Characterization of the Nonequilibrium Steady State of a Heterogeneous Nonlinear $q$-Voter Model with Zealotry

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by  Andrew Mellor

epl draft
Characterization of the Nonequilibrium Steady State of a
Heterogeneous Nonlinear q-Voter Model with Zealotry
6 Andrew Mellor1, Mauro Mobilia1 and R.K.P. Zia2,3
1
0 1 Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
2 2 Department of Physics & Astronomy, Iowa State University, Ames, IA 50011, USA
r 3 Department of Physics, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061, USA
a
M

9 PACS 89.75.-k–Complex systems
PACS 02.50.-r–Probability theory, stochastic processes, and statistics
h] PACS 05.40.-a–Fluctuation phenomena, random processes, noise, and Brownian motion
p
Abstract –We introduce an heterogeneous nonlinear q-voter model with zealots and two types
-
c of susceptible voters, and study its non-equilibrium properties when the population is finite and
o well mixed. In this two-opinion model, each individual supports one of two parties and is either
s
. a zealot or a susceptible voter of type q1 or q2. While here zealots never change their opinion,
s a qi-susceptible voter (i = 1,2) consults a group of qi neighbors at each time step, and adopts
c
their opinion if all group members agree. We show that this model violates the detailed balance
i
s whenever q1 6=q2 and has surprisingly rich properties. Here, we focus on the characterization of
y
the model’s non-equilibrium stationary state (NESS) in terms of its probability distribution and
h
currentsinthedistinctregimesoflowandhighdensityofzealotry. WeunveiltheNESSproperties
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in each of these phases by computing the opinion distribution and the circulation of probability
[
currents, as well as the two-point correlation functions at unequal times (formally related to a

2 “probability angular momentum”). Our analytical calculations obtained in the realm of a linear
v Gaussian approximation are compared with numerical results.
6
6
7
3
0
.
1 Introduction. – SinceSchelling’spioneeringwork[1] In this Letter, we focus on a variant of the VM known
0 there has beenincreasinginterestin using simple theoret- asthetwo-statenonlinearq-votermodel(qVM)[11]which
6
ical models to describe social phenomena such as the dy- has attracted much interest [12]. In the qVM, a voter
1
namics of opinions [2]. In this context, individual-based can be influenced by a group of q neighbors. The version
:
v models commonly used in statistical physics are particu- with q = 2 is closely related to the well-known models
i
X larlyinsightful,astheyrevealthemicro-macroconnection of Refs. [13]. Motivated by important psychology and so-
insocialdynamics [1,2]. The votermodel (VM) [3]serves ciology tenets [5,6], the basic ideas underlying the qVM
r
a as a reference to describe the evolution of opinions in so- and zealotry have been combined into the q-voter model
cially interacting populations. (See e.g. [2,4] and refer- with inflexible zealots (qVMZ) [14]. Indeed, social scien-
ences therein.) In spite of its paradigmatic role, the VM tistshaveestablishedthatconformitybyimitation,anim-
relies on a number of unrealistic assumptions, such as the portantmechanismfor collective actions,is observedonly
total lack of self-confidence of all voters and their per- whenthegroup-sizeislargeandcanbealteredbyindivid-
fect conformity, which invariably leads to a consensus. In uals who are able to resist the group pressure [6,7]. It is
fact,ithasbeenshownthatmembersofasocietyrespond understandablethatsocialconformityisunlikelyforsmall
differently to stimuli and this greatly influences the un- groupsandcanbesignificantlysuppressedbythepresence
derlying social dynamics [5–7]. An approach to model a ofzealots. IntheqVMZdynamics,bothgroup-sizelimited
population with different levels of confidence is to assume conformity and zealotry are accounted for. Furthermore,
that some agents are “zealots” who favor one opinion [8] in a well-mixed setting, this dynamics obeys detailed bal-
or maintain a fixed opinion [9]. Since the introduction of ance, so that the exact stationary distribution is easily
these simple types of behavior in the VM, a large variety found [14]. The system resembles one in thermal equi-
of zealot models have been studied, see, e.g., Refs. [10]. librium, characterized by two phases: As zealotry is low-
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A. Mellor et al.
eredthroughacriticallevel,theopiniondistributiontran- bour on a square lattice, our system behaves exactly as a
sitions from being single-peaked to being bimodal, with two-dimensionalrandomwalker,withinhomogeneousand
non-trivial switching dynamics (in finite populations). biased rates. Thus, our simulation runs consist of record-
Unlikeinthemodelsdescribedabove,thepopulationsin ing the trajectories of such a random walker. Meanwhile,
oursocietyarehighlyheterogeneous. Inprinciple,wemay the full stochastic process is defined by a master equation
describetherelevantsituationofdifferentresponsestoso- (ME) for the evolution of P (~n;T) [21], the probability to
cial stimuli by considering a distribution of q’s [5]. Does find our system in state ~n, T time steps (attempts) af-
this generalization modify the conclusions of the qVMZ? ter some initial configuration~n0. Since our main interest
If so, how? In this Letter, we explore the simplest way is the stationary distribution, P∗(~n), we suppress refer-
to this broader view, namely, a population with just two ences to ~n0. As T increases by unity, a walker at ~n steps
subgroups with different q’s (denoted by 2qVZ): q < q . to ~n′ with probability W (~n ~n′), a process represented
1 2 →
Similarto the qVMZ,phasetransitionsexistin the 2qVZ. by the ME P (~n;T +1) = ~n′G(~n,~n′)P (~n′;T). Here,
Unlike in qVMZ, the dynamics of this model does not ~n′ (n1 1,n2),(n1,n2 P1) is one of the four near-
∈ { ± ± }
obey detailed balance, so the system relaxes into a non- est neighbors of ~n, from which the transitions occur with
equilibrium steady state (NESS). Thus, from the stand- respective stepping probabilities W±(~n) and W±(~n):
1 2
point of statistical physics, 2qVZ is a highly non-trivial
extension. In general, there is no simple way to compute S n Z +n +n qi
W+(~n) = i− i + 1 2 (1)
aNESSdistribution[15], whilepersistentprobabilitycur- i N (cid:18) N 1 (cid:19)
rent prevails [16]. As a consequence, there are observable n Z +S +S− n n qi
quantitieswhicharetriviallyzerointheqVMZthatdonot W−(~n) = i + 1 2− 1− 2 . (2)
i N (cid:18) N 1 (cid:19)
vanish in the 2qVZ. For example, opinions among those −
with smaller q change more readily, and “drive” those in
From here, it is straightforward to write an explicit form
theothersubgroup. Thoughsubtle,directed“oscillations”
for , as well as joint probabilities (~n,T;~n′,T′) at two
associated with fluctuating quantities can be measured. G P
different times2. Much of our attention here will be de-
Here, we report results of a baseline study with the sim-
voted to the change, P (~n;T +1) P (~n;T), given by
plestcaseq1,2 =1,2. Ourmethodsincludestochasticsim- −
a sum of probability currents which account for transi-
ulationsandnumericalsolutionsofthemasterequationfor
tions into, or out-of, the configuration~n. Specifically, the
systemsfinitesizeN,aswellasacontinuumversionbased
net probability current from ~n to ~n′ (n + 1,n ) is
on the Fokker-Planckequation and its analysis through a K (~n;T)=W+(~n)P(~n;T) W−(~n′)P (≡~n′;T)1,anda2sim-
linear Gaussian approximation (LGA) [16–19]. 1 1 − 1
ilar expression for K (~n;T) for ~n to (n ,n +1). Thus,
2 1 2
Modelspecification and NESS .– Ourmodel,the P is intimately related to the current K~ =(K ,K ).
1 2
G
2qVZ, consists of a population of N voters who support To verify that this dynamics violates detailed balance
one of two parties, the opinion of each denoted by 1. (and time reversal), we may apply the Kolmogorov crite-
±
Some voters are inflexible zealots, never changing their rion [23] on any closed loop, the simplest being four ~n’s
opinions. Their numbers are denoted by Z±, the sub- aroundaplaquette[22]. Asaconsequence,oursystemset-
script showing the associatedopinion. The rest are swing tles into a NESS, with non-trivial P∗ and stationary cur-
voters of two types, denoted by q1 and q2. Known as qi- rentK~∗. Thoughthemainbehaviorofourmodelisquali-
susceptibles, their numbers are Si, with i = 1,2. During tativelythesameasintheqVMZ,thepresenceofK~∗leads
the evolution, each agent maintains its behavior, so that to important, distinguishing features, displayed through
AiZt±tiseaanacdhzSetaiilmaorteetashltleencpo,nnaoseravvcoettdieor(nwisiisthcthaSok1see+nn.Sa2Ht+oZrwa+nev+deoZrm,−ifa=nadNqi)i-f. Napnhodytseicctoahrlaroteblatshteierovnoasrb,dleehsrn,iosnufjciihnTdaisc≡emsePiann~nts,h,ne′hnnl′iianittj≡ePrP∗i(s~~nnc,nrTiuP;c~ni∗a′(,l~n:0))i.,
susceptible is chosen, then it collects the opinions from a (j) is associated with the earlier (later) variable when
random group of qi neighbors and adopts the opinion of T > 0. One key observable is the antisymmetric part
the group only if their opinion is unanimous 1 (see sup- of n n : Beingoddunder time reversal,ithighlights
plementary material (SM) [20]). For simplicity, we inves- thehuindjeirTly6=i0ngNESScharacteristic,andso,vanishesinthe
tigate this model on a complete graph (well-mixed pop- qVMZ. When the time difference is infinitesimal (T = 1,
ulation). Since there is no spatial structure, our system largeN), it can be identified as the total “probability an-
is completely specified by the number ni of qi-susceptible gular momentum” [24], in analogy with the classical an-
voters holding opinion +1 (also denoted by ~n=(n1,n2)). gular momentum associated with current loops in fluids.
Since configuration space is a discrete set of S1 S2 To illustrate these novel features, we examine in detail a
×
pointsandupdatesinvolveasinglesteptoanearestneigh- specific model – Z = Z , S = S , q = 1,2 – and
+ − 1 2 1,2
provide several explicit results below.
1AsinRefs.[11,12,14],weallowrepetition. NotethatinRef.[11]
avotercanchangeitsopinionwithafliprateǫevenintheabsence
ofconsensusamongitsqneighbors. Here,asinmostofRefs.[12,14], 2Assuming T > T′, P(~n,T;~n′,T′) = GT−T′(~n,~n′)P(~n′,T′).
wesetǫ=0. Detailswillbeprovidedin[22].
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Nonlinear q-voter model: heterogeneity & zealotry
Analytic results & simulation studies . – currents form closed loops, which lead us to identify
Mean-field Approximation (MFA): This approach offers
the most intuitive picture, valid when N with fixed L = x K∗(~x) x K∗(~x) d~x (4)
densities: (z±,si,xi) = (Z±,Si,ni)/N (c→on∞tinuous vari- ij Z~x(cid:2) i j − j i (cid:3)
ables subjected to z +z +s +s =1 and x [0,s ]).
In this limit, the rat+es W−± bec1ome2w+(~x) = (si ∈ x )µiqi as the total ‘probability angular momentum’ [24], by for-
and wi−(~x)=xi(1−µ)qi,iwhere µ≡zi++x1+xi2−cleiarly m( al~xanaJ~lodg~xy) iwnitflhuidthsewittohtaclurmreanstsdeannsgiutylaJr~.mAosmaenptsueum-
represents the fraction holding opinion +1. In the MFA, ~x ×
dRotensor in arbitrary dimensions, Lij has just a single in-
averagesofproductsarereplacedbyproductsofaverages,
dependent component, (say) L = . Since K~∗ is linear
and from the ME, we find the rate equations (REs) 12 L
inP∗, weidentify asthe steadystate averageofa func-
L
x˙ ∂ x =w+ w− =(s x )µqi x (1 µ)qi. (3) tion of ~x. Below we show that the simple approximation
i ≡ t i i − i i− i − i − K~∗ ~xP∗ providesmanyanalyticresults,e.g.,anexpres-
∝
sion for the two-point correlation at unequal times 3
As in the qVMZ [14], the REs admit one or three fixed
points, depending on z . At the fixed point(s), x∗, the
± i C (τ) x x (for τ =0). (5)
ratio ρ≡µ∗/(1−µ∗) satisfies z++ i=1,2si/(1+ρqi)= ij ≡h i jiτ 6
1/(1 + ρ), while x∗i = si/(1 + ρqiP). For models with Decomposing Cij into the symmetric and antisymmetric
z+ = z−, ρ = 1 is always a solution, associated with the parts,weseethatC˜ C C isodd inτ,highlighting
symmetric fixed point x∗ =s /2 (denoted by ~x(0) below). ij ≡ ij− ji
i i time reversal violation and serving as a principal charac-
Aboveacriticaldensityofzealots,z ,thisistheonlyfixed
c teristic of a NESS. Indeed, to lowest order (in ~x), L is
point and is stable. For z <z , ~x(0) turns unstable, while ij
two others (denoted by ~x(±))cemerge and both are stable. given by ∂τC˜ij .
(cid:12)0
Forourspecificmodel(z± =z,si =s=1/2−z,qi =i), Linear Gaussia(cid:12)(cid:12)n Approximation (LGA): In this scheme,
the MFA for zc is 1/6 (i.e., sc =1/3). When z <zc, ~x(±) we consider deviations from a fixed point, ~ξ ~x ~x∗,
are given by the two solutions to ρ+(1/ρ)+1 = 1/(2z), and, keeping the lowest non-vanishing order, w≡e fin−d the
associated with the spontaneously breaking of the Ising- linearized version of the FPE [21]
like symmetry (ρ 1/ρ). Thus, this MFA predicts the
⇔
same phase transition as in the qVMZ (pitchfork bifurca- ∂ P(~ξ,t)= ∂i D ∂j +Fj ξ P(~ξ,t), (6)
t ij i j
tion [14]). Of course, being deterministic, it cannot ac- Xi,j n o
count for fluctuations.
Fokker-Planck equation (FPE): In finite populations, de- where ∂i ≡ ∂/∂ξi, Dij = δijwi∗/N with wi∗ = wi+(~x∗) =
mographic fluctuations are important, as they drive wi−(~x∗), andFij ≡−(∂x˙i/∂xj)|~x=~x∗ is the linear stability
interesting time-dependent phenomena in the station- matrix at ~x∗. Thus, the LGA is defined by two matrices:
ary state. For large but finite N, these fluctua- Dij and Fij, or D and F for short. This linearized FPE
tions are embodied in the probability density P (~x;t), canbetranslatedintoLangevinequationswithlineardrift
the continuum version of P (~n;T). Here, t T/N F~ξplusGaussianwhitenoisecontrolledbyD[21]. Below
≡ −
also becomes continuous, as P (~n;T +1) P(~n;T) we show that the LGA provides much insight into the
− →
N−1∂ P(~x;t). To the leading, non-trivial order in non-equilibrium character of our model by allowing us to
t
1/N, the evolution is adequately captured by the FPE: find analytic expressions for various quantities. From the
∂ P (~x;t) = ∂ ∂ u (~x)P +v (~x)P [21], where explicit expressions [20], we find that detF 1 6z (in
uti ≡ wi++Pwi−i=1/,22N∂xiahn∂dxivii≡ wi− −iwi+. iClearly, the ianlltchaeserse)giaonndsiotfsienitgeernevsta.luAesls,odeexnpoteecdtebdy, Dλ±is,∝Oare(1−p/Nosi)t,ivsoe
right-h(cid:0)and-side c(cid:1)an be identified as the divergence of the
that fluctuations of ~ξ are O 1/√N , and we note that
probability current density
(cid:16) (cid:17)
D > D which confirms that the q -susceptibles are
11 22 1
K (~x;t)= ∂[u P]/∂x v P more likely to change their opinions.
i i i i
− − If D−1F is symmetric, then detailed balance is satisfied
(continuum version of Ki(~n;T)). The stationary prob- and P∗ exp ~ξD−1F~ξ/2 is a Gaussian distribution
ability density P∗(~x) is given by 0 = ~ K~∗. For a for which∝D−1Fn/−2 is the “pootential”. Here, we find that
∇ ·
dynamics which satisfies detailed balance, K~∗ necessarily D−1Fisnot symmetric. Nevertheless,thesolutionof(6)is
vanishes, leading to equations for P∗ that can be easily stillaGaussian[16–19]: P∗(~ξ) exp ~ξC−1~ξ/2 ,where
solved [21]. In a NESS, K~∗ = 0, i.e., non-trivial cur- ∝ (cid:16)− (cid:17)
rentspersist. Clearly,the curlo6f K~∗ does notvanish and, the elements of C are ξiξj 0, the truncated Cij(0)4. It
knowninfluiddynamicsasthe vorticity, ~ K~∗ isaone- canbeexpressedinterm(cid:10)sof(cid:11)theeigenvectorsandeigenval-
component field in two dimensions. Of∇co×urse, K~∗ can ues of F [16,17], or by solving [FC]=D [18,19], where
S
also be expressed as the curl of another field, the stream 3 Alsoknownasthelaggedcorrelationorlaggedcovariance.
function. (See [20] for some details). In other words, the 4(cid:10)ξiξj(cid:11)0≡hxixji0−hxiihxji=Cij(0)−hxiihxji.
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A. Mellor et al.
[FC] denotes the symmetric part of FC. Deferring the
eSxplicit forms of C to elsewhere [22], we present here the
implications for K~∗ and the physical observables.
Since~ξP∗ = C∂~P∗,thesteadycurrentsinEq.(6)can
−
be written as K~∗ = [FC D]~P∗ [16]. We emphasize
that, given [FC] = D, t−he m∇atrix FC D is precisely
the antisymmSetric part of FC , here den−oted by [FC].
A
Thus, K~∗ is manifestly divergence free.
Proceedingtoobservables,weconsiderL orsimply,L.
ij
Since K~∗d~x = 0, the ~x in Eq. (4) can be replaced by
~x
~ξ. In tRhe framework of the LGA, we readily find the re-
markably simple expression L = 2 [FC]. In this setting,
we see that FC = D+L/2, placinAg this angular momen-
tum on an equal footing with diffusion5. In our specific
2qVZ, the only independent component is , given by
L
s2/[(6 8s)N] and 4z2/[(2z+3)N] for ~x(0) and ~x(±),
respect−ively. Further, we recall that L is intimately re-
lated to the two-point correlation C(τ) ξ ξ 6. In
≡ i j τ
the LGA, C(τ) is explicitly Cexp FTτ [(cid:10)19]. H(cid:11)ere, its Fig.1: (Color online). Stationarydistributionandprobability
−
antisymmetric part is just one ind(cid:0)epende(cid:1)nt quantity and current obtained from theexact numerical solution of theME
so, we focus on (say) C˜12(τ). Dropping the subscript, we inthehigh/lowzealotryphase(left/rightpanels)withS =30.
∗
find the explicit expression Top: P as function of ~x = ~n/N; dark/light blue (grey) en-
codes a high/low probability. Bottom: The wind-field vectors
e−λ−τ e−λ+τ represent K~∗ at each point ~x. Parameters are Z = 20 and
C˜(τ)= − (7) N =100 in (a, c), and Z =13 and N =86 in (b,d).
L(cid:18) λ λ (cid:19)
+ −
−
which exposes some noteworthy features: Clearly,
∂ C˜(τ) = . Moreover, its long time behavior is gov- thatinn2,indicativeoftherelativeeasewithwhichtheq1-
τ
cearnlecdorbrey(cid:12)(cid:12)(cid:12)l0aλti−o,nLsthaetusmneaqllueasltteimigeens,vaC˜lu(eτ)oifsFn.onU-mnolinkeottoynpici-, stoustchepet“ibcleenstcehr”antgyepitchaelilry.opSininiocen,tshoestehawtitdhtehys sstcaayleclwositehr
√N ( 10 here), we should expect only a qualitative fit
with a peak at τˆ =(λ λ )−1ln(λ /λ ). ≈
+− − + − fromGaussians. (ii)Thealignmentoftheellipsesimpliesa
BelowweshowthatallthesepredictionsbytheLGAare
strong correlationbetween the two variables, an expected
borne out in simulations and exact numerical results (for
result of both groups of susceptibles following the whims
appropriateregions). SincetheLGAisformulatedaround
of the other.
a single fixed point, it clearly cannot describe double-
In the z < z case, we see that the q -susceptibles dis-
peaked distributions or escape times. By contrast, near c 2
play a larger spontaneous symmetry breaking than q -
eachpeak,thereliabilityoftheLGAimprovesinthelimit 1
voters, reflecting the same behavior as in the qVMZ.
of N with fixed z =0,z .
Exact→nu∞merical solution6 : Forc systems with small S, nu- Meanwhile, if left alone, the q1-susceptibles would reach
acoexistence state [9]. Thus,the brokensymmetry inthe
merical methods can be used to obtain P∗, by exploiting
the relation [ (~n,~n′)]∞ = P∗(~n) (independent of ~n′!). opinion of the former acts as an external imbalance on
G the latter, dragging them to lean toward one pole or the
For example, for S = 30, by iterating 2τ = τ τ just
G G G other. Now,duetothestrongfinitesizeeffects,theregion
64 times, we find changes at . 10−20. Illustrated in
aroundeachpeak appearsquite asymmetricandseriously
Fig.1(a,b)are heatmaps (contourplots) on30 30 grids
× non-Gaussian when N is small. We also note that the
for two cases: Z =20 and 13. Associated with above and
peaks are linked via a “ridge” , the lowest point along
belowz ,they clearlyshowthe expectedsinglevs. double
c which is often referred to as a “gap” by mountaineers. If
peaked distributions. In the SM [20], we show this tran-
we consider lnP∗, then this gap represents the lowest
sition in S = 50 systems with many such plots compiled −
barrier between two “wells.” Also known as the saddle
into a movie.
point, its height is expected to control the escape times
In Fig. 1a, we see that the contours resemble ellipses
faced by the random walker trapped in one or the other
typical of Gaussian distributions. Two other prominent
well. As we expect the height difference between the sad-
features are: (i) The width in n is much smaller than
1 dle and the well bottom to scale with N, we anticipate
escapetimes to scale witheN, as foundin the qVMZ [14].
5Note our distribution has unit “mass” (R P∗ = 1) so that the
unitsofLarex2/t,preciselythoseofdiffusion. Finally,note thatthe analysisofthe criticalregionz ∼=zc
6Specifically,L=2A(cid:2)∂τ C(τ)|0(cid:3). necessitates a detailed finite size scaling study, which is
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Nonlinear q-voter model: heterogeneity & zealotry
beyond the scope of this Letter.
From P∗, we have computed numerically other exact
quantities of interest. The wind-field like plots of K~∗
shown in Fig. 1(c,d) provide good impressions of the gen-
eral counterclockwise swirl. Other characteristics (vortic-
ity and stream function) are displayed in the SM [20].
More quantitatively, the physical observables are readily
obtainedandcanbecomparedwiththe predictionsofthe
LGA. Here, due to symmetry, the exact ~x is ~s/2, re-
h i
gardless of the location of the peak(s). The simplest non-
trivialquantities are two point correlations(C, ) and we
L
find that the predictions of the LGA are in qualitative
Fig.2: (Color online). C˜(τ)vs. τ inthehighzealotryphase,
agreementwithexactcalculations[20]inthehighzealotry
withN =100,Z =20, andS =30: Theresultsofsimulations
phase z >z . In principle, we access the exactdecay con-
c (black)obtainedbysamplingevery0.01timestepandaveraged
stants (λ±) by analyzing Gτ numerically, a task deferred over99×106datapoints. ComparisonwiththeLGAexpression
to a future study [22]. In the low zealotry phase, z < z ,
c (7) with λ+ =0.833 and λ− =0.091 (red/grey), see text.
weemphasizethatP∗ includesall trajectories,withvisits
nearbothx(±). Yet,theLGAcanbeexpectedtobegood
onlyaroundone ofthetwopeaks. Hence,blindlycomput- andforthbetweenthe wells,implyingthatP∗ isattained.
ingCor fromthisP∗ willnotallowustocomparethem As a result, the findings for C are also in excellent agree-
L
with the predictions of LGA. Indeed, it is a highly non- ment (within 1%) with the exact results [20]. However,
trivial task to interpret , since a detailed understanding in this case the comparison with the LGA predictions is
L
of the contributions from tunneling is necessary. As for pointless, since the LGA is devised for just one well.
comparisons with simulations results, the key is whether Turning to large systems, two examples (Z = 400 and
the runs are long enough to permit a good sampling of 800, S = 1000) are offered in the SM [20]. In the low
both wells. In summary, while the LGA succeeds in cap- zealotry case, the walker remains in one well for the en-
turing the essentials of the z systems here, the finite size tire run. Thus, it is meaningful to compare both sets of
effects are too large for good quantitative agreement. For data with MFA/LGA predictions. For z < z , the sim-
c
z < z cases, we expect that the predictive power of the ulation results of ~x compares well with the MFA ~x(±).
c
h i
LGA will improve when the zealotry is asymmetric since In both cases, the data for the correlations n n and
h i ji0
tunneling events then become extremely rare [14,22]. n n are in quantitatively good agreement with the
Simulation studies: While the above methods yield exact LhGiAjip1redictions for C andFC [20]. In summary,we have
results, they are restricted to small systems. To study found that the LGA is indeed quite reliable for large S,Z
larger N’s, we rely on stochastic simulations, based on (around each ~x∗). Meanwhile, the various methods pre-
running a random walker on a S S lattice with the bi- sented in this section all point to the presence of interest-
ased and inhomogeneous stepping×probabilities (1,2), and ing new phenomena associated with the NESS aspect of
performed using the Gillespie algorithm [25]. Using z’s 2qVZ, namely, the presence of observable quantities odd
not particularly close to either 0 or z , our runs are up to under time reversal.
c
108 steps, for systems as large as N = 3600. The entire
Summary and outlook. – In this Letter, we intro-
trajectory of each run is recorded, giving us the time se-
duced a generalization of the qVMZ (q-voter model with
ries~n(T)andso,~x(t)and~ξ(t). Examiningthese,wefind
zealots) [14] which takes into account expected inhomo-
that, within 1000time steps ofstarting at (0,0), the walk
geneities in the behavior of swing voters and the presence
appearstobeinasteadystate. Withthesetraces,wecan
of zealots. In arguably the simplest generalization of the
construct time averages and obtain ~x and the general
h i qVMZ, we have just two groups of swing voters, distin-
two-point correlation function C (τ) in the NESS .
ij guished by needing a consensus of q or q of its neigh-
1 2
First,asacheck,wecollecteddataforthesmallsystems bors to adopt their opinion. As in Ref. [14], our model
discussed above (S = 30, with Z = 20,13). For the for- is characterized by two phases: When the fraction z of
mer, we find the truncated Cij(0) to be (C11,C12,C22)= zealots is low, the long-time opinion distribution is bi-
(1.74,1.66,3.53) 10−3. Further, we compile C˜(τ) and modal whereas it is single-peaked when z is above a crit-
×
find the behavior predicted in Eq. (7), see Fig. 2. By ical value z . However, a major and far-reaching differ-
c
fitting with this form, we find λ = 0.833,0.091 and ence between the qVMZ and ours is that detailed balance
±
L∼=2.16×10−4. BothCij(0)andLareinexcellentagree- is violated here. Hence, though the qualitative features
ment(within0.6%)withtheexactresult[20]. Bycontrast, are similar to those recently reported in [14], our system
the LGA predictions are qualitatively acceptable (from a settles into a genuine NESS. As a result, there are persis-
few % to the right order of magnitude). For the Z = 13 tent probability current loops which are odd under time
case, the walker spends much of its time wandering back reversal. We investigate these currents and some observ-
p-5
A. Mellor et al.
ablemanifestationsthereof,inthe simplebutgenericcase [9] Mobilia M., Petersen A. and Redner S., Journal
q = 1,2. with S = S and Z = Z . Using numerical of Statistical Mechanics: Theory and Experiment, 2007
1,2 1 2 + −
methods for small systems, Gaussian approximation for (2007) P08029.
large ones, and simulation runs for both, we arrive at a [10] GalamS.andJacobsF.,PhysicaA:StatisticalMechan-
icsanditsApplications,381(2007)366;Sznajd-Weron
comprehensive picture for our system. Quite remarkably,
K., Tabiszewski M. and Timpanaro A. M., EPL (Eu-
weshowthatthissimplemodelexhibitsstationarymicro-
rophysicsLetters),96(2011)48002;XieJ.,Sreenivasan
scopic current loops (see Fig. 1), resulting in oscillations
S., Korniss G., Zhang W., Lim C. and Szymanski
in certain macroscopic observables (e.g., the antisymmet- B. K., Physical Review E, 84 (2011) 011130; Masuda
ric part of the two-point correlation function at unequal N., Scientific Reports, 2 (2012) ; Acemoglu D., Como
times, see Fig. 2). While the detailed relationships be- G., Fagnani F. and Ozdaglar A.,Mathematics of Op-
tween microscopic probability currents and macroscopic erations Research, 38(2013) 1; Nyczka P. and Sznajd-
socialphenomenaremaintobeexplored,ourstudyclearly Weron K., Journal of Statistical Physics, 151 (2013)
points to the presence,albeit subtle, of predator-preylike 174; Waagen A., Verma G., Chan K., Swami A.
oscillations. The overall counter-clockwiseflow in K~∗ im- and D’Souza R., Physical Review E, 91 (2015) 022811;
pliesthatfluctuationsintheq -susceptiblesfollowthoseof Arendt D. L. and Blaha L. M., Computational and
2 Mathematical Organization Theory, 21 (2015) 184; Ma-
the other group, much like the population of lynxs follow
suda N.,New Journal of Physics,17(2015) 033031; Bo-
thoseofhares. We believe this stemsfromthe presenceof
rile C., Molina-Garcia D., Maritan A. and Mun˜oz
“leaders” and “followers” in a society. Clearly, this study
M. A.,Journal of Statistical Mechanics: Theory and Ex-
providesonly the initial steps towardsa systematic inves-
periment, (2015) P01030; Mobilia M., Chaos, Solitons
tigation of multi-q VM’s, which are expected to display & Fractals, 56 (2013) 113; Mobilia M., Physical Re-
other interesting phenomena. For this particular model, view E, 86 (2012) 011134; Mobilia M., Physical Review
muchmorecanbeexamined,e.g.,finitesizeeffects,scaling E, 88 (2013) 046102; Mobilia M., Journal of Statistical
properties near zc, and full distributions of [24]. Be- Physics, 151 (2013) 69.
L
yond studying the 2qVZ and similarly tractable models, [11] Castellano C., Mun˜oz M. A. and Pastor-Satorras
thegoalofourlong-termeffortsistoexplorefundamental R., Physical Review E, 80 (2009) 041129.
issuesofNESS,inanattempttoformulateanoverarching [12] Slanina F., Sznajd-Weron K. and Przyby la P.,
EPL (Europhysics Letters), 82 (2008) 18006; Galam
framework for non-equilibrium statistical mechanics.
S. and Martins A. C., EPL (Europhysics Letters), 95
(2011) 48005; Przyby la P., Sznajd-Weron K. and
∗∗∗ Tabiszewski M., Physical Review E, 84 (2011) 031117;
Nyczka P., Cisl o J. and Sznajd-Weron K., Physica
We are grateful to B Schmittmann and J B Weiss A: Statistical Mechanics and its Applications, 391(2012)
for enlightening discussions. This research is supported 317; Timpanaro A. M. and Prado C. P., Physical Re-
partly by a US National Science Foundation grant: OCE- view E,89(2014)052808; Je¸drzejewskiA., ChmielA.
1245944. The support of the UK EPSRC and Bloom and Sznajd-Weron K., Physical Review E, 92 (2015)
Agency (Leeds, UK) via a CASE Studentship to AM is 052105.
gratefully acknowledged(Grant EP/L50550X/1). [13] Sznajd-WeronK.andSznajdJ.,InternationalJournal
ofModernPhysicsC,11(2000)1157;LambiotteR.and
RednerS.,EPL(Europhysics Letters), 82(2008)18007.
REFERENCES [14] Mobilia M., Physical Review E, 92 (2015) 012803.
[15] HillT.L.,JournalofTheoreticalBiology,10(1966)442.
[1] Schelling T. C., Micromotives & Macrobehavior (WW [16] Zia R.K.P.andSchmittmann B.,Journal of Statistical
Norton & Company) 2006. Mechanics: Theory and Experiment, (2007) P07012.
[2] Castellano C., Fortunato S. and Loreto V., Re- [17] Lax M., Reviews of Modern Physics, 38 (1966) 541.
views of Modern Physics, 81 (2009) 591. [18] Weiss J. B., Tellus A, 55 (2003) 208.
[3] Liggett T. M., Interacting Particle Systems (Springer) [19] Weiss J. B., Physical Review E, 76 (2007) 061128.
1985. [20] Mellor A., Mobilia M. and Zia R.K.P.,
[4] Galam S., Sociophysics: A Physicist’s Modeling of Figshare (2016), Supplementary Material:
Psycho-political Phenomena(SpringerScience&Business https://dx.doi.org/10.6084/m9.figshare.2060595
Media) 2012; Sen P. and Chakrabarti B. K., Socio- [21] Van KampenN. G.,Stochastic Processes inPhysics and
physics: An Introduction (OxfordUniversityPress) 2013. ChemistryVol.1(Elsevier)1992;GardinerC.W.etal.,
[5] Granovetter M., American Journal of Sociology, 83 Handbook of Stochastic Methods Vol. 3 (Springer Berlin)
(1978) 1420. 1985.
[6] Latan´e B.,American Psychologist, 36 (1981) 343. [22] Mellor A., Mobilia M. and Zia R.K.P., In prepara-
[7] Asch S. E., Scientific American, 193 (1955) 35; Mil- tion.
gram S., Bickman L. and Berkowitz L., Journal of [23] Kolmogorov A.,Math. Ann., 112 (1936) 155.
Personality and Social Psychology, 13 (1969) 79. [24] Shkarayev M. S. and Zia R., Physical Review E, 90
[8] Mobilia M., Physical Review Letters, 91 (2003) 028701; (2014) 032107.
Mobilia M.andGeorgievI. T.,Physical Review E,71 [25] Gillespie D. T.,The Journal of Physical Chemistry, 81
(2005) 046102. (1977) 2340.
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