree,thusG = 4. InbothcasestheclusteringcoefficientC is
zero.Yetthemembershipofunconnectedplayersinthesamegroups
introduceseffectivelinksbetweenthem,whichmayevokebehavior 2.1.Groupversuspairwiseinteractions
that is characteristic for lattices with closed triplets [20]. On the
otherhand,thekagome´ (c)andthetriangularlattice(d)bothfeature
For games governed by pairwise interactions, such as the
percolatingoverlappingtriangles,whichmakesthemlesssusceptible
prisoner’sdilemmagame,thedependenceofthecriticaltemp-
toeffectsintroducedbygroupinteractions. Thekagome´ latticehas
G = 5andC = 1/3, whilethetriangularlatticehasC = 2/5and tationtodefectbconKisdeterminedbythepresenceofover-
G=7. lappingtriangles. Notably,herebc isthetemptationtodefect
babovewhichcooperatorsareunabletosurvive(seealsoTa-
ble I). If an interaction network lacks overlapping triangles,
andaccordinglyhastheclusteringcoefficientC =0,asisthe
1withoutlossofgenerality,tothecommonpoolwhiledefec-
caseforthesquareandthehoneycomblattice, thenthereex-
tors contributenothing. Finally, the sum of all contributions
ists anintermediateK atwhichb is maximal. On the other
in each group is multiplied by the synergy factor r and the c
hand,ifoverlappingtrianglespercolate,asisthecasefor the
resultingpublicgoodsaredistributedequallyamongstall the
triangular and the kagome´ lattice (see Fig. 2), then the de-
groupmembers.Thepayoffofplayeriineverygroupgis
terministic limit K → 0 is optimal for the evolution of co-
Pig =r j∈Ggsja −sia=rNGCga −sia, (2) odpifeferaretinotnly[,3h0ig,h3l1ig].htiTnhgethsaptagtiraoluppuibnltiecragcotoiodnssgaaremmeobreehathvaens
P justthesumofthecorrespondingnumberofpairwiseinterac-
where Ng is the numberof cooperatorsin group g. The net tions. As demonstratedin [20], groupinteractionsintroduce
C
payoffitherebyacquiresisthesumofthepayoffsreceivedin effectivelinksbetweenplayersthatarenotdirectlyconnected
allthegroupsitparticipatesin: Pi = gPig. bymeansoftheinteractionnetwork. Topologicaldifferences
Themicroscopicdynamicsinvolvesthefollowingelemen- betweenlatticesthereforebecomevoid,andthedeterministic
P
tarysteps. First,arandomlyselectedplayeriplaysthepublic limit K → 0 becomes optimal for the evolution of cooper-
goods game as a member of all the g = 1,...,G groups. ation regardless of the type of the interaction network. Re-
Next,playerichoosesoneofitsneighborsatrandom,andthe sults for pairwise and group interactions are summarised in
chosenplayer j also acquiresits payoffPj in the same way. Fig.3. Thisimpliesthatbygroupinteractionstheuncertainty
Finally, player i enforces its strategy si onto player j with bystrategyadoptionsplaysatmostasiderole,asitdoesnot
someprobabilitydeterminedbytheirpayoffdifference. One influencetheoutcomeoftheevolutionaryprocessinaquali-
ofthepossiblechoicesforthisupdateprobabilityistheFermi tativeway.
function, The fact that membership in the same groups effectively
connectsplayersthat are not linked by means of directpair-
1
Π(si →sj)= , (3) wiselinksnaturallybringsforththegroupsizeasakeysystem
1+exp[(P −P )/GK]
j i parameter.In[32]itwasshownthatincreasingthegroupsize
whereK quantifiestheuncertaintybystrategyadoptionsand does not necessarily lead to mean-field behavior, as is tradi-
G normalises it with respect to the number and size of the tionallyobservedforgamesgovernedbypairwiseinteractions
5
0.16
(a)

square
0.12 honeycomb
triangular
G
G -r ) / c 0.08
(
0.04

0
0 0.2 0.4 0.6 0.8 1
K / G FIG.4:Characteristicsnapshotofevolutionaryprocessforsmall(up,
G = 5) and large(down, G = 301) groups. Cooperators are de-
0.4
pictedbluewhiledefectorsaredepictedred. Forsmallgroups, the
(b) evolution of strategiesproceeds withthecharacteristic propagation
square
0.3 honeycomb of the fronts of the more successful strategy (in this case D) until
triangular eventuallythemaladaptivestrategyCgoesextinct.Forlargegroups,
G
( G - r ) / c 0.2 ddheeocfwereecvatoesrers,s,,thetheveecinoropifpaeyrroaifstfivsveuedrcydluessnmtleyarlsbl.eacrSeotmsiltler,osanvsgetrahynecdodcmeannpseiottyiuttiovpfeer,dfaeonfrdemctthothuress
theycaninvadetheseeminglyinvinciblecooperativeclusters. Such
0.1
analternatingtimeevolutioniscompletely atypical andwas previ-
ouslyassociatedwithcooperatorsonly.
0
0 0.2 0.4 0.6 0.8 1
K / G
forthefinaloutcomeofthepublicgoodsgame.
FIG.3: Borders between themixed C +D and thepure D phase
independence onthenormalizeduncertaintybystrategyadoptions
K/G, as obtained on different lattices for pairwise (a) and group 2.2.Heterogeneitiesinthedynamics
(b)interactions. Verticalaxisdepictsthedefectiontemptationrate,
i.e., the higher its value the smaller the value of r that still allows
Group interactions on structured populations are thus dif-
the survival of at least some cooperators. By pairwise interactions
(G=2),theabsenceofoverlappingtrianglesiscrucial(squareand ferent from the corresponding sum of pairwise interactions.
honeycomb lattice), as then there exists and intermediate value of Consequentlynotjustthegroupsize,butalsothedistribution
K atwhichtheevolutionofcooperation isoptimallypromoted. If of payoff within the groups becomes important. As shown
triangles do percolate (triangular lattice), the K → 0 limit is op- in [35, 36], heterogeneous payoff distributions do promote
timal. This behavior is characteristic for all social dilemmas that the evolution of cooperation in the public goods game, yet
arebasedonpairwisegames,mostfamousexamplesbeingthepris- unlikeasgamesgovernedby pairwiseinteractions[37], uni-
oner’s dilemma and the snowdrift game (see Fig. 3 and 5 in[30]).
formdistributionsoutperformthe moreheterogeneousexpo-
Conversely,whengroupinteractionsareconsidered(seeFig.2forG
nential and power law distributions. The setup may also be
values)thetopologicaldifferencesbetweenthelatticesbecomevoid.
reversedinthatnotthepayoffsbutratherthecontributionsto
Accordingly,thedeterministicK →0limitisoptimalregardlessof
the groupsare heterogeneous. In [38, 39] it was shown that
thetopologyofthehostlattice[20].
correlatingthe contributionswith the levelof cooperationin
each group markedly promotes prosocial behavior, although
the mechanism may fail to deliverthe same results on com-
plexinteractionnetworkswherethesizeofgroupsisnotuni-
[33], but rather that public cooperation may be additionally form. Conceptuallysimilarstudieswithlikewisesimilarcon-
promoted by means of enhanced spatial reciprocity that sets clusionsarealso[40–42],althoughtheyrelyondifferencesin
in forverylarge groupswhereindividualshave the opportu- thedegreeofeachplayertodeterminepayoffallocation. The
nitytocollectpayoffsseparatelyfromtheirdirectopponents. latter will be reviewed in Sec. 3 where the focus is on pub-
However, very large groups also offer very large benefits to lic goods games that are staged on complex networks. An-
invading defectors, especially if they are rare, and it is this otherpossibilitytointroduceheterogeneitytothespatialpub-
back door that limits the success of large groups to sustain licgoodsgameisbymeansofdifferentteachingactivitiesof
cooperation and puts a lid on the pure number-in-the-group players,aswasdonein[43]. Inthiscase,however,theresults
effect[34]. Figure4featurestwocharacteristicsnapshotsand aresimilartothosereportedpreviouslyforgamesgovernedby
furtherdetailstothateffect. Itisalsoworthemphasisingthat pairwiseinteractions[44],inthatthereexistsanoptimalinter-
thejointmembershipinlargegroupswillindirectlylinkvast mediatedensityofhighlyactiveplayersatwhichcooperation
numbersofplayers,thusrenderinglocalaswellglobalstruc- thrivesbest.
tural properties of interaction networks practically irrelevant Asidefromheterogeneousdistributionsofpayoffsandini-
6
1.0
0.8
0.6
S)
B (
0.4
0.2
0.0
0 1 2 3 4 5
S / G
FIG. 6: Time evolution of strategies on a square lattice having
FIG.5:DifferentrealizationsofthepublicbenefitfunctionB(S)= G = 25,forthecriticalmassM = 2(up)andM = 17(down)at
1 ,whereT representsthethresholdvalueand βthe r/G=0.6[46].Defectorsaremarkedred,whilecooperatorsarede-
1+exp[−β(Si−T)]
steepness of the function [45]. For β = 0 the benefit function is pictedblueiftheirinitialcontributionsareexaltedorwhiteiftheygo
a constant equalling 0.5, in which case the produced public goods towaste.Accordingly,cooperatorscanbedesignatedasbeingeither
are insensitive to the efforts of group members. Conversely, for “active”or“inactive”.WhenM islowallcooperatorsareactive,yet
β =+∞thebenefitfunctionbecomesstep-likesothatgroupmem- theydon’thaveastrongincentivetoaggregatebecauseanincrease
berscanenjoythebenefitsofcollaborativeeffortsvia r onlyifthe intheirdensitywillnotelevatetheirfitness.Hence,onlyamoderate
total amount of contributions in the group S exceeds a threshold. fractionofcooperatorscoexistswiththeprevailingdefectorsinthe
Otherwise,theyobtainnothing. Thedepictedcurveswereobtained stationary state. If the critical mass is neither small nor large, the
forT =2.5andβ=0.1(dottedred),1(dashedgreen)and10(solid status of cooperators varies depending on theirlocation on the lat-
blue). tice:thereareplaceswheretheirlocaldensityexceedsthethreshold
andtheycanprevailagainstdefectors. Therearealsoplaceswhere
thecooperators areinactivebecause theirdensity islocallyinsuffi-
cient and looseagainst defectors. Thesurviving domains of active
tial investments,groupinteractionsare also amenableto dif- cooperatorsstartspreading,ultimatelyrisingtoneardominance.
ferent public benefit functions, as demonstrated in Fig. 5.
While traditionally it is assumed that the productionof pub-
licgoodsislinearlydependentonthenumberofcooperators
within each group, it is also possible to use more complex behaviorofothersinthegroup. Correlatingthecontributions
benefitfunctions.Theideahasbeenexploredalreadyinwell- witheitherthelevelofcooperationineachgroup[38,39]or
mixed populations [15, 45, 47, 48], and in structured popu- thedegreeofplayers[40–42]canthusbeseennotjustashet-
lationsthepossibilitiesaremore. Oneistointroducea criti- erogeneouscontributing, but also as conditionalcooperation
calmassof cooperatorsthathaveto bepresentin a groupin [52]. Anexplicitformofthiswasstudiedin[53],wherecon-
order for the collective benefits of group membership to be ditional cooperator of the type C only cooperate provided
j
harvested[46]. If the criticalmass is notreached, the initial there are at least j other cooperators in the group. It was
contributionscaneithergotowaste,ortheycanalsobedepre- shown that such strategies are the undisputed victors of the
ciatedbyapplyingasmallermultiplicationfactorinthatpar- evolutionaryprocess,evenatverylowsynergyfactors. Snap-
ticulargroup[49,50]. Althoughsuchmodelsinevitablyintro- shotsofthespatialgridrevealthespontaneousemergenceof
duceheterogeneityinthedistributionofpayoffs[51],theycan convexisolated “bubbles” of defectorsthat are contained by
also lead to interesting insightsthat go beyondad hoc intro- inactive conditional cooperators. While the latter will pre-
ducedheterogeneity.In[46],forexample,itwasshownthata dominantlycooperatewith the bulkof activeconditionalco-
moderatefractionofcooperatorscanprevailevenatverylow operators,theywillcertainlydefectintheoppositedirection,
multiplicationfactorsif the criticalmass M is minimal. For wheretherearedefectors.Consequently,defectorscannotex-
largermultiplicationfactors,however,thelevelofcooperation ploit conditional cooperators, which leads to a gradual but
was found to be the highest at an intermediate value of M. unavoidable shrinkage of the defector quarantines. Notably,
Figure 6 features two characteristic scenarios. Notably, the conditionalstrategies introducedin this way have no impact
usageofnonlinearbenefitfunctionsisuniquetogroupinter- onthemixedstateinunstructuredpopulationsandarethusof
actions,andingeneralitworksinfavorofpubliccooperation interestonlyonstructuredpopulations.
[49,50]. Apart from conditional strategies, the impact of loners,
sometimes referred to as volunteers, has also been studied
in the realm of the spatial public goods game [57]. While
2.3.Strategiccomplexity inwell-mixedpopulationsvolunteeringleadstocyclicdomi-
nancebetweenthethreecompetingstrategies[58,59],onlat-
Besidesheterogeneityinpayoffsandnonlinearityinpublic tices the complexityofthe emergingspatial patternsenables
benefitfunctions,introducingstrategiccomplexityisanother the observationof phase transitionsbetween one-, two-, and
way of bringingthe public goodsgame closer to reality. As three-strategystates[57],whichfalleitherinthedirectedper-
notedabove,thewillingnesstocooperatemaydependonthe colationuniversalityclass [60] orshow interesting analogies
7
populationsthat is due to pattern formation. The aptness of
structured populationsfor explaining the stability and effec-
tiveness of punishment can in fact be upgraded further by
meansofcoevolution[69],aswewillreviewinSec.4.Onthe
contrary,whileexperimentsattesttotheeffectivenessof both
punishment [70] and reward [71] for elevating collaborative
efforts,thestabilityofsuchactionsinwell-mixedpopulations
isratherelusive,asreviewedcomprehensivelyin[72].
2.4.Statisticalphysics:avoidingpitfalls
FIG.7:Indirectterritorialbattlebetweenpurecooperators(blue)and
peer-punishers (green) (upper row), and between pure cooperators Beforeconcludingthissectionanddevotingourattentionto
(blue)andrewardingcooperators(lightgray)(lowerrow). Intheup- morecomplexinteractionnetworksandcoevolutionarymod-
perrow pure cooperators andpeer-punishers formisolatedclusters els,itisimportanttoemphasizedifficultiesandpitfallsthatare
thatcompeteagainstdefectors(red)forspaceonthesquare lattice.
frequentlyassociatedwithsimulationsofthreeormorecom-
Since peer-punishers are more successful in competing against de-
peting, possibly cyclically dominating, strategies on struc-
fectorsthanpurecooperators(alsofrequentlyreferredto assecond-
turedpopulations.Heremethodsofstatisticalphysics,inpar-
orderfree-riders[54]),eventuallythelatterdieouttoaleaveamixed
ticular that of Monte Carlo simulations [61, 73, 74], are in-
two-strategy phase (peer-punishers and defectors) as a stationary
state (see [55] for further details). In the lower row defectors are valuablefora correcttreatment. Foremost,it isimportant to
quicktoclaimsupremacyonthelattice,yetpureandrewardingco- chooseasufficientlylargesystemsizeandtouselongenough
operatorsbothformisolatedcompactclusterstotryandpreventthis. relaxation times. If these conditions are not met the simu-
Whilerewarding cooperators can outperform defectors, purecoop- lationscanyieldincorrectone-and/ortwo-strategysolutions
eratorscannot. Accordingly,thelatterdieout,leavingqualitatively that are unstable against the introduction of a group of mu-
thesameoutcomeasdepictedintheupperrow(see[56]forfurther tants. Forexample,thehomogeneousphaseofcooperatorsor
details).
pool-punisherscanbeinvadedcompletelybytheoffspringof
a single defector inserted into the system at sufficiently low
valuesofr [64]. Atthesame time, defectorscanbeinvaded
by a single group of pool-punishers (or cooperators) if ini-
toIsing-typemodels[61]. tially they form a sufficiently large compactcluster. In such
Thecomplexityofsolutionsinspatialpublicgoodsgames casesthecompetitionbetweentwo homogeneousphasescan
with three or more competing strategies is indeed fascinat- becharacterisedbytheaveragevelocityoftheinvasionfronts
ing, which can be corroborated further by results reported separatingthetwo spatialsolutions. Note thata system with
recently for peer-punishment, [55, 62, 63], pool-punishment three(ormore)strategieshasalargenumberofpossiblesolu-
[64, 65], the competition between both [66], and for reward tionsbecauseallthesolutionsofeachsubsystem(comprising
[56]. In general, the complexity is largely due to the spon- only a subset of all the originalstrategies) are also solutions
taneous emergence of cycling dominance between the com- of the whole system [24]. In such situations the most sta-
peting strategies, which can manifest in strikingly different blesolutioncanbededucedbyperformingasystematiccheck
ways. By pool-punishment,forexample, if the value of r is of stability between all the possible pairs of subsystem so-
withinanappropriaterange[64],thepool-punisherscanout- lutions that are separated by an interface in the spatial sys-
performdefectorswhointurnoutperformcooperatorswhoin tem. Fortunately, this analysis can be performed simultane-
turn outperform the pool-punishers, thus closing the loop of ouslyifonechoosesasuitablepatchystructureofsubsystem
dominance. Interestingly, in the absence of defectors peer- solutionswhereallrelevantinterfacesarepresent. Thewhole
punishersandpurecooperatorsreceivethe same payoff,and gridisthendividedintoseveraldomainswithdifferentinitial
hencetheirevolutionbecomesequivalentto thatofthevoter strategydistributionscontainingone, two orthree strategies.
model[61]. Notablyhowever,thelogarithmicallyslowcoars- Moreover,thestrategyadoptionsacrosstheinterfacesare ini-
ening can be effectively accelerated by adding defectors via tiallyforbiddenforasufficientlylonginitialisationperiod.By
rarerandommutations[63]. Similarlycomplexsolutionscan using this approach one can avoid the difficulties associated
be observed for rewarding [56]. There, if rewards are too either with the fast transients from a random initial state or
high, defectors can survive by means of cyclic dominance, with the differenttime scales that characterizethe formation
but in special parameter regions rewarding cooperators can ofpossiblesubsystemsolutions.Itiseasytoseethatarandom
prevail over cooperators through an indirect territorial bat- initialstatemaynotnecessarilyofferequalchancesforevery
tle with defectors, qualitatively identically as reported for solution to emerge. Only if the system size is large enough
peer-punishment [55]. Figure 7 features two sequences of the solutions can form locally, and the most stable one can
snapshots that demonstrate both evolutionaryscenarios. Al- subsequentlyinvadethewholesystem. Atsmallsystemsizes,
together, these results indicate that second-order free-riding however,onlythosesolutionswhosecharacteristicformation
[54,67,68],referringtocooperatorswhorefraineitherfrom timesareshortenoughcanevolve.Theseminalworksconsid-
punishingorrewarding,findsanaturalsolutiononstructured eringpunishmenton structuredpopulations[75, 76], as well
8
comepossibletodeterminetheactualcontactpatternsacross
varioussocio-technicalnetworks[82–84]. Thesestudieshave
shown that the degree distribution P(k) of most real-world
networks is highly skewed, and that most of the time it fol-
lows a power law P(k) ∼ k−γ [85]. The heterogeneity of
degrees leads to social diversity, which has important con-
sequences for the evolution of cooperation. Although many
seminal works concerning evolutionary games on networks
have focused on pairwise interactions [24, 25], games gov-
ernedbygroupinteractionsarerapidlygainingonpopularity.
FIG. 8: When the public goods game is staged on a complex net-
work,cooperatorscaneitherbareafixedcostpergamez(leftpanel),
orthiscostcanbenormalizedwiththenumberofinteractions,i.e.,
z/(ki+1),whereki isthenumberofneighborsofeachparticular 3.1.Socialheterogeneity
cooperatori. Inthelattercase,oneeffectivelyrecoversafixedcost
per individual (right panel). Thisdistinction has significant conse-
Due to the overwhelming evidence indicating that social
quences for the evolution of public cooperation on complex inter-
heterogeneitypromotesthe evolution of cooperationin pair-
action networks, as originally reported in [81]. Only if the cost is
normalizedwiththenumberofneighborsdoessocialheterogeneity wisesocialdilemmagames[86–90],itisnaturaltoaskwhat
significantlypromotetheevolutionofpubliccooperation. isitsimpactongamesgovernedbygroupinteractions.Santos
etal.[81]havethereforereformulatedthepublicgoodsgame
tobestagedoncomplexnetworks.Everyplayeriplaysk +1
i
publicgoodsgames,asdescribedbeforeforlattices,onlythat
as a most recently anti-social punishment[77], could poten- herethedegreek ofeveryplayercanbeverydifferent.Since
i
tially benefitfromsuch an approach,as it couldrevealaddi- thegroupswillthusalsohavedifferentsize, cooperatorscan
tionalstable solutionsbeyondthe well-mixedapproximation contributeeithera fixed amountpergame c = z, ora fixed
i
[58,78–80]. amountpermemberofthegroupc =z/(k +1),asdepicted
i i
in Fig. 8. Identicalto the traditionalsetup, the contributions
withindifferentgroupsaremultipliedby r andaccumulated.
3.COMPLEXNETWORKS However,thepayoffofanotherwiseidenticalplayerisnotthe
sameforthetwodifferentoptions. Bydefiningtheadjacency
With the maturity of methods of statistical physics, the matrixofthenetworkasA =1whenindividualsiandjare
ij
availability of vast amounts of digitised data, and the com- connectedandA =0otherwise,weobtainthefollowingnet
ij
putational capabilities to process them efficiently, it has be- benefitP forbothversionsofthegame
i
P = N A r( Nl=1Ajlslcl+sjcj) + r( Nj=1Aijsjcj +sici) −(k +1)s c , (4)
i ij i i i
k +1 k +1
j=1 P j P i
X
where however the precise value of c is set depending on the critical value obtained on lattices. Moreover, heteroge-
i
whethercooperatorsbareafixedcostpergameorafixedcost neousnetworks enable complete cooperatordominancewell
perplayer.Aftereachfullroundofthegameallplayersdecide before cooperative behavior even emerges on regular net-
synchronouslywhetherornottheywillchangetheirstrategy. works. Phenomenologically,thepromotionofcooperationis
This is done by following the finite population analogue of due to the diversity of investments, which is a direct conse-
the replicatorrule. An individual i with payoffP randomly quence of the heterogeneity of the underlying network. As
i
selects one neighbor j amongst its k contacts. If P ≥ P cooperatorspay a cost that dependson their degree, namely
i i j
nothingchanges, butif P < P playeri adoptsthe strategy c/(k + 1), the fitness landscape becomes very rich and di-
i j
ofthemoresuccessfulneighborj with aprobabilitythatde- verse–afeatureabsentforlattices. Infact,forasinglepublic
pendsonthedifference∆P =P −P . goods game the difference between the payoff of a cooper-
j i
ator and defector is no longer proportional to c, but rather
Resultspresentedin[81]showthatheterogeneousnetworks
inversely proportional to the number of games each player
promotetheevolutionofpubliccooperation.Yetthisispartic-
plays. This gives and evolutionaryadvantageto cooperative
ularly true when cooperatorspay a fixed cost perindividual.
hubs,i.e.,playerswithahighdegree.
Cooperationisthenviablealreadyatη = r/(hki+1) = 0.3
(normalizedmultiplicationfactor), which is less than half of
9
TheseminalstudybySantosetal.[81]motivatedmanyoth- claredalreadyintheseminalpaperbySantosetal.[81].
ers to study the evolution of public cooperation on complex
networks.Asevidencedbyprecedingworksconsideringpair-
wise social dilemmas, the degree distributionis not the only 3.2.Accountingforgroupstructure:bipartitegraphs
propertythat affects the outcome of an evolutionaryprocess
[91–95]. Other properties, like the average path length, the Theimplementationofthepublicgoodsgameasintroduced
clusteringcoefficient,orthepresenceofcorrelationsamongst in[81]makesanimportantassumptionregardingthecompo-
high-degreenodescanbejustasimportant[24].RongandWu sitionofgroupsinwhichthegamestakeplace. Thisassump-
[96] have explored how the presence of degree correlations tionreliesonthefactthateachgroupisdefinedsolelyonthe
affectsthe evolutionof public cooperationon scale-freenet- basis of connectionsmakingup the complexinteractionnet-
works. Theyfoundthatassortativenetworks–thoseinwhich work.However,itisratherunrealisticthatthisdefinitionholds
alikenodesarelikelyconnectedtoeachother–actdetrimental inrealsocialnetworks,suchasforexamplecollaborationnet-
asheterogeneitynolongerconfersanaturaladvantagetoco- works[103].Figure9featuresaschematicpresentationofthis
operativehubs. Conversely,ifplayerswithdissimilardegree situation.Supposeweknowtheactualinteractionstructureof
aremorelikelyconnected,theonsetofcooperationoccursat asystemcomposedof6individualsperformingcollaborative
lowervaluesofr. Similarly,Rongetal.[97]haveinvestigated tasksarrangedinto4groups(centralpanel). Ifwemergethis
theevolutionofpubliccooperationonhighlyclusteredhetero- structure into a projected (one-mode) complex network, the
geneousnetworks,discoveringthatclusteringhasabeneficial collectionofgroupsistransformedintoastar-likegraph(left
effecton the evolutionofcooperationasit favorstheforma- panel)havingacentralhub(node6)withfiveneighbors. By
tion and stability of compact cooperative clusters. Yang et makingthiscoarse-graining,wehavelostalltheinformation
al. [98], on the other hand, adopted a different approach by aboutthegroupstructureofthesystem,anditiseasytorealise
trying to optimise the number of cooperative individuals on thatfollowing[81]toconstructthegroupswerecoveravery
uncorrelatedheterogeneousnetworks. Theyhaveconsidered differentcompositionmadeupof6groupsofsizes6,4(2),3
avariationoftheoriginalmodel[81],inwhichpotentialstrat- (2)and2,respectively. Moreover,itisimportanttonotethat
egydonorsarenolongerchosenrandomlybutratherpropor- ascale-freedistributionofinteractionsP(k)∼k−γ mapsdi-
tionally to their degree. It was shown that the promotion of rectlytoascale-freedistributionofgroupsizes P(g) ∼ g−γ.
cooperationisoptimaliftheselectionofneighborsislinearly However,in realityindividualstendtoperformcollaborative
proportionaltotheirdegree. Whiletheseresultsindicate that tasksingroupsofaratherhomogeneoussize[104],regardless
correlationsareveryimportantfortheevolutionofpublicco- of the size of the set of their overall collaborators. Accord-
operation,furtherexplorationsareneededtofullyunderstand ingly, the distributionofgroupsize is betterdescribedby an
allthedetailsofresultspresentedin[96–98],whichwehave exponentialdistributionP(g)∼exp(−αg).
hereomitted. Topreserveinformationaboutboththestructureofpairwise
tiesandthestructureofgroups,Go´mez-Garden˜esetal. have
We end this subsection by revisiting the role of hetero-
introducedtheuseofbipartitegraphs[102,105]. Abipartite
geneitiesinthedynamicsofinvestmentsandpayoffdistribu-
representation, as depicted in the right panel of Fig. 9, con-
tions, as reviewed before in subsection 2.2. Unlike lattices,
tains two typesof nodes. One denotingindividuals(circular
complexnetworksmakeitinterestingto correlatethedegree
nodes),andtheotherdenotinggroups(squarenodes),whereas
ofplayerswitheither(i)theinvestmentstheymakeascoop-
linksconnectthemasappropriate.Suchabipartiteframework
erators[40,99],(ii)thepayoffstheyarereceivingfromeach
iswell-suitedforstudyingdynamicalprocessesinvolvingN-
group [41, 100], or with (iii) both (i) and (ii) together [42].
playerinteractions.
Thesestudiesexploittheheterogeneityofscale-freenetworks
The setup of the publicgoodsgameon bipartite networks
toimplementdegree-basedpoliciesaimedatpromotingcoop-
is similar to that on one-mode networks, with deviations as
eration.In[40],forexample,ithasbeenshownthatpositively
describedin[102,105]. Thegraphiscomposedof N agents
correlatingthecontributionsofcooperatorswiththeirdegree
playingthegamewithinG(notnecessarilyequaltoN)groups
actsstronglydetrimentalonthe evolutionofpubliccoopera-
whoseconnectionsareencodedinaG×N matrixB . Thei-
tion. On the otherhand, if cooperatorswith onlya few con- ij
throwofthismatrixaccountsforalltheindividualsbelonging
nectionsare those contributingthe most, cooperationis pro-
togroupi,sothatB =1whenagentj participatesingroup
moted.Anoppositerelationhasbeenestablishedwithrespect ij
iwhile B = 0 otherwise. Alternatively,the informationin
tothecorrelationsbetweenthedegreeofplayersandtheallo- ij
thei-thcolumnencodesallthegroupscontainingagenti,i.e.,
cationofpayoffs[41,100]. Inparticular,cooperationthrives
B = 1 when agent i participates in group j and B = 0
ifplayerswiththehighestdegreereceivethebiggestshareof ji ji
otherwise. At each time step player i plays a round of the
thepayoffwithineachgroup.Moreover,theimpactofdegree-
gameineverygroupitismember.Thetotalpayoffafterbeing
correlatedaspirationlevelshasalsobeenstudied[101],andit
involvedinq = G B groupscanbeexpressedas
wasshownthatapositivecorrelation,suchthatthelargerthe i j=1 ji
degreeofaplayerthehigheritsaspirationlevel,promotesco-
PG N
rB
operation. Together,these resultsindicatethatfavoring hubs ji
P = B s c −s c q , (5)
i jl l l i i i
byeitherdecreasingtheirinvestmentsorincreasingtheirpay- j=1 mj "l=1 #
offs or aspiration promotes the evolution of public coopera- X X
tion,whichinturnstrengthenstheimportanceofhubsasde- wherem = N B isthenumberofindividualsingroup
j i=1 ji
P
10
FIG. 9: Schematic presentation of the two different forms of encoding collaboration data. In thecentral plot several collaborating groups
representtheoriginaldata. Theinteractionsamongstplayerscanbetranslatedintoaprojectedcomplexnetwork(left). However,ifoneaims
atpreservingalltheinformationaboutthegroupstructure,arepresentationasabipartitegraph(right)ismoreappropriate.Figurereproduced
withpermissionfrom[102].
j. Althoughinprincipleonecouldtake furtheradvantageof fined by the networkitself imposesa high degreeof overlap
the group structure in order to define different scenarios for betweenthegroups,especiallyforscale-freenetworks.
theupdateofstrategies,theevolutionarydynamicsisdefined
identically as on one-mode projected networks [102, 105].
The updating can rely on the usage of a replicator-like rule 3.3.Othernetwork-basedframeworks
[81],ortheFermiruleintroducedinEq.3.
Results presented in [102] indicate that, regardless of the In addition to the distributions of individual contacts and
updateruleandthedetailsofthepublicgoodsgame,theactual groupsizes,theimpactofothertopologicalfeaturesofsocial
groupstructureofcollaborationnetworkspromotestheevolu- networkshasalsobeenstudied. Inparticular,in[107]theau-
tion of cooperation. One arrives to this conclusion by com- thorsstudiedahierarchicalsocialstructurecomposedofcom-
paringthecooperationlevelonthebipartiterepresentationof munitiesormodulesinwhichseveralpublicgoodsgamesare
a real collaborationnetwork (containingauthor-articlelinks) playedsimultaneously. Forasetupwith 2hierarchicallevels
with the cooperationlevel on a projected one-modenetwork wethushavethefollowingframework:playeriismemberin
thatiscomposedsolelyofauthor-authorties(seeFig.10). On one groupof size m at the lowest level and, simultaneously,
theotherhand,bycomparingtheperformanceoftwobipartite itisalsoamemberinalargergrouptogetherwiththerestof
structures having different social connectivity – one having thepopulation.Thissetupcanbegeneralisedtosystemscom-
scale-freeandtheotheraPoissoniandistributionofdegree– posedofnhierarchicallevels,asshownintheupperpanelof
butthesamegroupstructure[105],wefindthatitisthegroup Fig.11forn=3. Thecouplingbetweentheevolutionarydy-
structureratherthanthedistributionofdegreethatdetermines namicsin each of the levelsis accomplishedby splitting the
theevolutionofpubliccooperation. Inparticular,thepromo- contribution c of each cooperator by the number of groups,
tion of cooperationdue to a scale-free distribution of degree andbychoosingadifferentprobabilityfortheupdatingrules
asreportedin[81]ishinderedwhenthegroupstructureisdis- withinandbetweenmodules. Results reportedin[107]indi-
entangledfromthesocialnetworkofcontactsbymeansofthe cate that public cooperation is promoted when imitation be-
bipartiteformulation. tweenplayersbelongingtodifferentmodulesisstrong,while
Notably, the bipartite formulationhas recentlybeen revis- atthesametimetheimitationbetweenplayerswithinthesame
ited byPen˜aandRochat[106], whocomparedtheimpactof lowest level module is weak. This combination of strengths
differentdistributionsusedseparatelyforgroupsizesandthe leadsto theonsetofgroupscomposedsolelyofcooperators,
numberof individualcontacts. They showed that a key fac- but it also enables cooperatorsthat coexist with defectors to
torthatdrivescooperationonbipartitenetworksisthedegree avoidextinction.
of overlap between the groups. The later can be interpreted Another important structural feature recently addressed is
as the bipartite analogueof the clustering coefficientin one- multiplexity[109–111], orthecouplingbetweenseveralnet-
modenetworks,whichasreviewedabove,ishighlybeneficial worksubstrates. Althoughthisstructuralingredienthasonly
forthe evolutionof publiccooperation. The results reported recently been tackled in the field of network science, some
in [106] also help to understand the high level of coopera- worksonthesubjecthavealreadyappearedinthecontextof
tionobservedinone-modescale-freenetworks[81], because evolutionarygames[108,112]. In[108],wherethefocuswas
the assumption that the structure of groups is implicitly de- ongroupinteractions, the authorshavestudied a simple lay-