Co-evolution of strategy and structure in complex networks with dynamical linking

0.1 MB·English

by  Jorge M. Pacheco

Co-evolutionofstrategy andstructure incomplex networks
withdynamical linking
Jorge M. Pacheco,1,2 Arne Traulsen,1 and Martin A. Nowak1,3
1Program for Evolutionary Dynamics, Harvard University, Cambridge MA 02138, USA
2CentrodeF´ısicaTeo´ricaeComputacional,DepartamentodeF´ısicadaFaculdadedeCieˆncias,P-1649-003LisboaCodex,Portugal
3Department of Organismic and Evolutionary Biology,
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
(Dated: February5,2008)
Hereweintroduceamodelinwhichindividualsdifferintherateatwhichtheyseeknewinteractionswith
others,makingrationaldecisionsmodeledasgeneralsymmetrictwo-playergames. Oncealinkbetweentwo
7
individualshasformed,theproductivityofthislinkisevaluated. Linkscanbebrokenoffatdifferentrates. We
0
provideanalyticresultsforthelimitingcaseswherelinkingdynamicsismuchfasterthanevolutionarydynamics
0
andvice-versa,andshowhowtheindividualcapacityofformingnewlinksorseveringinconvenientonesmaps
2
into the problem of strategy evolution in a well-mixed population under a different game. For intermediate

n ranges,weinvestigatenumericallythedetailedinterplaydeterminedbythesetwotime-scalesandshowthatthe
a scopeofvalidityoftheanalyticalresultsextendstoamuchwiderratiooftimescalesthanexpected.
J

3 PACSnumbers: 87.23.-n87.23.Kg89.75.Fb

]
E
Networks pervade all sciences [1, 2, 3, 4, 5]. During re- rangeofvalidityoftheanalyticalresultsismuchwider. The
P
centyears,researchershavedevelopedmethodstocharacter- presentmodelleadstosingle-scalenetworksasdefinedin[4],
.
o ize such networks, providing novel insights into the proper- withassociatedcumulativedegreedistributionsexhibitingfast
i ties accruing to those networked systems and organizations. decayingtails[4],asshowninFig.1. Suchtailswhichdecay
b
- The classical social networkmetaphor[6] placesindividuals exponentiallyor faster than exponential, leading to what are
q
atthe nodesof a network,the networklinksrepresentingin- knownas“broad-scale”and“single-scale”networks,respec-
[
teractionsorconnectionsbetweenthoseindividuals. Citation tively[4],arefeatureswhich,togetherwithalargevariability

1 networks, collaboration networks, co-authorship and movie in the average connectivity [1, 2, 5], characterize most real-
v co-acting networks, as well as the networks of sexual rela- world social networks [3, 4]. We start by characterizing the
8
tions,allfallintothismetaphoricrepresentation[1,2,3,4,5]. networksemergingfromourmodel. Subsequently,weintro-
0
Most analytical studies on this type of networks carried out duceindividualswhoadoptdefinitestrategiesandmakeratio-
0
1 todatehaveaimedtoexplaintheemergenceoftheobserved naldecisionsbyengaginginagamewithothers,studyinghow
0 topological properties, as deduced from the empirical data. strategy and structure co-evolve. Finally, we strenghten the
7 Networks, however,are dynamicalentities, andin this sense couplingbetweenstrategyandstructurebylettingindividuals
0
the empirical information often only provides a fixed-time evaluatetheproductivityoflinksinwhichtheyparticipate.
/
o snapshotof networkswhich are continuouslyevolving. Fur- Let us first consider the structural evolution in a popula-
bi thermore, dynamical features of networks have been stud- tionoftwotypesofindividuals(players),AandB,occupying
- ied in connection with their growth, modeled in terms of thenodesofanetwork. Thetotalpopulationsize isconstant
q
the preferentialattachment(or cumulative advantage)mech- N = N +N . Linksdefine interactionsbetweenindivid-
: A B
v anism [1, 6, 7], via random addition and removal of nodes uals, being formed at certain rates and having specific life-
Xi [2] orby imposingdifferentformsofconnectivitysaturation times. The maximumpossible numberof AA, AB and BB
[4, 8, 9, 10]. Moreover, individual decisions to establish or linksisrespectivelygivenbyN = N (N −δ )/(1+δ )
r ij i j ij ij
a remove/rewire a given link have been studied by numerical (i,j = {A,B}). SupposeAandB playershaveapropensity
simulations [4, 8, 9, 10, 11, 12, 13, 14, 15]. Here we de- to formnew linksdenotedbyα and α , such thatij links
A B
velop a new model which incorporates decisions of individ- are formed at rates α α . The death rates are given by γ
i j ij
ualswhenestablishingnew linksorgivingupexistinglinks. (withassociatedlifetimesτ = γ−1). Withthesedefinitions
ij ij
Individualsare capable of making rationalchoices, modeled the mean field equations governing what we call the Active
in terms of a game, associated with well defined strategies. Linking(AL)dynamicsofthisnetworkare
We use evolutionary game theory [16, 17, 18, 19, 20] and
X˙ =α α (N −X )−γ X . (1)
studythedynamicalco-evolutionofindividualstrategiesand ij i j ij ij ij ij
networkstructure. Werestrictouranalysistosymmetrictwo-
These differential equations lead to an equilibrium distri-
player games, although the model can be easily extended to bution of links given by X∗ = N φ , where φ =
gameswith anarbitrarynumberof strategies. We obtainan- ij ij ij ij
α α (α α +γ )−1 denotes the fraction of active links. In
alyticalresultswhichareformallyvalidintheextremelimits i j i j ij
Fig.1weshowhowthismodelleadstostationaryregimesof
when one of the dynamics (strategy or structure) dominates
complexnetworkswhichcanexhibitdifferentdegreesofhet-
the other, although our numerical simulations show that the
erogeneity. Inparticular,thethedependenceofthestationary
2
networksonthefrequencyofindividualsofagiventypewill p=[1+e−β(fA−fB)]−1. Thereversewillhappenwithprob-
automatically couple network dynamics with the frequency- ability1−p.Thequantityβcontrolstheintensityofselection.
dependentevolutionarydynamicsweintroduceinthefollow- Forβ → ∞theindividualwiththelowerpayoffdeterminis-
ing. ticallyadoptsthestrategyoftheotherindividual. Forβ ≪1,
Let us now introduce a game between A and B leading we recoverthe weak selection limit of the frequencydepen-
to frequency-dependentevolution of strategies. The game is dentMoranprocess[22].
givenbythepayoff-matrixM When T ≪ T , AL proceeds much faster than strategy
ij a s
update on each node. Hence, the stationary regime of AL-
A B dynamics determines the average payoff and fitness of indi-
A a b viduals. This means that strategy evolution proceedsas in a
.
B (cid:18) c d (cid:19) well-mixedpopulationofA andB players(completegraph)
engaged in a game specified by the payoff-matrix M′ =
ij
In the stationary regime of Active Linking (AL) dynamics, M φ . SinceAL-dynamicsisfast,thedynamicsofthesys-
ij ij
the average fitness of A and B individualsis given by f = tem does not depend on the starting condition, and we can
i
M φ (N −δ ). Itisnoteworthythattheseexpressions computeanalyticallythefixationprobabilitiesofstrategiesA
j ij ij j ij
Pare equivalent to the average payoffs of NA and NB play- and B. The probability ρA(k) that k A-players introduced
erswhoplayagamespecifiedbythe(rescaled)payoff-matrix intoapopulationofB-playerswilltakeovertheentirepopu-
M′ =M φ onacompletegraph. lationisgivenby[21]
ij ij ij
erf[ξ ]−erf[ξ ]
ρ (k)= k 0 , (2)
A
erf[ξ ]−erf[ξ ]
N 0
1
where erf(x) is the error function and ξ = β (ku+v).
bution 0.1 Wb′Ne h−avce′N2u+ c=′. Fao′r−ub=′ −0,cE′q+. 2ds′imaknpdlifi2eqvs tuo=ρ −(ka)′ +=
stri (e−2βvk−1)/(e−2βvN −1). A
di
e On the other hand, in the opposite limit where Ta ≫ Ts,
e
r 0.01 evolution will proceed according to the usual game dynam-
g
e
d ics [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]
e
v on a static network reflecting the initial configuration. If we
ulati 0.001 start from a complete graph then Eq. (2) remains valid, ex-
um NA / N αA αB γAA= γAB= γBB cept that u and v must be calculated employing the original
C 00..5500 01..2000 01..2000 1500 payoff-matrixMij. If we start fromanothergraphtopology,
0.0001 0.35 1.10 0.75 50 analyticalandnumericalresultsforstatic networksapplyin-
10 20 30 40
stead[37,38,39]. WheneverT ∼T ,oneexpectsadetailed
Degree a s
interplay between these two processes to drive co-evolution.
FIG.1: (coloronline)Cumulativedegreedistributionsfornetworks ThisregimecanbeexploredbycomputersimulationsofAL-
generatedwiththepresentmodel,forpopulationsofsizeN = 103 dynamics. As anexample, we investigatethe interactionbe-
andtwodifferenttypesofindividuals. Thefastdecayingtailscorre- tween cooperators and defectors in the Prisoner’s Dilemma
latewellwiththeobservedtailsofrealsocialnetworks[1,2,3,4,5]. (PD). A cooperator, C, pays a cost c for every link, and the
Ontheother hand, thedependence ofthefinalnetwork onthefre-
partnerofthislinkreceivesabenefitb>c. Defectors,D,pay
quency of each type of individuals leads to a natural coupling be-
nocostanddistributenobenefits.Thepayoff-matrixbecomes
tween network dynamics and frequency-dependent strategy evolu-
tion.
C D
C b−c −c
.
SofarwehavedealtwithAL-dynamics. Letusnowstudy D (cid:18) b 0 (cid:19)
howthefrequenciesofstrategiesAandB changeunderevo-
lutionarygame dynamics. We assume that the characteristic On complete graphs, cooperators are never advantageous
time-scale associated with AL-dynamics is Ta, whereas that compared to defectors. This means if Ta ≫ Ts coopera-
associatedwithstrategyupdatingisT . tors are never favored by selection. On the other hand, if
s
Reproductioncanbegeneticorcultural.Weadoptthepair- T ≪ T , the effective payoff-matrix is different, and may
a s
wise comparison rule, which provides a convenient frame- notcorrespondanymoretoaPD,thatis,whenALdominates,
work of game dynamics at all intensities of selection [21]. the problem becomes equivalentto the evolutionarydynam-
Two individuals from the population, A and B are ran- ics of a different game in a complete graph. The advantage
domly chosen for update. The strategy of A will replace of cooperators from AL can be captured by the parameter
that of B with a probability given by the Fermi function r = (φ − φ )/φ which provides a measure of the
CC CD CC
3
1.00 1.00
Prisoner’s dilemma
(a) Coordination game
0.75 s
r
cooperators 00..2550 cnoCDe mt 01 w ..C50po l er -0k 0t .eD0.5 active linking aliCDcn’’t k 00i vi ..Cn31e1’7g - 00 D..00’80 of cooperato00..5705 Pdrili es Cmo nm Dear’s Cgaomo r e dCin’ aDti’on
y of 0.00 on CD 45 23 CD’’ 52/ 6 1 /13
abilit 1.00 Snowdrift game acti0.25 β=20.0
prob 0.75 (b) Harmony game Fr β=0.05
Fixation 0.50 cnoe mt w Cpo l er kt De ctive linking lainctk i iv nCeg’ D’ 0.00 0.01Time scale of activ0e.1 linking (T /T ) 1
0.25 C 0.6 0.2 a C’ 0.37 0.03 s a
D 1.0 0.0 D’ 0.17 0.00
FIG. 3: (color online). Co-evolutionary dynamics of strategy and
0.00
structure. Thecurves drawn correspond tothe resultsof computer
0 20 40 60 80 100
Initial number of cooperators, N simulations carried out for networks of size N = 100. Parame-
C
ters and payoff matrices are the same as in Fig. 2. The rescaled
FIG.2: (coloronline)Fastactivelinking(AL)changesthedynam- payoff-matrix leads to a fixed point at a fraction of cooperators
icsofthesocialnetworkandthepayoffmatrix. a)ALtransformsa NC/N ≈ 35%. For each value of the ratio Ts/Ta, we ran 100
Prisoner’sDilemmawithc = 0.5andb = 1.0intoacoordination simulations, starting from 50% cooperators and a complete graph.
game. Whilefixationof cooperators isnegligible inthePrisoner’s The values plotted correspond to the fraction of runs which ended
Dilemma (solid line), cooperators can take over with AL (dashed with100%cooperators. WefixTs = 1andvaryTa. Ineachtime
line)b)Asnowdriftgamewithc = 0.8andb = 1.0inwhichfix- step,synchronousupdatingofstrategiesiscarriedoutwithprobabil-
ationofdefectorsiscertainistransformedintoaharmonygamein ityTa/(Ts +Ta) using Fermi-update, AL beeing carriedout oth-
whichcooperatorsprevail.TheverticalarrowsshowhowALaffects erwise. Fortheextremelimitsweobtainperfectagreementwiththe
thefixationprobabilitywheninitially50%cooperatorsarepresent: analyticpredictions.However,theanalyticresultsremainvalidfora
whiletheyhavenochances onacompletegraph despitetheirhigh muchlargerrangeofvalues0.01 ≤ Ta/Ts ≤ 0.1thanonewould
abundance,ALmakesfixationofcooperationalmostcertaininboth expect from pure theoretical considerations. Indeed, only between
systems (β = 0.1, N = 100, αC = αD = 0.4, γCC = 0.1, thesetwolimitsacrossovertakesplace,whichdependsontheinten-
γCD =0.8,γDD =0.32). sityofselectionβasillustrated.
advantageofassortativeinteractions(CC-links)withrespect dynamics. An interestingcouplingarises when we associate
to disassortative ones (CD-links). In terms of r, the PD is thepropensitytoformnewlinksandthelifetimeofdifferent
transformed into a coordination game whenever r > c/b, typesoflinkswiththeproductivityofthoselinksassessedin
which is formally equivalent to Hamilton’s rule of kin se- termsofpayoffs.Manypossibilitiescanbereadilyenvisaged,
lection [40]. In strategy phase space, an unstable interior whichwillleadtodifferentcontext-basedjustificationsforthe
fixed point develops at a frequency of cooperators given by choicesofparametersα andγ . Hereweexplorethecasein
i ij
N /N ≈(1−r)c/[r(b−c)].Inotherwords,forT ≪T fix- whichcooperatorsanddefectorssharethesamepropensityto
C a s
ationofcooperatorsisalmostcertainif theinitialfractionof form new links α = α , whereas the lifetimes of ij-links
C D
cooperatorsinthepopulationexceedsthisratio.InFig.2awe areproportionaltotheaverageprofitexpectedfromthatlink.
providenumericalexamplesofthisscenario,whereasinFig.3 Otherlinearaswellasnon-linearalternativesarepossible. A
we investigate the behavior of our co-evolutionarymodel as simpleaveragerelation,basedontheexpectedoutcomefrom
Ts/Ta increases. The results of Fig. 3 show how the ratio differenttypesofinteractionsleadstoτij = (Mij +Mji)/2
of time scales affects co-evolution of strategy and structure. which yieldsτ = τ /2, whereasτ = 0. More gen-
CD CC DD
In all cases we start from well-mixed populations(complete erally, we may assume that τ = τ /p with p > 1,
CD CC
graphs). Clearly, notonlytheasymptoticbehaviorcoincides maintainingτ = 0(thisresultsfromthezero-entryinthe
DD
with the analytic prediction but, perhaps more importantly, payoff-matrix). We may now express r in terms of the con-
Fig. 3showsthatonlyfor0.01 ≤ Ts/Ta ≤ 0.1doesthe in- stantθ =τCCα2C,obtainingr = pp+−θ1,anincreasingfunction
terplaybetweenthetwotimescalesdeviatesignificantlyfrom of p. Theintuition behindthis resultis clear: The largerthe
theanalyticpredictions. valueofp,thatis, thesmallerthelifetimeofCD-linkscom-
Finally,letusfurthercouplethedynamicsoflinksandthe paredtoCC-links, thesmallerthevalueofb/cabovewhich
dynamicsof strategiesby introducingpayoffdependentAL- cooperation will thrive. Moreover, the larger the value of p
4
thesmallerthefractionofcooperatorsthatisnecessarytobe
initiallypresentinthepopulationforcooperationtodominate
overdefectionintheresultingcoordinationgame.
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