Brazilian Journal of Physics, vol. 33, no. 3, September, 2003 1

Diverging Tendencies in Multidimensional Secession

Arne Soulier, Natalie Arkus, and Tim Halpin-HealyPhysics Department, Barnard College, Columbia University, NY, 10027-6598

Received on 2 May, 2003

We review mean-field and fluctuation-dominated behaviors exhibited by the Seceder Model, which movesan evolving population to various critical states of self-organized segregation, delicately balancing opposedsociological pressures of conformity & dissent, and giving rise to rich ideological condensation phenomena.The secession exponent and finite societal Seceder limits are examined.

This paper discusses recent research [1] on theSeceder Model [2], an intriguing far-from-equilibriumstochastic model of opinion dynamics in which thecompeting tendencies of conformity and dissent cause anevolving population tofragment, disperse, and coalesce,forming distinct groups characterized by free and frequentinterchange of individuals. We imagine a populationthat is ultimately economic, sociological, or political incharacter, with interactions based upon shared investmentstrategies, cultural opinions, or electoral inclinations. Thisis in sharp contrast to many well-known examples ofinanimate cluster formation found in Nature (e.g., in theastrophysical context- Saturnian rings, globular clusters,Virgo galactic supercluster, etc.), where the underlyingforces are physical in origin; here, the interactions are basedupon signaling, opinion, and information exchange. TheSeceder Model bears close spiritual kinship to a numberof distinguished evolutionary minority games, among themthe Zhang & Challet variant [3] of Arthur’sEl Farol Barproblem [4], as well as its stochastic generalization byJohnson and coworkers [5], where it was discovered thatthe introduction of chance brought the population to astate of self-organized segregationin which two groupsadopted diametrically opposed strategies. Further work byHod & Nakar [6] revealed a dynamical phase transition inthis setting, between 2-group segregation and single-groupclustering, driven by the economic cost-benefit ratio implicitin the model. Relevant, too, is the work of Hauert andcollaborators [7, 8], who studied 3-group dynamics amidstcooperation, defection, and abstention in a noncompulsorypublic goods game, mischievously dubbed by popularscience pundits [9] as the “Physics of Loners”! In itsinitial rendering, they considered a well-mixed formulationin which participants could interact readily with any othermember of the population. Afterwards, they introducedspatial structure, finding peculiar traveling wave phenomenawithin the model; see Hauert’s web page. Finally, ina paper titled “Meet, Discuss, and Segregate!” Weisbuchet al. [10] consider a model of opinion dynamics inwhich a population of interacting agents adjust continuous

positions via randombinary interactions subject to thresholdconstraints. High thresholds (large inertial barriers) producea societal convergence of opinion to a single group inoverall agreement, whereas low thresholds result in multipleopinion clusters. These, and related consensus models, suchas those of Bonabeau [11], Sznajd [12], and others, havebeen discussed in wry pedagogical fashion by Stauffer in anice review [13]. Issues on a grander scale were addressedby Stauffer and his close Brazilian colleagues [14].

The Seceder mechanism was devised to demonstratethat a dynamics favoring individuality, yet permittingconformity, cannot only create distinct groups, but alsoyields a rich diversity of cluster-forming dynamics. Theessential trick was to benefit individuals that distinguishthemselves from others. This is natural, for one recallsfrom an epidemological point of view, genetic differencescan enhance long-term survival probabilities. Similarly,for players in a minority game or traders on the stockmarket floor, distinct strategies can yield large returns. TheSeceder Model is succintly stated: Within a populationof N individuals, each described by ad-dimensionalopinion vector (representing a multicomponent investmentstrategy, ideological or political position), we implement thefollowing update algorithm-

• A member (the “voter”) of the population is chosen atrandom to revise his position.

• Pick m individuals (a “polling group”) from thepopulation; calculate the mean vector of this subset.

• From among this selection multiplet of sizem, choosethe mostdistinct member; i.e., the fellow farthest from themean- that is, the subset center of mass.

• The “voter” adopts an ideological position nearby thedissenter; aside from a superposed, but small, deviate, thisadjustment is perfect. [15]

Clearly, the model as defined induces two intrinsicallyopposed tendencies. If the polled group is tightly-knit inthe first place, the result is to enhancehomogeneity,sincethe voter, potentially at some ideological distance fromthat subset, vacates his position and effectively conforms.On the other hand, should there be an outlier among the

2 Arne Soulier, Natalie Arkus, and Tim Halpin-Healy

selection multiplet, the voter choses the underdog, ratherthan the majority view. Therefore, dissent, and by extension,secession,is implicitly encouraged. Of course, thereis a catch, since distinct individuals frequently, but notalways, inspire a following, thereby becoming part of themainstream themselves. In dissenting, they guarantee thedemise of their distinction. In what follows, we employthe standard convention thatN -updates of the populationcorresponds to one generational time-step. Using thisdefinition, each individual will, on average, be replaced onceper generation. Our initial condition is always the same,an entirely homogeneous population with all individualsideologically located at the origin.

For convenience, we consider first the cased=1, asingle ideological axis, and investigate them-dependenceof this Seceder Model. We expect the limitm → N,which guarantees enhanced cross-correlation throughout thesociety, to elicit inevitably, a simpler “mean-field” typeof behavior, if only in the extreme case whenm = N,when we’re averaging over the entire population using thesocietal mean to determine the most distinct, and ultimately,adopted position. In fact, this is what we discovered first,numerically. The shock, though, comes with the suddennessof the transition. There is, surprisingly, a marked changeof behavior as we switch from a triplet(m = 3) toquartet(m = 4) polling groups. In Figure 1a, we showsingle runs of the Seceder Model using multiplet selectionm = 3 − 8, within a large population,N = 512. Form = 3, we have classic Seceder pattern formation, withself-similar branching characterized by three dominant, butfluctuating arms, with ample small-scale stochastic structureassociated with the transient appearance of variously short-lived subbranches. Rather than a gradual crossover, we findthat for mc = 4, the typical stable configurationabruptlyinvolves two groups, not three. In addition, these twobranches exhibit only minor fluctuations. Settingm = 5further diminishes the fluctuations (one barely notices anysubbranches here...), but hardly affects the tilt of what seemsto be the nearlylinear space-time trajectories of the twodivergent groups. Next, form = 6&7, there is, strangely,a discrete jump to an altogether different, but closely alliedpair of segregating groups. Withm = 8(&9), another jumpin velocity is evident in our space-time plot. So it goes witheach successive even-odd pair of multiplet selection group.Thus we see that the dominant dynamic of Seceder involves,for m ≥ mc = 4, self-organized segregation into two evenlypopulated opposing groups with continuous interchange ofindividuals over the course of time. Similar dynamicalsegregation was reported by Johnsonet al., within theirstochastic evolutionary minority game.

Ensemble averaging over many realizations, we cansystematically study the growth of the population diameterover time. Figure 1b shows a a double-log plot of thatdiameter for multiplet selection groupsm = 3 − 12,drawn from a populationN = 512, with 3000 runsguaranteeing decent statistics. Inferences made previously

from the single runs are quickly corroborated here- form ≥4, we retrieve essentially linear behavior, with successiveeven/odd selection groups paired up; indeed, the logarithmrenders them nearly indistinguishable already at timest >100 for m = 6&7, though they can be separated easilyenough at earlier times. The figure renders apparent thedecreasing gap size in the discrete velocity spectrum. Thescaling exponents are 0.99+ for m = 8 − 10, fitting datafor t ∈ (1000, 3000). Only for the case of triplet selection,m = 3, where the segregative dynamics involves stablemultiple-branch solutions, do we see a very different power-law scaling behavior. Indeed, form = 3, which correspondsto the lowest curve in Figure 1b, the diameter asymptoticallyscales with asecession exponentof 3/4, our extracted valuebeing 0.74±0.01.

It’s interesting to consider finite-size effects within theextremesocietal Secederlimit. In Figure 1c, we showthe generic behavior when the entire population serves asthe selection group. A glance at this figure reveals thatfor the smallest populations,N = m = 3 − 6, thedominant effect is a single, fairly tightly knit group thatis essentially localized near the origin; i.e, the trajectorywanders little from the horizontal time axis. While thereis, of course, no branch formation per se forN = m =3, we see already forN = m = 4, the occasionalappearance of a short-lived branch budding off of theprimary Brownian path. The effect is amplified forN =m = 5&6 where the transient branching phenomena ismore apparent, especially in the latter case where we noticethat there is an effectively entropic repulsion between thenewly created branch and its ancestor; i.e., they headoff at some characteristic angle (in this case, roughly 45degrees)w.r.t. each other. At these population sizes, there’slittle difference typically between trajectories generated bytriplet and higher multiplet selection dynamics. ForN =m = 7&8, however, we gain a glimpse of the true mean-field asymptotic behavior- a strong tendency to bifurcateinto two dominant groups, heading off symmetrically fromthe origin. Occasionally, branch-splitting occurs; thoughthis happens less and less frequently for increasingm,the branches are longer lived. Note, especially the clearprogression in this manner forN = m = 6, 7, 8. Asbefore, we notice a strong temporal correlation betweenthe statistical death of one branch, marked by the nearlyimmediate birth, or splitting off of another, so as preservethe sanctity of choice. In this context, were a branch todie, without replacement elsewhere, the result would bea single group (nearly all identical individuals), anathemato a secessional dynamic which encourages difference, atleast for a while. For larger population sizes,N = m =10&25, we see essentially deterministic divergence of thetwo groups. Of course, on larger time scales, splitting wouldinevitably occur for these smallish mesoscopic populations,but the asymptotic behavior is clear. For infinitem, thetrajectories would be linear, symmetrically splayed aboutthe origin, in a V-pattern, suggested already in Figure 1a.

Brazilian Journal of Physics, vol. 33, no. 3, September, 2003 3

0 100 200 300 400 500−75

−60

−45

−30

−15

0

15

30

45

60

75

d=1 Stochastic Seceder Model

m=3

4

5

6 78

1 10 100 1000 100001

10

100

1000

10000

m=3

m=4m=5

m=6,7m=8,9m=10

m=12

(b)

3/4

0 200 400 600 800 1000time

−150

−100

−50

0

50

100

150

25 10

8

7

6

5

4

3

(c)

Figure 1. a) Single runs for the 1d stochastic Seceder model, usingselection multipletsm=3,4&5, 6&7, 8, within a population of sizeN = 512, b) Scaling plot of the divergent population diameter,averaged over 3000 realizations. Form ≥ mc = 4, the behavior isessentially linear, whereas a triplet selection produces a divergentpopulation that scales ast3/4, and c) Finite-size effect within thesocietal Secedermodel, wherein the entire population serves as theselection group. Shown are the casesm=N=3-8,10,25.

Investigating higher dimensions,d=2 and 3, we discoverthat the critical multiplet selection group size remainsfixed atmc=4, but that the rate of secession scales nearlylinearly in these cases even form=3. More specifically,

we show in Figure 2a, the time trace of the three divergentgroups in a typical two dimensional simulation. Noticethe roughly equilateral arrangement of the three branches.A careful examination of the temporal evolution of thepopulation diameter reveals, see Figure 2b, that at earlytimes (t < 100) the group separation grows as a squareroot- the solid line in the figure has slope 1/2. By contrast,later on, the separation rate is just a hair sublinear- thedashed line has slope 1. Fits to the final 2000 timesteps yields exponents 0.889,0.941,0.992 for populationsizesN=512,1024,2048, respectively. Simulations ind=3produce very similar results; e.g., the secession exponentbeing 0.948 for the intermediate population size. Perhapsthe most stunning conclusion from the higher dimensionalnumerics is that additional degrees of freedom associatedwith extra ideological axes are, from the point of view of thetriplet Seceder dynamic, mainlyirrelevantsince the typicaldynamical situation involves collapse to an hyperplanecontaining just three groups. If one starts, ind=3, withfour equally-sized parties at the corners of a tetrahedron,it simply won’t last- one group always goes extinct! It isan intrinsicallyunstableconfiguration, eventually sufferingdimensional-reductionto a flat triangular arrangement.

−50 −30 −10 10 30 50 70−60

−40

−20

0

20

40

d=2 Triplet Seceder Model

(a)

100

101

102

103

10410

0

101

102

103

104

1/2

1

(b)

Figure 2. a) Short-time trace in the xy-plane of 2dm=3 tripletSeceder Model, and b) Scaling plot of the divergent populationdiameter for the same model, averaged over 1000 realizations.Slopes of the fit lines are indicated.

4 Arne Soulier, Natalie Arkus, and Tim Halpin-Healy

Interestingly, much can be gleaned about thisstochasticSeceder Model by considering a deterministic, discretizedversion written as a system of coupled, nonlinearreplicatorequations, [16] well-known to some members of themathematical biology/game theory community. Forexample, if we consider the simplest case of the triplet(m=3) selection dynamics and study the stability ofB=3branches (opinion groups), the relevant first-order rateequations read:

ẋ1 = x31 + 3x1(x22 + x

23) + 3x1x2x3 − x1

ẋ2 = x32 + 3x2(x21 + x

23)− x2

where x1,2,3 represent population fractions of the threeclusters. We’ve not bothered with the equation forẋ3,since it is identical to the first (modulo interchange ofindices); probability conservation requiresx1+x2+x3=1,and since group 2 lies midway between 1 & 3 on the1D ideological axis, it is fundamentally disadvantaged inmatters of distinction. If we examine the equation forẋ1,the nature of the RHS should be apparent- the cubic termpromotes homogeneity [17], the next- secession [18], whilethe trilinear piece, involving equal representation in thepolled group by each of the three branches, is shared equallyby the growth variables of the outer two groups [19]. Theseequations are trivially solved (though less and less so asB increases...), and we find several unstable fixed points-(1, 0, 0), (12 ,

12 , 0), and permutations therein, as well as a

singlesuperstablefixed point (FP):( 25 ,15 ,

25 ). Investigating

the situation forB=4 groups, we have:ẋ1 = x31+3x1(x

22+x

23+x

24)+3x1x2x3+6x1x3x4−x1

ẋ2 = x32 + 3x2(x21 + x

23 + x

24) + 3x2x3x4 − x2

Note, again, that all terms are third order, because we’restill looking at triplet selection,m=3. In any case, becausethe stable branches are symmetric about its central axis,the flow equations for the remaining variables are easilyobtained via the interchangex1 ↔ x4 and x2 ↔ x3and, indeed, the globally stable fixed point (no negativeeigenvalues!) must lie within this reduced subspace, mirrorvariables identified. For the case at hand, invoking theconstraintx2 = 1/2 − x1 and demandinġx1 = 0 leadsto the cubic equation7x31 − 6x21 + 5/4x1 = 0, yielding(x1, x2, x3, x4) = ( 514 ,

17 ,

17 ,

514 ), in addition to the 2-

branch solutions( 12 , 0, 0,12 ) and (0,

12 ,

12 , 0). Of course, if

we brute force numerically integrate the coupled ODEs onthe computer, following the trajectories from a randomlygenerated initial condition, we flow with 100% probabilityto this unique superstable fixed point. The situation forB = 5 recalls the 3-branch case; try it! You’ll find the stableFP( 413 ,

213 ,

113 ,

213 ,

413 ), though there’s an unstable solution,

( 14 ,14 , 0,

14 ,

14 ), self-similar to the 2-branch(

12 , 0,

12 ). The 6-

branch is eaily seen to be irrational, involving solution of thecubic equation20x32 + 80x

22 + 133x2 − 17 = 0, possessing

one real, 2 imaginary roots. Don’t bother looking for an 8-branch FP; it doesn’t exist! Instead the flows will bring youto ( 1140 ,

18 , 0,

110 ,

110 , 0,

18 ,

1140 ), exactly self-similar to theB=3

superstable FP( 25 ,15 ,

25 ). For additional details regarding the

self-similarhierarchical gapped spectrum intrinsic to tripletselection, see [1]. Finally, the replicator formalism provides

some hidden extra benefits, among them an explanationfor the critical critical selection multiplet sizemc = 4.Assuming arbitrarym, one performs a stability analysis ofthe 2-branch FP( 12 , 0,

12 ), focussing on the Seceder term

arising from the multinomial expansion [1]; in this fashion,we discoverẋ2 =

[2(m1

)(12

)m−1 − 1]x2 + O(x22), so weflow back to vanishingx2 for 2m ≤ 2m−1; i.e., m ≥4, as the prefactor of the quadratic term is easily shownto be negative. Lastly, with the replicator equations, wesee why the greatest degree of population fragmentationoccurs for the case of one ideological dimension wherethree groups are the norm though 4, 5, 6 groups are notuncommon. By contrast, ind ≥ 3, the population self-organizes to a coarsened trio of three tightly-knit opinionclusters, effectively collapsing to a 2D ideological plane [1].

Our present efforts focus on the political perspective-understanding the kinetic symmetry breaking phenomena,intermittent dynamics, and subpopulation fluctuationsintrinsic to the Seceder mechanism [20].

THH is grateful to P.M.C. & S. M. de Oliveira forhospitality during his brief visit to UFF-Niteroi (BoaViagem, Brasil), and to D. Stauffer for advice and a much-remembered introduction to percolation theory. N. Arkus*& THH were supported by NSF DMR-0083204, CondensedMatter Theory. Finally, many thanks to John Lee for helpwith a multitude of computational issues.

*NSF-RUI Undergraduate Summer Research Asst.

References

[1] A. Soulier and T. Halpin-Healy, Phys. Rev. Lett.90, 258103(2003).

[2] P. Dittrich, F. Liljeros, A. Soulier and W. Banzhaf, Phys. Rev.Lett. 84, 3205 (2000).

[3] D. Challet and Y.-C. Zhang, Physica A246, 407 (1997).

[4] W.B. Arthur, Am. Econ. Assoc. Proc.84, 406 (1994).

[5] N.F. Johnson, P.M. Hui, R. Jonson and T.S. Lo, Phys. Rev.Lett 82, 3360 (1999).

[6] S. Hod and E. Nakar, Phys. Rev. Lett.88, 238702 (2002).

[7] C. Hauert, S. De Monte, J. Hofbauer, and K. Sigmund,Science296, 1129 (2002).

[8] G. Szab́o and C. Hauert, Phys. Rev. Lett.89, 118101 (2002).

[9] Phys. Rev. Focus, http://focus.aps.org/v10/st10.html

[10] G.Weisbush, G. Deffuant, F. Amblard, and J.-P. Nadal,Complexity7, 55 (2002).

[11] E. Bonabeau, G. Theraulaz, and J. L. Deneubourg, PhysicaA217, 373 (1995).

[12] K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C11, 1157(2000).

[13] D.Stauffer,Sociophysics Simulations,cond-mat/0210213.

[14] S. Moss de Oliveira, P. M. C. de Oliveira, and D. Stauffer,Evolution, Money, War and Computers,(Teubner, Leipzigand Stuttgart, 1999).

Brazilian Journal of Physics, vol. 33, no. 3, September, 2003 5

[15] To be more explicit- ifr denotes a position vector in thed-dimensional opinion space, thenr′voter = rdissenter + ε,where the components ofε are randomly drawn from theinterval{− 1

2,− 1

2}. The voter’s new ideological position is

close to that of the dissenter, but not an exact replica. The twoindividuals thus become of the same mindset, though theiropinions are not identical.

[16] J. Hofbauer and K. Sigmund,Evolutionary Games &Population Dynamics,(Cambridge University Press, UK,1998).

[17] Here, like begets likesince the polled multiplet,{1,1,1}, isuniformly represented by group 1. Withx1 the probability ofa randomly chosen member of the population being in group1, the likelihood of a selection triplet beingstrictly group 1 isjustx31.

[18] In this case, a group 1 member is a solitary minority in the

polled subset, outnumbered 2-to-1 by either a group 2 (or3) majority; even so,the underdog that wins the vote.Theprobability of this happening isx1x22 (or x1x

23), there being

three ways, however, in either scenario.

[19] This presents ademocratic dilemmaof sorts, since the pollinggroup cannot decide between the two outmost groups, 1 and3. In this paper, we simply flip a coin. The combinatoricprefactor, 6, is thus split in two, equal contributions of3x1x2x3 going to the rateṡx1 andẋ3. In ref [20], below, thedynamical effects ofideological symmetry breaking, whichremoves this degeneracy, are considered in some detail,particularly in regard to strong fluctuations within partymemberships of the different groups.

[20] A. Soulier and T. Halpin-Healy, cond-mat/0305356; aSeceder applet is available via this abstract.