Evolutionary Prisoner's Dilemma on heterogeneous Newman-Watts small-world network

0.41 MB·English

by  Feng Fu

EPJ manuscript No.
(will be inserted by the editor)
Evolutionary Prisoner’s Dilemma on heterogeneous
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0 Newman-Watts small-world network
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a F. Fua, L.-H. Liu, and L. Wangb
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2 Center for Systems and Control, College of Engineering, Peking University,Beijing 100871, P. R. China

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Received: date/ Revised version: date
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.
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t Abstract. Inthis paper,we focus on theheterogeneity of social networksand its role to theemergence of
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prevailingcooperationandsustainingcooperators.Thesocialnetworksarerepresentativeoftheinteraction
[

2 relationships between players and their encounters in each round of games. We study an evolutionary
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6 Prisoner’sDilemmagameonavariantofNewman-Wattssmall-worldnetwork,whoseheterogeneitycanbe
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6 tuned by a parameter. It is found that optimal cooperation level exists at some intermediate topological
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0 heterogeneity for different temptations todefect. That is, frequency of cooperators peaks at some specific
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0 values of degree heterogeneity — neither the most heterogeneous case nor the most homogeneous one
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t would favor the cooperators. Besides, the average degree of networks and the adopted update rule also
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affect thecooperation level.
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PACS. 02.50.Le Decision theoryandgametheory–89.75.Hc Networksandgenealogical trees–87.23.Ge
r
a
Dynamics of social systems
1 Introduction well-mixed population by using mean-field theory of sta-
tistical physics. In classic game theory, the players are
Evolutionary game theory has been well developed as an
assumed to be completely rational and try to maximize
interdisciplinary science by researchers from biology, eco-
their utilities according to opponents’ strategies. Under
nomics,socialscience,computerscienceforseveraldecades.
these assumptions, in the Prisoner’s Dilemma game (PD
In pastfew years,it alsogainedthe interests of physicists
game), two players simultaneously decide whether to co-
to study some phenomena and intriguing mechanisms in
operate (C) or to defect (D). They both receive R upon
a e-mail: [email protected]
b e-mail: [email protected] mutual cooperation and P upon mutual defection. A de-
2 F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network
fector exploiting a C player gets T, and the exploited conclude that both well-mixed population and spatially-
cooperator receives S, such that T > R > P > S and structured population are represented by regular graphs.
2R>T +S. As a result, it is best to defect regardless of It is, however, unrealistic to assume the real world is as
the co-player’s decision. Thus, defection is the evolution- homogeneous as regular graphs. Furthermore, as afore-
arily stable strategy (ESS), even though all individuals mentioned, many real-world networks of interactions are
would be better off if they cooperated. Thereby this cre- heterogeneous,namely,differentindividualshavedifferent
atesthe socialdilemma,becausewheneverybodydefects, numbers of average neighbors with whom they interact.
the mean population payoff is lower than that when ev- In [6], Abramson and Kuperman studied a simple model
erybodycooperates.However,cooperationisubiquitousin ofanevolutionaryversionofthePrisoner’sDilemmagame
natural systems from cellular organisms to mammals. In playedinsmall-worldnetworks.Theyfoundthatdefectors
the past two decades, some extensions on PD game have prevail at some intermediate rewiring probability. By in-
been considered to elucidate the underlying mechanisms troducing the volunteering participation, Szabo´ et al. in-
boosting cooperation behaviors by which this dilemma vestigatedthe spatio-temporaldiagramsof the evolution-
could be resolved. For instance, depart from the well- ary process within the context of rewired lattices which
mixed population scenario, Nowak and May considered have small-world property [7]. Other paradigms of game
PD game on spatially structured population [1]. All indi- with spatially-structured population or on graphs, such
viduals areconstrainedto play only withtheir immediate assnowdriftgame,rock-scissors-papergame,publicgoods
neighbors.Theyfoundthatspatialstructureenhancesthe game, etc, have been studied [8,9,10].
ability of cooperators to resist invasion by defectors.
Recently,SantosandPachecodiscoveredthatinscale-
The successful development of network science pro- freenetworks,thecooperatorsbecomedominantforentire
vides a systematic framework for studying the dynam- range of parameters in evolutionary PD game and snow-
ics processes taking place on complex networks [2,3]. For driftgame[11].Theyalsodemonstratedthattheenhance-
mostnetworks,includingthe WorldWideWeb,the Inter- mentofcooperationwouldbeinhibitedwheneverthecor-
net, and the metabolic networks, they have small-world relations between individuals are decreased or removed.
property and are demonstrated to be scale-free [4,5]. The BeingsimilartoHamilton’srule,Ohtsukietaldescribeda
network theory is a natural and convenient tool to de- simplerulefor evolutionofcooperationongraphsandso-
scribe the population structure on which the evolution cialnetworks—thebenefit-to-costratioexceedstheaver-
of cooperation is studied. Each vertex represents an in- agenumber ofneighbors[12].In[13,14],Santossuggested
dividual and the edges denote links between players in that the heterogeneity of the population structure, espe-
terms of game dynamical interaction. Trivially, one could cially scale-free networks, provides a new mechanism for
F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network 3
cooperatorstosurvive.Moreover,cooperatorsarecapable someexplanationstotheresults.Conclusionsaremadein
of exploiting the heterogeneity of the network of interac- Sec. IV.
tions, namely, finally occupy the hubs of the networks.
Besides, Vukov and Szabo studied the evolutionary PD
2 The model
game on hierarchical lattices and revealed that the high-
eststationaryfrequencyofcooperatorsoccursinsomein- 2.1 The Heterogeneous Newman-Watts (HNW)
termediate layers [15]. Some researchers have studied mi- model
noritygame [16] ,with small-worldinteractions[17], with
Weconsiderone-dimensionallatticeofN verticeswithpe-
emergentscale-freeleadershipstructure[18],etc.Howbeit,
riodic boundary conditions, i.e., on a ring, and join each
theseworksaremostlystudiedoncrystalized(static)net-
vertex to its neighbors κ or fewer lattice spacing away.
works, i.e., the topology of network is not affected by the
Instead of rewiring a fraction of the edges in the regu-
dynamics on it. Therefore, adapting the network topol-
lar lattice as proposed by Watts and Strogatz [4], some
ogy dynamically in response to the dynamic state of the
“shortcuts” connecting randomly chosen vertex pairs are
nodes should be taken into account. Evolutionary games
added to the low-dimensional lattice [21]. To introduce
with“adaptive”networkstructurehavebeeninvestigated
certain type of heterogeneity to this NW model [22], we
by some researchers [19,20].
chooseN verticesatrandomfromallN nodeswithequal
h
In this paper, we consider evolutionary PD game on probabilities and treat them as hubs. Then we add each
a variant of Newman-Watts (NW) small-world network. of m shortcuts by connecting one node at random from
As is known, the NW small-world network is moderately all N nodes to another node randomly chosen from the
homogeneous. In order to investigate the heterogeneity’s N centered nodes (duplicate connections and self-links
h
role in the emergence of cooperationfor different tempta- arevoided),seeFig.1asanillustration.Insuchheteroge-
tionstodefect,someartificialheterogeneityinnetworkde- neous Newman-Watts (HNW) model, nodes can be nat-
greesequencesisintroduced.Namely,somenodesareran- urally divided into two groups: hubs, which tend to have
domlychosenashubsandatleastoneendpointofeachof higher connectivity, and the others that have lower con-
theaddedmshortcutsislinkedtothesehubs.Theremain- nectivity. Accordingly, the heterogeneity is controlled by
derofthepaperisorganizedasfollows.SectionIIdiscusses the parameter N : small N leads to higher degree of
h h
the heterogeneousNewman-Watts (HNW) model andthe the hubs, which in turn results in increased heterogene-
evolutionary PD game, including payoff matrix and the ity. Trivially, one could see that when N =N, shortcuts
h
microscopic updating rule. And then Sec. III gives out arejustsimplyaddeduniformlyatrandom,effectivelyre-
the simulation results for different parameters and makes ducing the network to the homogeneous Newman-Watts
4 F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network
model. If N =1, all shortcuts are linked to a single cen- where the Ω denotes the neighboring sites of x, and the
h x
ter, making the network an extremely heterogeneousone. sum runs overneighbor set Ω of the site x. Without loss
x
Thedegreeofheterogeneityofthenetworkiscomputedas of the generic feature of PD game, the payoff matrix has
h=N−1 k2N(k)−hki(thevarianceofthenetworkde- a rescaled form, suggested by Nowak and May [24]
Pk
gree sequcence), where N(k) gives the number of vertices
1 0
M = (3)
withk edges,hkidenotestheaveragedegree.FromFig.2,  
b 0
2
it shows that variance σ (= h) monotonically decreases
where1<b<2.Inevolutionarygamestheplayersareal-
as the fraction number of hubs N /N increases. Thus we
h lowedtoadoptthestrategyofoneoftheirmoresuccessful
simply use the fraction N /N to indicate the degree of
h neighbors.Thentheindividualxrandomlyselectsaneigh-
heterogeneity.
bor y for possibly updating its strategy.Since the success
ismeasuredbythetotalpayoff,wheneverP >P thesite
y x
2.2 Evolutionary Prisoner’s Dilemma (PD) game x will adopt y’s strategy with probability given by [11]
P −P
y x
Since the pioneering work on iterated games by Axel- Wsx←sy = bk> (4)
rod[23], the evolutionaryPrisoner’sDilemma (PD) game where k> is the largestbetween site x’s degree kx andy’s
hasbeen a generalmetaphor forstudying the cooperative ky. This microscopic updating rule is some kind of imita-
behavior. In the evolutionary PD game, the individuals tionprocessthatissimilartoWin-Stay-Lose-Shiftrulesin
are pure strategists, following two simple strategies: co- spirit.Unlikethecaseofboundedrationality,suchpropor-
operate (C) and defect (D). The spatial distribution of tionalimitation (the take-overprobability is proportional
strategies is described by a two-dimensional unit vector tothepayoffdifferenceofthetwosites)doesnotallowfor
for each player x, namely, an inferior strategy to replace a more successful one.
1 0
s= and (1)
3 Simulation results and discussions
   
0 1
for cooperators and defectors, respectively. Each individ- EvolutionaryPD game is performed analogicallyto repli-
ual plays the PD game with its “neighbors” defined by catordynamics:in eachgeneration,all directly connected
their who-meets-whom relationships and the incomes are pairsofindividualsxandy engageinasingleroundgame
accumulated. The total income of the player at the site x and their accumulated payoffs are denoted by Px and Py
can be expressed as respectively. The synchronous updating rule is adapted
here. Whenever a site x is updated, a neighbor y is cho-
P = sTMs (2)
x X x y
senatrandomfromallk neighbors.Thechosenneighbor
y∈Ωx x
F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network 5
takesoversite x with probabilityWsx←sy as Eq.(4),pro- around small Nh/N near the zero. In Fig. 5, we report
vided that P > P . Simulations were carried out for a the result of frequency of cooperation as function of the
y x
population of N =2001 players occupying the vertices of parameters space (b,N /N). It also shows that for fixed
h
the HNW network. Initially, an equal percentage of co- b, optimum cooperation level exits at certain intermedi-
operators and defectors was randomly distributed among ate N /N. In addition, for certain intermediate N /N,
h h
the population. We confirm that different initial condi- the density of cooperation is higher than the cases when
tionsdonotqualitativelyinfluencetheequilibriumresults. N /N →0 or N /N →1. As a result, it reveals that op-
h h
In Fig. 3, with fixed chosen hub numbers N = 41, the timalcooperationleveloccurs atintermediatetopological
h
equilibriumfrequenciesofcooperatorswithrespecttodif- heterogeneityforsomefixedb,that’sneitherthemosthet-
ferent b are almost the same when started from different erogeneouscasenorthemosthomogeneousonefavorsthe
initial conditions. Equilibrium frequencies of cooperators cooperators best. Actually, under the most homogeneous
were obtained by average over 2000 generations after a case of N /N = 1, the cooperation behavior is almost
h
transient time of 10000 generations. The evolution of the extinct for large values of b. While under the most het-
frequency of cooperators as a function of b and N /N erogeneous case of N /N = 1/N, the cooperation level
h h
has been computed. Furthermore,each data point results is also low. The reason for this is that, in the most het-
from averagingover 100 simulations, corresponding to 10 erogeneous case, all shortcuts are linked to a single node,
runs foreachof10differentrealizationsofagiventype of making the network star-like with sparse connections be-
network of contacts with the specified parameters. In the tween the neighbor sites of the centered node. The pe-
followingsimulations,N =2001,κ=2arekeptinvariant. ripheral nodes’ imitation strategies heavily depend upon
the centered individual’s choice. Once the unique largest
To investigate the influence of network degree hetero-
hub is occupied by a defector, defection will spread over
geneitytotheevolutionofcooperation,wefixedthevalue
the entire network. Then the system can hardly recover
of b when increased fraction of hubs N /N from 1/N to
h
from the worst case of all defectors. Therefore the most
1. Fig. 4 shows the equilibrium frequency of cooperators
heterogeneous case with N /N = 1 is not the ideal case
h
as function of the fraction number of hubs N /N by dif-
h
for arising cooperation.The most homogeneouscase with
ferentvalues ofb forcomparisonwhenm=1000.We find
N /N = 1 does not favor cooperation either. In this sit-
h
the ρ ’s curve exhibits non-monotonous behavior with a
c
uation, all nodes almost have the same number of neigh-
peak at some specific value of N /N. As aforementioned,
h
bors with some long range shortcuts making the average
the quantity N /N can be regarded as a measurement
h
lengthshort.Therefore,thedefectionofsomenodeiseas-
of heterogeneity. For certain b, it has been illustrated in
ilyspreadfromonenodetoanotherthroughtheshortcuts.
Fig.4,thereis aclearmaximumfrequencyofcooperators
6 F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network
Withinsomegenerations,thedefectorsprevailonthenet- that increasing average degree (larger m) promotes co-
work. Hence the most homogeneous case with N /N = 1 operation on the network. Nevertheless, we should point
h
does not promote the evolution of cooperation. out that with sufficient largem which makes the network
nearly fully-connected, the cooperation will be inhibited
Let’s consider the case of intermediate heterogeneity
due to mean-fieldbehavior[25]. Onthe other hand, when
where some certain number of nodes are selected as hubs
m changes, our above conclusion on the effect of hetero-
(densityofcooperatorspeaksatsomespecificvalueN /N
h
geneity to cooperation is still valid. We can see in Fig. 7
for fixed b). These hubs are connected to each other by
that, for different fixed m and b, ρ still peaks at certain
shortcuts or placed on the same ring linked by the reg- C
intermediate value of N /N.
ular edges. This provides the protection to each other in h
resisting the invasion of defectors. Even if one hub is oc- We further explore the situation where we adopt an-
cupied by the defector, the cooperation level will tem- other microscopic update rule different from Eq. 4. In
porarily decrease due to the diffusion of defection from steadofupdating the strategiesaccordingto the accumu-
the hub. However, other hubs occupied by cooperators lated payoff of individuals, we consider using the average
could reciprocally help each other and promote cooper- payoffofindividualx,Px = Pkxx.Thusthenewupdaterule
ation behaviors on the network. Finally, cooperation is is:
recovered to the normal level. Accordingly, it in part re-
P −P
sult from the ability of cooperators taking advantage of Wsx←sy = y b x (5)
hubs. It is reported in [11] that cooperation dominates in
Baraba´si-Albert (BA) scale-free network due to the de-
By such update rule, we report the results in Fig. 8 cor-
greeheterogeneitywherecooperatorsaremorecapableto
responding to b = 1.1 and b = 1.3. It is found that the
occupy the hubs in the network. Though the cooperation
equilibrium frequency of cooperation is not sensitive to
level on our HNW model is not as high as it on the BA
the heterogeneity as N /N increases from 1/N to 1 (the
h
scale-freenetwork(seeFig.6andthefigureinref.[11]for
curves of average payoff are almost flat). It is because
comparison),bothofthem benefit fromthe heterogeneity
that normalizing individual’s payoff diminish the role of
of the network in the evolution of cooperation.
heterogeneity.Consequently,the difference in individual’s
TheaveragedegreeofHNWnetworkhki=2κ+2m/N. accumulated payoff arising from the degree heterogeneity
It is believed that average degree affects the evolution of will be positive to evolution of cooperation. We should
cooperation.Fig.7plotsthefrequencyofcooperatorsver- be careful to come to a conclusion with different update
sus N /N corresponding to different shortcuts number rules. Moreover,all these phenomena and conclusions are
h
m = 800,1000,1200 respectively with b = 1.1. We find also valid under the cases for different population size N.
F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network 7
4 Conclusion remarks and future work 3. M. E. J. Newman, SIAMReview 45, (2003) 167.
4. D. J. Watts and S. H.Strogatz, Nature393, (1998) 440.
In conclusion, we have studied the effect of topological
5. A.-L. Barab´asi and R. Albert, Science 281, (1999) 509.
heterogeneity to the evolution of cooperation behavior in
6. G.AbramsonandM.Kuperman,Phys.Rev.E63,(2001)
evolutionary Prisoner’s Dilemma game. It is found that
030901(R).
frequency of cooperation peaks at some specific value of
7. G. Szab´o and J. Vukov,Phys.Rev. E 69, (2004) 036107.
N /N, that is, neither the most heterogeneous case nor
h
8. M. Tomassini, L. Luthi, and M. Giacobini, Phys. Rev. E
the most homogeneous one would favor cooperators. We
73, (2006) 016132.
found that the network degree heterogeneity is one of the
9. A.SzolnokiandG.Szab´o,Phys.Rev.E70,(2004)037102.
factors affecting the emergence of cooperation. Besides, 10. G. Szab´o and C. Hauert, Phys. Rev. Lett. 89, (2002)
the averagedegree and the adopted update rule also play 118101.
aroletocooperation.Therefore,itismeaningfulandnec- 11. F. C. Santos and J. M. Pacheco, Phys. Rev. Lett. 95,
essary to explore the underlying factors that do matter (2005) 098104.
the emergence of cooperation behaviors. In addition, the 12. H. Ohtsuki, C. Hauert, E. Lieberman and M. A. Nowak,
interplaybetweengamedynamics andnetworktopologies Nature441, (2006) 502.
13. F. C. Santos and J. M. Pacheco, J. Evol. Biol. 19, (2006)
is an interesting topic for further investigations.
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14. F.C.Santos,J.M.PachecoandTomLenaerts,Proc.Natl.
Acknowledgments
Acad. Sci. USA103, (2006) 3490.
F. F. would like to acknowledge Jie Ren for stimulat- 15. J. Vukovand G. Szab´o, Phys.Rev. E 71, (2005) 036133.
ing discussions. The authors are partly supported by Na- 16. D. Challet and Y.-C. Zhang, Physica A 246, (1997) 407.
tional Natural Science Foundation of China under Grant 17. M. Kirley, Physica A 365, (2006) 521.
Nos.10372002and60528007,National973Programunder 18. M. Anghel, Z. Toroczkai, K. E. Bassler and G. Korniss,
Phys. Rev.Lett. 92, (2004) 058701.
Grant No.2002CB312200, National 863 Program under
19. M. G.Zimmermann and V.M.Egu´ıluz, Phys.Rev.E 72,
Grant No.2006AA04Z258 and 11-5 project under Grant
(2005) 056118.
No.A2120061303.
20. J. Ren, X. Wu, W.-X. Wang, G. Chen, and B.-H. Wang,
arXiv:physics/0605250.
References
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1. M. A.Nowak and R. M May, Nature359, (1992) 826. (1999) 341.
2. R.AlbertandA.-L.Barab´asi,Rev.Mod.Phys.74,(2002) 22. T. Nishikawa, A.E.Motter, Y.-C.Lai and F.C. Hoppen-
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8 F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network
23. R.AxelrodandW.D.Hamilton,Science211,(1981)1390. Fig.1IllustrationoftheheterogeneousNewman-Watts
24. M.A.Nowak,S.BonhoefferandR.M.May,Int.J.Bifur- small-world model. A one-dimensional lattice with con-
cation Chaos, 3, (1993) 35. nections between all vertex pairs separated by κ or fewer
25. C.-L. Tang, W.-X. Wang, X. Wu, and B.-H. Wang, Eur. lattice spacing, with κ = 2 in this case. The red dots are
Phys.J. B, 53, (2006) 411.
chosenascentersorhubs andthe addedm shortcuts(the
dashedline)haveatleastoneendpointbelongingtothese
hubs.
Fig.2Thelog-logplotofthevarianceofnetworkdegree
sequenceversusfractionofhubsN /N forN =2001,m=
h
1000. Each data point is averaged over 10 times different
realization corresponding to each N /N.
h
Fig.3Theplotoffrequenciesofcooperatorsasfunction
oftemptationtodefectbcorrespondingtothedifferentini-
tial fractions of cooperators 20%, 50%, 80%, respectively.
N =2001,κ=2,m=1000,N =41.
h
Fig.4Frequencyofcooperatorsasfunctionofthe frac-
tion of hubs N /N, for different values of the temptation
h
to defect b as shown in the legend. Note that when b=1,
the game is not a proper PD game.
Fig.5 The frequency of cooperators ρ as function of
c
the parameter space (b,N /N). The red region indicates
h
the cooperation of high level corresponding to the inter-
mediate N and b.
h
F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network 9
Fig.6 The frequencies of cooperators ρ vs. the temp-
c
tation to defect b with different levels of heterogeneity.
Fig.7 Plot of the fraction of cooperators vs. N /N for
h
different values of m with b=1.1.
Fig.8Plotoftheresultswithupdaterulescorrespond-
ingtoaverageandaccumulatedpayoffrespectively.Fig.8- Fig. 1.
A plots the case when b=1.1 and Fig. 8-B with b=1.3

Degee Variance
100

10
1
1E-3 0.01 0.1 1
Nh/N
Fig. 2.

1.0 Nh=41 0.2
0.5
0.8 0.8
0.6
C
0.4
0.2
0.0
1.0 1.2 1.4 1.6 1.8 2.0
b
Fig. 3.
10 F. Fu et al.: Evolutionary Prisoner’s Dilemma on heterogeneous Newman-Wattssmall-world network
1.1
1.0 1.0
m=800
0.9 0.9 m=1000
0.8 b=1.00 0.8 m=1200
0.7 b=1.02 0.7
b=1.08
0.6 b=1.20 0.6
c0.5 bb==11..3400 C 0.5
0.4 b=1.5 0.4
0.3 b=1.8 0.3
0.2 0.2
0.1 0.1
0.0 0.0
-0.1 -0.1
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Nh/N Nh/N
Fig. 4. Fig. 7.

1.0 0 0.8
1.0 Accumulated Accumulated
Average 0.7 Average
0.8 0.2000
0.8 0.6
b=1.1 0.5 b=1.3
0.6 0.4000 0.6
/Nh C 0.4
N
0.4 0.6000 0.4 0.3
0.5000
0.2
0.6000
0.2
0.2 0.8000 0.1
0.9000 0.0 0.0
1.000 A B
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 0.0 0.2 0.4 0.6 0.8 1.0 -0.1 0.0 0.2 0.4 0.6 0.8 1.0
b Nh/N Nh/N
Fig. 5.
Fig. 8.

1.1
1.0
0.9 Nh= 1
0.8 Nh= 10
0.7 Nh= 20
0.6 Nh= 41
c0.5 Nh= 241
0.4 Nh= 2001
0.3
0.2
0.1
0.0
-0.1
1.0 1.2 1.4 1.6 1.8 2.0
b
Fig. 6.