Heterogeneous pair approximation for voter models on networks

0.25 MB·English

by  Emanuele Pugliese

epl draft
Heterogeneous pair approximation for voter models on networks
Emanuele Pugliese1,2 and Claudio Castellano3
0
1 1 Dipartimento di Fisica, “Sapienza” Universita` di Roma, P.le Aldo Moro 2, I-00185 Roma, Italy
0 2 S. Anna School of Advanced Studies, P.za Martiri della Liberta` 33, I-56100 Pisa, Italy
2
3 SMC, INFM-CNR and Dipartimento di Fisica, “Sapienza” Universita` di Roma, P.le Aldo Moro 2, I-00185 Roma,

n Italy
a
J

6
2 PACS 87.23.Ge–Dynamics of social systems

PACS 64.60.aq–Networks
]
h PACS 05.40.-a–Fluctuation phenomena, random processes, noise, and Brownian motion
c
e Abstract.- Formodelswhoseevolutiontakesplaceonanetworkitisoftennecessarytoaugment
m the mean-field approach by considering explicitly the degree dependence of average quantities
(heterogeneous mean-field). Here we introduce the degree dependence in the pair approximation
-
t (heterogeneous pair approximation) for analyzing voter models on uncorrelated networks. This
a
t approachgivesanessentiallyexactdescriptionofthedynamics,correctingsomeinaccurateresults
s
ofpreviousapproaches. Theheterogeneouspairapproximationintroducedherecan beappliedin
.
t full generality to many other processes on complex networks.
a
m
-
d
n
o
c Introduction. – The voter model is one of the sim- this imitates a randomly chosen neighbor. In the reverse
[ plest possible models of nonequilibrium dynamics, as wit- voter dynamics [13] or invasion process the order of selec-

nessed by the number of different contexts where it has tion is opposite: it is the neighbor that imitates the node
3
v been considered, ranging from probability to theoretical selected first. An intermediate case is the link-update dy-
9 ecology and heterogeneous catalysis [1–3]. namics[12],whereonehastochooseatrandomfirstalink
8 andthenwhichofthetwonodesattheendofthelinkimi-
4 Eachnode ofagraphisendowedwitha binaryvariable
tatestheother. Onlythelink-updaterulepreservesacru-
5 (a spin) s =±1. At each time step a node and one of its
cial property of voter dynamics on lattices, conservation
. neighborsarechosenatrandomandthefirstnodebecomes
3 onaverageofmagnetization[12],whereasotherquantities
0 equalto the second. This microscopicdynamics givesrise
are conserved in the two other cases [11,13]. On regu-
9 to a nontrivial macroscopic phenomenology [3] with the
lar geometries the three types of dynamics coincide, but
0 formationofcorrelateddomainsthattendtogrowintime.
: when the topology is strongly heterogeneous important
v In finite systems this eventually leads to one of the two
differences arise: on scale-free networks [19] the scaling of
i possible ordered states (consensus) with either all spins
X averageconsensus time T with the systemsize N changes
up (+1) or down (−1). In the thermodynamic limit full
r depending on which of the three versions is considered.
a order is reached only in euclidean dimension d≤2.
From the point of view of statistical physics voter dy- Theoretical progress on the understanding of these is-
namicshastheveryinterestingfeaturethatitsoverlysim- sueshasbeenmadepossiblebytheapplicationofahetero-
plerulesallowforanalyticaltreatmentinmanycases. On geneousmean-fieldapproach[11,13,16]. The basic idea is
euclidean lattices many exact results are available [4,5]. todefinethedensitiesσ ofupspinsrestrictedtonodesof
k
In the past few years, in the framework of the huge in- degree k and to write down evolution equations for them.
terest around complex networks and dynamics on top of Inthiswayitispossibletopredictthatforthedirectvoter
them[6–8], the effectof nontrivialtopologiesonvoterdy- the average time T to reach consensus grows sublinearly
namics has been explored[9–18]. One interestingeffect of as a function of the system size N if the exponent γ of
the presence of a disordered connectivity pattern is that the tail of the degree distribution is smaller than 3, while
slightly different definitions of the model, perfectly equiv- it is linear for γ > 3. Instead, for the reverse and link-
alentonlattices,becomeinequivalentonnetworks. Inthe update versions of the dynamics the growth is linear for
direct voter case, one picks up a vertex at random and any γ > 2. Very recently, the theoretical picture for the
p-1
E. Pugliese et al.
voter model has been further refined by the application pothesis the equationofmotion for ρ is derived,involving
of an improved mean-field approach, a homogeneous pair the magnetization m and the first moment of the degree
approximation [20], taking into account dynamical corre- distribution µ. As mentioned before, the behavior pre-
lations amongnearestneighbors. This approachallowsto dictedchangesdramaticallyasafunctionofµ. Forµ<2,
follow in detail the temporal evolution of the system and ρgoestozeroexponentiallyfastintime,indicatingthata
to compute with remarkable accuracy even the prefactor fullyorderedconfiguration(consensus)isquicklyreached:
of the law that expresses the average consensus time as a theaveragetimeT toreachconsensusdoesnotdependon
power of N. the system size N so that T/N →0. For µ>2 instead it
Despite these achievements, some open problems re- ispredictedthat,overaveryshorttemporalscale,aquasi-
main, calling for further improvement. The first has to stationary state is reached, characterized by a density of
dowiththe predictionofatransitionforaveragedegreeµ active links
µ−2
equalto 2. Accordingto Ref.[20], whileforµ>2consen- ρS = (1−m2). (1)
2(µ−1)
sus is attained via a quasi-stationary intermediate state
over a long temporal scale, when µ < 2 order is reached ThesuperscriptS indicates,throughouttheLetter,quan-
exponentially fast. Contraryto this prediction, numerical titiesinthequasi-stationarystate. Theeventualbehavior
simulations on an Erdo¨s-Renyi graph do not display any of the system consists in the erratic variation of m and
singular behavior in correspondence of µ = 2 [20]. An- ρS that jointly fluctuate in time while obeying Eq. (1),
other issue that requires a clarificationhas to do with the until a fluctuation leads to the absorbing state |m| = 1,
use of the same approach for reverse voter dynamics. As ρ = 0, corresponding to full consensus. Other dynamical
it will be explained below, the application is straightfor- observables,describingtheorderingprocess,areexpressed
ward,buttheagreementwithnumericalsimulationsisnot using Eq. (1), as for example the average consensus time
complete. that, for uncorrelated initial conditions with magnetiza-
In this paper we go beyond the homogeneous pair ap- tion m=0, is
proximation introduced in Ref. [20] by allowing correla- 2 2
µ N (µ−1)µ N
tions between nearest neighbor nodes to depend on their T = ln(2)= ln(2), (2)
2ρS(m=0)µ2 (µ−2)µ2
degree (heterogeneous pair approximation). In practice
welift the assumptionofRef. [20]thatthe probabilityfor whereµ2 isthe secondmomentofthedegreedistribution.
a link between two nodes to be active (i.e. to connect While Eq. (1) agrees with remarkable precision with the
nodes in different spin state) does not depend on the de- outcome of numerical simulations for large values of µ,
gree of the nodes. This is analogous to the introduction significant discrepancies arise on many types of networks
ofthe heterogeneousmean-fieldapproach[8,21]necessary whenµgetscloseto2. Forexample,consideringasasub-
fordealingwiththeSISmodel(andmanyothers)onscale- strate an Erdo¨s-Renyi graph, the divergence of T/N for
freenetworks. Withtheheterogeneouspairapproximation µ→2 predicted by Eq. (2) is not seen in numericalsimu-
we find, for the direct voter dynamics, results in general lations [20] (Fig. 1). This indicates that, despite the very
very close to those obtained with the homogeneous pair good performance for large values of µ, the homogeneous
approximation but we get rid of the spurious transition pair approximation of Ref. [20] qualitatively fails in the
at µ = 2, in agreement with computer simulations. Also limit µ→2.
for the reverse voter and link-update dynamics we obtain The same indication emerges when the approach is
results in very good agreement with numerics, showing applied to the reverse voter dynamics. For the quasi-
that the heterogeneous pair approximation fully captures stationaryvalueofthedensityofactivelinksitisstraight-
thebehaviorofvotermodels. Thesesuccessfulresultscall forwardto obtain
for the application of the heterogeneous pair approach to ρS = µ2−2µ (1−m2). (3)
other nontrivialdynamical processes on networks,includ-
2(µ2−µ)
ingalsoepidemiologicalmodelsorevolutionarygames[22].
While the detailed implementation will differ depending Thisformulacoincides,asexpected,withtheresultforthe
on the case, the idea is completely general and may lead direct voter model [Eq. (1)] when the degree distribution
to substantial progress with respect to the heterogeneous is a delta-function so that µ2 = µ2. However, a care-
mean-field approach. ful comparison with numerics shows that Eq. (3) is not
correct when µ2 ≫ µ2 (see Figure 2). The parabolic de-
Problems with the homogeneous pair approxi- pendence on magnetization remains (not shown), but the
mation. – The homogeneous pair approximation intro- value for m = 0 predicted by Eq. (3) (diamond symbols)
ducedinRef.[20]isbasedonthederivationoftheequation isconsiderablylargerthanwhatsimulationsgive(circles),
of motion for the quantity ρ, the fraction of active links, unless γ is very large. Here, as in the rest of the paper
i. e. edgesconnecting nodesinoppositespinstate. Inthe unless specified otherwise, we compare the results of the
derivation, it is assumed that the probability for a link in theoretical treatment with simulations performed on the
the system to be active does not depend on the degree of uncorrelatedconfigurationmodel(UCM) [23],thatgener-
the nodes it connects, and is equal to ρ. Under this hy- ates graphs with power-law degree distribution P ∼k−γ
k
p-2
Heterogeneous pair approximation for voter models on networks
4
0.55
Heterogeneous mean-field approximation
Homogeneous pair approximation
Numerical, N = 1000 0.5
3 Numerical, N = 10000 0)
Heterogeneous pair approximation =
m 0.45
S(
k
ρ
N
T/2 0.4 NHuommeorgiecnaleous pair approximation
Heterogeneous pair approximation
0.35
1
0.3
10 100 1000
k
0
1 2 3 4 5
µ
Fig. 3: Plot of theprobability that alink connected to anode
ofdegreek isactiveinthequasi-stationarystate(andform=
0) for reverse voter dynamics taking place on an uncorrelated
Fig. 1: Average time to reach full consensus for direct voter networkwithγ =2.5andsizeN =106. Theblacksymbolsare
dynamics on an Erd¨os-Renyi graph, as a function of the av- the results of a numerical simulation. The dashed red line is
erage degree µ, starting from a symmetric uncorrelated ini- thepredictionofthehomogeneouspairapproximation,Eq.(3),
tial configuration σk(t = 0) = 1/2. The curve for the het- the blue solid line is the prediction of the heterogeneous pair
erogeneous mean-field approach is obtained using the formula approximation.
T/N = µ2/µ2ln(2) from Ref. [11]. The curve for the homo-
geneous pair approximation is obtained using Eq. (2). The
curvefortheheterogeneouspairapproximationisobtainedus- andupperboundofthedegreedistributionequaltoN1/2,
ing Eq. (10). For finite N and µ very close to 1, the network so that automatically there are no self-loops nor double-
is separated in many nonextensive disconnected components. links . The lower-cutoff of the degree distribution is 3.
Thisistheoriginofthediscrepancybetweennumericalresults
TheoriginofthediscrepanciesinFigs.(1)and(2)isnot
and the theoretical prediction in an interval that shrinks to
hardtoidentify. Itissufficienttoevaluatenumericallythe
zero as the system size diverges. Each point is the average
quantityρS,definedasthe fractionofneighborsofanode
over1000 realizations. k
of degree k that are in a different spin state in the quasi-
stationary state, and plot it as a function of k (Fig. 3).
While the approach of Ref. [20] assumes this quantity to
0.5 be the same for all nodes, a substantial dependence on k
shows up. Interestingly, a numerical check indicates that
0.45 also for direct voter dynamics ρSk has a sizeable depen-
dence on the degree (Fig. 4). Based on this evidence, we
now generalize the pair approximation approach by con-
0.4
0) sidering ρk,k′, i.e. allowing the density of active links to
= depend explicitly on the degree on the connected nodes.
m0.35
sρ( From ρk,k′ it is possible to determine all other quantities
of interest. The density ρ of active links connected to a
0.3 k
node ofdegreek is ρk = k′Qk′ρk,k′, where Qk =Pkk/µ
is the degreedistributionPofthe neighboursofa randomly
0.25
Numerical chosennode. The totaldensityofactivelinksρislikewise
Heterogeneous pair approximation
0.2 Homogeneous pair approximation given by ρ= kQkρk = k,k′QkQk′ρk,k′.
103 104 105 106 P P
N Heterogeneouspairapproximation forthe direct
voter model. – Thefirstgoalistodeterminetheequa-
tion of motion for ρk,k′. This quantity is modified if the
flipping node has degree k and one of its neighbours has
Fig.2: Reversevoterdynamics: quasi-stationaryvalueρS(m= degree k′ (or vice versa). Let us assume that the flipping
0) as a function of the system size for three different values of (firstselected)nodehasdegreek andcallk′′ the degreeof
γ (top tobottom, γ =2.5, γ =4, γ =8).
the copied (second selected) node. It is useful to consider
separately the two cases where k′′ 6=k′ or k′′ =k′.
p-3
E. Pugliese et al.
k’
0.6
j=3
Numerical k k"
Homogeneous pair approximation
Heterogeneous pair approximation
0) 0.5 k’
= n=1
m
S(k k’
ρ
0.4
Fig. 5: Direct voter dynamics: Illustrative representation of a
0.3
10 100 1000 configuration contributingto dρ3,7/dt. The first node selected
k ′′
has degree k = 7 and the second has degree k = 4, different
′
fromk =3. Filled(empty)circlesdenotespinsinstates=+1
(s = −1). The variation to the density ρk,k′ is due to the
j = 3 links that connect the flipping node to nodes of degree
Fig. 4: Plot of theprobability that alink connected to a node ′
k. Before the flipping of the first node selected, n= 1 of the
ofdegreek isactiveinthequasi-stationarystate(andform= links are active and j−n= 2 are inactive. After the flipping
0) for direct voter dynamics taking place on an uncorrelated
the state of links is reversed and the variation in the number
networkwithγ =2.5andsizeN =106. Theblacksymbolsare of activelinksbetween nodes of degreek and k′ is j−2n=1.
the results of a numerical simulation. The dashed red line is
thepredictionofthehomogeneouspairapproximation,Eq.(1),
the blue solid line is the prediction of the heterogeneous pair 1/σ(s)=4/(1−m2)) the formula becomes
s
approximation. P
∆ρk,k′ (k−1) 2
∆t = Qk′′ρk,k′′ k (cid:18)1− 1−m2ρk,k′(cid:19).
kX′′6=k′
Inthefirstcasethevariation∆ρk,k′ forasingledynami- (5)
calstep(occurringoveratime∆t=1/N)isdeterminedas When k′′ = k′, the value of ∆ρk,k′ is similar to Eq. (4)
follows. The probabilitythatanode instate s anddegree with (obviously)Qk′ instead of Qk′′, no sum over k′′, and
k flips is given by the probability P that the first node inthenumeratorofthelastfactorj+1−(n+1)−(n+1)=
k
selected has degree k times the probability σ(s) that it is j−2n−1,becausetherearej+1linkstonodesofdegreek,
in state s, times the probability Qk′′ that the second has n+1ofwhichareactiveintheinitialstateandinactivein
degree k′′ multiplied by the probability ρk,k′′/[2σ(s)] that the final. Summing up the two contributions and adding
thelinkconnectingthetwoisactive. Onehasthentomul- the symmetric terms with k and k′ swapped, we get
ttiiponlyotfhaisctqivueanlitnitkysbbyettwheeeansksoacinadtekd′v(aFriiga.t5io)n. Aofmthonegfrtahce- dρdkt,k′ =ρkk−k 1 +ρk′k′k−′ 1 + (6)
k−1otherlinks ofthe flipping node, the number ofthose
connecting to a node of degreek′ will be j distributed ac- −ρk,k′(cid:18)k1 + k1′ + 1−2ρmk 2k−k 1 + 12−ρkm′2k′k−′ 1(cid:19).
cording to a binomial R(j,k−1) with probability of the
singleeventequaltoQk′. Intheirturn,onlynoutofthese Fromthisequation,bysummingoverkandk′ oneobtains
j links will be active, with n binomially distributed with the equation of motion for the total density ρ
msinugltleipelyvebnyttphreobvaarbiialittiyonρok,fkρ′/k[,2k′σ(wsh)]e.nFninoaultlyofonjelinhkassgtoo dρ = QkQk′dρk,k′
dt dt
from active to inactive as a consequence of the flipping of Xk,k′
thenodeink. Thisisgivenbythevariationofthenumber π 4 π∗
of active links [(j −n)−n] divided the total number of = 2(cid:18)ρ−2µ(cid:19)− 1−m2 (cid:18)ρ∗2− µ2(cid:19), (7)
links between nodes of degree k and k′ (NµQ Q′). One
has then to sum over k′′ 6=k′, s, j and n, obtakininkg where π = kPkρk, π2∗ = kPkρ2k and ρ∗2 = kQkρ2k.
It is easy toPsee that, if onePassumes ρk =ρ, i.e.Pthat the
ρk,k′′ density does not depend on the degree (and as a conse-
∆ρk,k′ = Pk σ(s) Qk′′2σ(s) · (4) quence π = ρ, ρ∗2 = π2∗ = ρ2), Eq. (7) coincides with the
Xs kX′′6=k′ analogous equation of Ref. [20].
k−1 j j−2n After a time scale of order unity the quasi-stationary
R(j,k−1) B(n,j) .
Xj=0 nX=0 NµQkQk′ state is established. From dρk,k′/dt=0 we obtain
ρSk−1 +ρS k′−1
By performing explicitly the summations (and using ρSk,k′ = 2 ρSkk−1k+ρSkk′′−k1′ + 1 + 1 (8)
1−m2 k k k′ k′ k k′
(cid:0) (cid:1)
p-4
Heterogeneous pair approximation for voter models on networks
From this equation it is evident that all ρS are pro- is slightly modified
k,k′
portional to 1−m2. Imposing the consistency condition
πS(k−1)+πS(k′−1)
ρρkSk =forPakll′kQ.kT′ρhke,k′n,uwmeerhiacvaeliatesreattiovfecsoouluptleiodneoqfutahtiisonsestfoorf ρSk,k′ = 1−2m2 πkS(kk−1)+πkS′(k′k′−1) +µ/k+µ/k′.(12)
equationsallowsthedeterminationofthequasi-stationary (cid:2) (cid:3)
values of all ρS and hence of all ρS . where πk = k′Pk′ρk,k′. Much in the same way as for
k k,k′ the direct voPter dynamics a set of consistency equations
Fig.4showsthattheheterogeneouspairapproximation
for πS is solved iteratively. The values of πS, inserted
captures extremely well the detailed dependence of ρS on k k
k into Eq. (12) provide ρS , from which in turn ρS and ρS
k. Alsothecomparisonofthequasi-stationaryvalueofρS k,k′ k
can be computed. Fig. 3 shows that the agreement with
(recoveredby summing Eq. (8) over k and k′) with simu-
the values of the ρS obtained numerically is very good.
lationsforseveralvaluesofγ andN indicatesanexcellent k
Moreover, Fig. 2 shows that the quasi-stationary value
agreement (not shown).
of ρS is in fully satisfactory agreement with numerics for
The knowledge of the ρS allows to determine also the
k severalvalues of γ and N.
consensus time T to reach a fully ordered configuration.
Also in this case the consensus time is computed us-
Following the formalism of Ref. [16], using the backward
ing the backward Kolmogorov equation for T [16]. The
Kolmogorov equation, one simply needs to express the
transition probabilities in the quasi-stationary state are
transition probabilities R and L in terms of the ρ .
k k k obtained from those of the direct case with the replace-
In the quasi-stationary state the probability RS that the
number of nodes of degree k in state s=+1 inkcreases by ment PkQk′′ →Pk′′Qk:
1 is πS
RS = Q k (13)
k k 2
ρS
RkS =Pk(1−σk) Qk′′ρSk,k′′/[2(1−σk)]=Pk 2k. (9) LSk = RkS.
Xk′′
Using these quantities and the fact [13,16] that the dy-
TLSkhe=prRokSb.abUilsiitnygLtkhtehseatetxhperensusmiobnesroinsereodbutcaeindsisthesame, wnahmeriecsµc−o1n=serveksPtkh/ekq,uoannetiotbytaωi−ns1 = PkPkσkk−1/µ−1,
P
T =−τ(N)[(1−ω)ln(1−ω)−ωlnω] (10) T =−τ(N)[ω−1ln(ω−1)+(1−ω−1)ln(1−ω−1)], (14)
where ω = Q σ (t =0) is conserved by the dynamics and the temporal scale is
k k k
and P Nµ2 τ(N)= Nµµ2−1 . (15)
τ(N)= 2 kPkk2ρSk(m=0) (11) 2 k PkkπkS(m=0)
P P
sets the temporal scale over which consensus is reached. For uncorrelated initial conditions σ (t = 0) = 1/2 →
k
Noticethatwiththemean-fieldassumption,ρSk(m=0)= ω−1 =1/2,
1/2,Eq.(11)coincideswiththeresultofSoodandRedner T =τ(N)ln(2). (16)
τ(N)=Nµ2/µ2. With the homogeneouspairapproxima-
ThecomparisonofthepredictionsofEq.(16)withnumer-
tionρS(m=0)=(µ−2)/[2(µ−1)]itreturnstheresultof
k ical results is displayed in Fig. 6 and again confirms the
VazquezandEgu´ıluz[20]. WithEq.(10),validforanyde-
correctness of the theoretical treatment.
greedistribution,wehavecomputedthe valueofthe ratio
For link-update dynamics, all calculations proceed ex-
T/N for an Erdo¨s-Renyi graph of average degree µ. The
actlyasinthedirectvotercase,nowwiththereplacement
plotinFig.1presentsthecomparisonofthisvaluewiththe
predictionofpreviousapproachesandwiththeoutcomeof of PkQk′′ with QkQk′′. It is straightforwardto obtain
numerical simulations. The singular behavior for µ=2 is ρS(k−1)+ρS(k′−1)
an artifact of the homogeneous pair approximation, that ρSk,k′ = 2 ρkS(k−1)+ρkS′ (k′−1) +2. (17)
isremovedbytheheterogeneouspairapproximation,lead- 1−m2 k k′
(cid:2) (cid:3)
inginsteadtoasmoothbehavioraroundµ=2andavery The consistency condition ρk = k′Qk′ρk,k′ is solved it-
good agreement between theory and simulations. eratively. Once again, the valuesPof ρSk and ρSk,k′ obtained
inthiswayareinverygoodagreementwithnumerics(not
Heterogeneous pair approximation for the re-
shown). The transition probabilities for the Kolmogorov
verse and link-update voter dynamics. – The ap-
backward equation are RS = LS = Q ρS/2, that yield,
plication of the heterogeneous pair approximation to the k k k k
for the consensus time
reverse voter model proceeds along the same lines of the
direct dynamics. The only difference is that the factor
T =τ(N)[ω0ln(ω0)+(1−ω0)ln(1−ω0)], (18)
PkQk′′ is replaced by Pk′′Qk, because the role of the
flipping and the copied nodes is swapped. As a conse- where the conserved quantity ω0 is σ(s = +1), the total
quence, the formula for the quasi-stationaryvalue of ρk,k′ densityofnodesinstates=+1andτ(N)=N/[2ρS(m=
p-5
E. Pugliese et al.
1e+06 a promising avenue for future work. In particular,its role
Heterogeneous pair approximation could be crucial for understanding cases where heteroge-
γ = 2.5
γ = 8 neous mean-field approaches do not work [24].
1e+05 Heterogeneous mean-field approximation
∗∗∗
T10000 WethankA.BarratandR.Pastor-Satorrasforacritical
reading of the manuscript.
1000
REFERENCES
[1] Liggett T. M., Interacting Particle Systems (Springer,
100
100 1000 10000 1e+05 Berlin Heidelberg) 1985.
N [2] Durrett R. and Levin S., J. Theor. Biol., 179 (1996)
119
[3] Castellano C., Fortunato S. and Loreto V., Rev.
Mod. Phys., 81 (2009) 591.
Fig. 6: Reverse voter dynamics: Plot of the consensus time T [4] Krapivsky P. L., Phys. Rev. A, 45 (1992) 1067.
as a function of the system size N for two values of gamma. [5] FrachebourgL.andKrapivskyP.L.,Phys.Rev.E,53
Filledsymbolsarefornumericalresults,solidlinesarethethe- (1996) R3009.
oreticalpredictionsofEq.(16),dashedlinesarethemean-field [6] Dorogovtsev S. N. and Mendes J. F. F., Adv. Phys.,
predictions of Ref. [13]. 51 (2002) 1079.
[7] Newman M. E. J., SIAM Rev., 45 (2003) 167.
[8] Barrat A., Barth´elemy M. and Vespignani A., Dy-
0)]. Notice that using ρS(m = 0) = 1/2 one recovers namical Processes on Complex Networks (Cambridge Uni-
the mean-field result τ(N) = N, i.e. the independence of versity Press, Cambridge) 2008.
consensus time of network features [13]. However, since [9] Castellano C., Vilone D. and Vespignani A., Euro-
ρS(m = 0) actually depends on the network, this inde- phys. Lett., 63 (2003) 153.
pendence is just an artifact of the mean-field approach. [10] ViloneD.andCastellanoC.,Phys.Rev.E,69(2004)
016109.
Conclusions. – In this Letter, we have introduced a [11] Sood V. and Redner S., Phys. Rev. Lett., 94 (2005)
heterogeneous pair approximationto deal with the order- 178701.
ingdynamicsofvotermodelsonnetworks. This approach [12] Suchecki K., Egu´ıluz V. M.andSan Miguel M.,Eu-
rophys. Lett., 69 (2005) 228.
allows to determine, with great precision, not only the
[13] Castellano C., AIP Conf. Proc., 779 (2005) 114.
scaling but also the prefactors of the relevant dynamical
[14] Castellano C., Loreto V., Barrat A., Cecconi F.
observables (quasi-stationary density of active links, con-
and Parisi D., Phys. Rev. E, 71 (2005) 066107.
sensus time). The heterogeneouspair approximationthus
[15] Suchecki K., Egu´ıluz V. M. and San Miguel M.,
constitutesaverygoodapproximatetheoryforvotermod- Phys. Rev. E, 72 (2005) 036132.
elsonuncorrelatednetworks. Theonlypointthatremains [16] Sood V., Antal T. and Redner S., Phys. Rev. E, 77
to be understood is the origin of the very small discrep- (2008) 041121.
ancy(atmostofthe orderof1%)thatstillexists between [17] Baxter.G.J.,BlytheR.A.andMcKaneA.J.,Phys.
numerical and analytical results. Rev. Lett., 101 (2008) 258701.
While the treatment presented here is valid for un- [18] Baronchelli A. and Pastor-Satorras R., J. Stat.
Mech., (2009) L11001.
correlated networks, generalization to correlated nets is
straightforward. In particular, if P(k′|k) is the probabil- [19] Baraba´siA.-L.andAlbertR.,Science,286(1999)509.
[20] VazquezF.andEgu´ıluzV.M.,NewJ.Phys.,10(2008)
ity that a neighbor of a randomly chosen node of degree
063011.
k has degree k′, the stationarity conditions for the three
[21] Pastor-Satorras R. and Vespignani A., Phys. Rev.
types of dynamics, Eqs. (8), (12) and (17), remain the Lett., 86 (2001) 3200.
same, provided P(k′|k) replaces its expression Qk valid [22] Lieberman E., Hauert C. and nowak M. A., Nature,
in the uncorrelated case, in the definitions of ρk and πk: 433 (2005) 312.
ρk = k′P(k′|k)ρk,k′, πk = k′µP(k′|k)/k′ρk,k′. While [23] Catanzaro M., Bogun˜a´ M., and Pastor-Satorras
thegePneralizationoftheanalyPticalapproachisimmediate, R.,Phys. Rev. E, 71 (2005) 027103.
an interesting question deserving further investigations is [24] Castellano C. and Pastor-Satorras R., Phys. Rev.
whether the agreement between theory and numerics re- Lett., 96 (2006) 038701.
mains good even in the correlated case.
The idea behind the heterogeneous pair approximation
introducedhereisfullygeneralanditsapplicationtoother
dynamicalmodelsevolvingoncomplexnetworksisindeed
p-6