We introduce the vacillating voter model in which each voter consults two
neighbors to decide its state, and changes opinion if it disagrees with either
neighbor. This irresolution leads to a global bias toward zero magnetization.
In spatial dimension d>1, anti-coarsening arises in which the linear dimension
L of minority domains grows as t^{1/(d+1)}. One consequence is that the time to
reach consensus scales exponentially with the number of voters.Comment: 4 pages, 6 figures, 2-column revtex4 forma

Hegselmann R

Liggett T M

R Lambiotte

Redner S

S Redner

Vazquez F

Wolfram S

English

arXiv

We introduce the vacillating voter model in which each voter consults two neighbors to decide its state, and changes opinion if it disagrees with either neighbor. This irresolution leads to a global bias toward zero magnetization. In spatial dimension d&gt;1, anti-coarsening arises in which the linear dimension L of minority domains grows as t. One consequence is that the time to reach consensus scales exponentially with the number of voters

Lambiotte, R.

Redner, S.

Repository of the University of Namur

RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHEAuthor(s) - Auteur(s) :Publication date - Date de publication :Permanent link - Permalien :Rights / License - Licence de droit d’auteur :Bibliothèque Universitaire Moretus PlantinInstitutional Repository - Research PortalDépôt Institutionnel - Portail de la Rechercheresearchportal.unamur.beDynamics of vacillating votersLambiotte, R.; Redner, S.Published in:Journal of Statistical Mechanics: Theory and ExperimentDOI:10.1088/1742-5468/2007/10/L10001Publication date:2007Document VersionPublisher's PDF, also known as Version of recordLink to publicationCitation for pulished version (HARVARD):Lambiotte, R & Redner, S 2007, 'Dynamics of vacillating voters', Journal of Statistical Mechanics: Theory andExp riment, no. 10. https://doi.org/10.1088/1742-5468/2007/10/L10001General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 23. Jun. 2020Dynamics of vacillating votersThis article has been downloaded from IOPscience. Please scroll down to see the full text article.J. Stat. Mech. (2007) L10001(http://iopscience.iop.org/1742-5468/2007/10/L10001)Download details:IP Address: 138.48.20.157The article was downloaded on 30/12/2012 at 07:21Please note that terms and conditions apply.View the table of contents for this issue, or go to the journal homepage for moreHome Search Collections Journals About Contact us My IOPscienceJ.Stat.Mech.(2007)L10001ournal of Statistical Mechanics:An IOP and SISSA journalJ Theory and ExperimentLETTERDynamics of vacillating votersR Lambiotte1,2 and S Redner31 GRAPES, Universite´ de Lie`ge, Sart-Tilman, B-4000 Lie`ge, Belgium2 INMA, Universite´ catholique de Louvain, 4 avenue Georges Lemaitre,B-1348 Louvain-la-Neuve, Belgium3 Center for Polymer Studies and Physics Department, Boston University,Boston, MA 02215, USAE-mail: Renaud.Lambiotte@uclouvain.be and redner@bu.eduReceived 5 October 2007Accepted 8 October 2007Published 23 October 2007Online at stacks.iop.org/JSTAT/2007/L10001doi:10.1088/1742-5468/2007/10/L10001Abstract. We introduce the vacillating voter model in which each voterconsults two neighbors to decide its state, and changes opinion if it disagreeswith either neighbor. This irresolution leads to a global bias toward zeromagnetization. In spatial dimension d > 1, anti-coarsening arises in which thelinear dimension L of minority domains grows as t1/(d+1). One consequence isthat the time to reach consensus scales exponentially with the number of voters.Keywords: coarsening processes (theory), critical phenomena of socio-economicsystems, scaling in socio-economic systems, probability theoryArXiv ePrint: 0710.0914c©2007 IOP Publishing Ltd and SISSA 1742-5468/07/L10001+8$30.00J.Stat.Mech.(2007)L10001Dynamics of vacillating votersThe voter model [1] gives an appealing, albeit idealized, description for the opiniondynamics of a socially interacting population. In this model, each node of a graphis occupied by a voter that has one of two opinions, ↑ or ↓. The population evolvesby: (i) picking a random voter; (ii) the selected voter adopting the state of a randomlychosen neighbor; (iii) repeating these steps ad infinitum or until a finite system necessarilyreaches consensus. Descriptively, each voter has no self confidence and follows one of itsneighbors. With this dynamics, a voter chooses a state with a probability equal to thefraction of neighbors in that state, a feature that renders the voter model soluble in alldimensions [1, 2].In this work, we investigate a variation that we term the vacillating voter model.By vacillating, we mean that a voter very much lacks confidence in its state. In anupdate, if a voter happens to select a random neighbor of the same persuasion, the voteris still not convinced that this state is right. Thus the voter selects another randomneighbor and adopts this state. This vacillation causes a voter to change state with alarger probability than the fraction of disagreeing neighbors, and leads to a bias towardthe zero-magnetization state in which there are equal densities of voters of each type.Thus vacillation inhibits consensus, but due to a different mechanism than that in theprototypical Axelrod model [3], the bounded compromise model [4] and its variants [5]. Forthese latter models, consensus is hindered because of the absence of interaction whenevertwo agents become sufficiently incompatible. For vacillating voters, it is individualuncertainty that forestalls consensus. The vacillating voter model also differs from modelsthat incorporate ‘contrarians’ [6] because voters still try to imitate their neighbors.The update steps in the vacillating voter model are as follows.(1) Pick a random voter.(2) The voter picks a random neighbor. If the neighbor disagrees with the voter, thevoter changes state.(3) If the neighbor and the voter agree, the voter picks another random neighbor andadopts its state.(4) Repeat steps 1 and 2 ad infinitum or until consensus is reached.For example, the probability that a vacillating voter on the square lattice flips is0, 12, 56, and 1, respectively, when the number of anti-aligned neighbors is 0, 1, 2, and ≥ 3(figure 1). In contrast, for the classic voter model, the flip probability is k4, where k is thenumber of neighbors of the opposite opinion. We now explore the consequences of thisvacillation on voter dynamics.Consider first the mean-field limit. Here the density x of ↑ voters obeys the rateequationx˙ = −x[1− x2] + (1− x)[1− (1− x)2]= x(1− x)(1− 2x). (1)The first term on the right accounts for the loss of ↑ voters in which a ↑ voter is firstpicked (factor x), and then the neighborhood cannot consist of two ↑ voters (factor 1−x2).Similarly, in the second (gain) term, a ↓ voter is first picked, and then the neighborhoodmust contain at least one ↑ voter. The factorized form shows that there are unstable fixeddoi:10.1088/1742-5468/2007/10/L10001 2J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 1. Illustration of an update for the vacillating voter on the square lattice(left and middle). For the configuration on the right, the central voter flips withprobability 5/6 because out of the six ways of selecting two neighbors, only onechoice leads to both neighbors agreeable (dashed).points at x = 0, 1 and a stable fixed point at x = 1/2. Thus a population is driven to thezero-magnetization state.However, because consensus is the only absorbing state of the stochastic dynamics, afinite population ultimately reaches consensus. To characterize the evolution to this state,we first study the exit probability En, defined as the probability that a population of Nvoters ultimately reaches ↑ consensus when there are initially n ↑ voters. Then En obeysthe backward equation [7]En = wn→n+1En+1 + wn→n−1En−1 + wn→nEn, (2)where wn→m is the probability for the transition from the state with n ↑ voters to m ↑ votersin an update. This equation expresses the probability to exit from n as the probabilityto take one step (the factors w) times the probability to exit from the point reached afterone step. In the large-n limit, we write x = n/N , and the transition probabilities becomewn→n+1 = (1− x)[1− (1− x)2]wn→n−1 = x(1− x2)wn→n = x3 + (1− x)3.Substituting these in (2), writing En±1 → E(x± δx), and expanding to second orderin δx, gives3x(1− x)2N∂2E∂x2+ x(1− x)(1− 2x)∂E∂x= 0, (3)with solutionE(x) =∫ x−1/2−1/2e2Ny2/3 dy/∫ 1/2−1/2e2Ny2/3 dy. (4)Notice that E(x) approaches the constant value 1/2 for increasing N (figure 2), reflectingthe bias towards the zero-magnetization state. Almost all initial states are driven to thepotential well at x = 1/2, so that the exit probability becomes independent of the initialdensity of ↑ voters.doi:10.1088/1742-5468/2007/10/L10001 3J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 2. Exit probability E(x) versus the density of ↑ voters x for the caseN = 16, N = 25 and N = 100.Similarly, we study the time to reach consensus as a function of the initial compositionof voters. Let tn denote the time to reach consensus (either all ↑ or all ↓) when startingwith n ↑ voters in a population of N voters. Similar to (2), tn obeys the backwardequation [7]tn = δt + wn→n+1tn+1 + wn→n−1tn−1 + wn→ntn, (5)where δt = 1/N is the time elapsed in an update. In the large-n limit, this equationbecomes3x(1− x)2N∂2t∂x2+ x(1− x)(1− 2x) ∂t∂x= −1. (6)The formal solution is again elementary, but the result can no longer be expressed inclosed form. The main result is that the consensus time scales as eaN , with a a constantof order 1. In contrast to the classical voter model, the global bias drives the system intoa potential well that must be surmounted to reach consensus. Thus the consensus time isanomalously long.In one dimension, a voter changes its opinion if at least one of its neighbors is indisagreement. For example, a ↑ voter flips with rate 1 if the neighborhood configurationsare ↑↑↓, ↓↑↑, and ↓↑↓. As an amusing side-note, this dynamics is equivalent to rule178 of the one-dimensional cellular automaton [8], except that this rule is implementedasynchronously in the vacillating voter model. In the framework of the Ising–Glaubermodel [9], the flip rate of a voter at site i, whose states are now represented by σi = ±1,isw({σ} → {σ′}i) = − [σi(σi+1 + σi−1) + σi−1σi+1 − 3]4, (7)with {σ} denoting the state of all voters and {σ′}i the state where the ith voter flips.The first two terms correspond to conventional Glauber kinetics, but as mentionedparenthetically in [9], the presence of the σi−1σi+1 term couples the rate equation forthe mean spin to three-body terms and the model is not exactly soluble.doi:10.1088/1742-5468/2007/10/L10001 4J.Stat.Mech.(2007)L10001Dynamics of vacillating votersThe mean spin, sj ≡ 〈σj〉 =∑{σ} σjP ({σ}; t), evolves according to∂sj∂t=∑{σ}σj[∑iw({σ′}i → {σ})P ({σ′}i; t)− w({σ} → {σ′}i)P ({σ}; t)], (8)which reduces to, after straightforward but tedious steps,∂sj∂t=12(sj+1 + sj−1 + 〈σj−1σjσj+1〉 − 3sj) . (9)In a similar spirit, the rate equation for the nearest-neighbor correlation function,〈σjσj+1〉, is∂〈σjσj+1〉∂t=12[〈σj−1(σj + σj+1)〉+ 〈(σj + σj+1)σj+2〉] + 1− 3〈σjσj+1〉. (10)We can simplify equation (10) by considering domain walls—nearest-neighbor anti-aligned voters—whose density is given by ρ = (1− 〈σiσi+1〉)/2. According to the flip ratein equation (7), an isolated domain wall diffuses freely, just as in the pure voter model.However, when two domain walls are adjacent, they annihilate with probability 1/3 orone hops away from the other with probability 2/3. This process is isomorphic to single-species annihilation, A + A → 0, but with a reduced reaction rate compared to freelydiffusing reactants because of the nearest-neighbor repulsion. The domain wall densitystill asymptotically decays as t−1/2 with an amplitude that depends on the magnitude ofthe repulsion.Because domain walls are widely separated at long times, the second-neighborcorrelation function is〈σjσj+2〉 = +prob (0 or 2 walls between j and j + 2)− prob(1 wall between j and j + 2) ≈ 1− 2ρ.Using the approximation of widely separated domain walls, 〈σjσj+2〉 ≈ 〈σjσj+1〉 ≡ m2,and the rate equation for the nearest-neighbor correlation function m2 becomes ∂m2/∂t =1−m2, with solutionm2(t) = 1 + [m(0)2 − 1]e−t. (11)Here we chose the uncorrelated initial condition, so that m2(0) = m(0)2, where m(0) ≡〈sj(0)〉 is the average magnetization at t = 0.Let us now return to the rate equation (9) for the mean spin. For a spatiallyhomogeneous system, the 〈sj〉 are all identical and the magnetization is m ≡ 〈sj〉. Also,we follow [10] and decouple the three-spin correlation function as 〈σj−1σjσj+1〉 ≈ mm2.Then by averaging over all sites, the rate equation, equation (9), becomes∂m∂t=12(mm2 −m) = m2e−t(m(0)2 − 1), (12)whose solution, for the initial condition m(0), ism(t) = m(0)e1/2(1−e−t)(m(0)2−1). (13)Thus we obtain a non-trivial relation between the final magnetization m(∞) and m(0):m(∞) = m(0)e1/2(m(0)2−1). (14)doi:10.1088/1742-5468/2007/10/L10001 5J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 3. Exit probability E(x) as a function of the initial density of ↑ votersx for a one-dimensional system composed of 25, 36 and 1000 voters respectively.The voter model result, E(x) = x, that follows from magnetization conservationis shown for comparison.Since the density of ↑ voters is x = (1+m)/2, while m(∞) = 2E(x)−1, the exit probabilityE(x) becomesE(x) = 12[(2x− 1)e2x(x−1) + 1]. (15)This result is in excellent agreement with our simulation results (figure 3). For smallsystems (N = 25 and 36), we directly measure the probability E(n) that the populationultimately reaches a ↑ consensus when there are initially n ↑ voters by averaging over 5000realizations of the dynamics. We also verified equation (15) for large systems (N = 1000nodes) by a different approach that avoids the need to measure E(n) directly by simulatinguntil ultimate consensus. Instead, we run the dynamics up to 1000 time steps and measurethe magnetization at this time. We then average over 200 realizations of the process toobtain m(∞) and finally obtain E(x) from E(x) = (1+m(∞))/2. We again find excellentagreement with our prediction (15).The vacillating voter model in greater than one dimension has the new qualitativefeature that small minority domains tend to grow. This anti-coarsening is a manifestationof the bias toward the zero-magnetization state. To appreciate how this anti-coarseningarises, consider a circular two-dimensional island domain of ↑ voters of linear dimension Land area A in a sea of ↓ voters. For large L, each voter at the interface has the same localenvironment, so that there is no environmental bias. However, there are slightly more ↓voters just outside the circle than ↑ voters just inside. In a time of the order of δt ∼ Leach interface voter is updated once, on average, so that the island area increases by anamount δA that is of the order of the difference in the number of ↑ and ↓ voters at theinterface. Thus δA/δt ∼ 1/L, which gives L ∼ t1/3. In d dimensions, this same reasoninggives L ∼ t1/(d+1). We probed for this anti-coarsening by simulating the evolution of aninitial small square domain of ↑ voters in a ↓ background in two dimensions (figure 4).Although such domains do not remain contiguous, the data suggest that the number, oroccupied area, of ↑ voters grows as tα, with α around 0.73, in reasonable agreement withour expectation α = 2/3.doi:10.1088/1742-5468/2007/10/L10001 6J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 4. Double logarithmic plot of the number of ↑ voters versus time on thesquare lattice starting from a 4×4 square of ↑ voters in a background of ↓ voters.Figure 5. Exit probability E(x) as a function of the initial density of ↑ voters x fora square lattice of 16, 25, 36 and 49 voters, respectively, with periodic boundaryconditions.A system with non-zero initial magnetization is therefore again drawn to the attractorwhere the density x of ↑ voters equals 1/2 before final consensus is eventually reached.It is only for x initially very close to 0 or 1 that the system achieves consensus withoutfirst being drawn to this attractor. Thus the exit probability E(x) should be nearlyindependent of x for almost all x, just as in the mean-field limit. Simulations of thevacillating voter model on the square lattice (figure 5) confirm that E(x) approaches1/2 for a progressively wider range of x as L increases. Simulations also show that thecorrelation function C1 ≡ 〈σi,jσi,j+1〉 does not approach 1 in the long-time limit, as inone dimension or in the pure voter model in two dimensions. Rather, C1 reaches thestationary value 0.31, so that domains of opposite opinions coexist (figure 6), and only arare macroscopic fluctuation allows consensus to be reached.doi:10.1088/1742-5468/2007/10/L10001 7J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 6. Snapshots of the vacillating (left) and pure (right) voter model on a50 × 50 lattice starting with a random zero-magnetization state after 100 timesteps. The correlation function C1 equals 0.31 (left) and 0.59 (right) respectively.In summary, when vacillation is incorporated into the voter model, consensus isinhibited but not prevented. In the mean-field limit, the vacillation drives a populationaway from consensus and toward the zero-magnetization state. A finite system ultimatelyachieves consensus only via a macroscopic fluctuation that allows the system to escape thisbias-induced potential well. Because of the bias, the probability to reach ↑ consensus isessentially independent of the initial composition of the population. In one dimension, thesystem coarsens, albeit more slowly than in the pure voter model because of the repulsionof neighboring domain walls, and the probability to reach the final state of ↑ consensus hasa non-trivial initial state dependence. In two and higher dimensions, the domains slowlyanti-coarsen to drive the system to the zero-magnetization state. The overall behavior isqualitatively similar to that of the mean-field vacillating voter model, and very differentfrom the pure voter model.We gratefully acknowledge the support of the European Commission Project CREENFP6-2003-NEST-Path-012864 (RL), the ARC ‘Large Graphs and Networks’ (RL) andNSF grant DMR0535503 (SR), and the hospitality of the Ettore Majorana Center wherethis project was initiated.References[1] Liggett T M, 1999 Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes (New York:Springer)[2] Krapivsky P L, 1992 Phys. Rev. A 45 1067[3] Axelrod R, 1977 J. Confl. Res. 41 203Axtell R, Axelrod R, Epstein J and Cohen M D, 1996 Comput. Math. Organ. Theory 1 123[4] Weisbuch G, Deffuant G, Amblard F and Nadal J P, 2002 Complexity 7 55Hegselmann R and Krause U, J. Artif. Soc. Social Simul. 5 (3)Ben-Naim E, Krapivsky P L and Redner S, 2003 Physica D 183 190[5] Vazquez F and Redner S, 2004 J. Phys. A: Math. Gen. 37 8479[6] Galam S, 2004 Physica A 333 453[7] See e.g. Redner S, 2001 A Guide to First-Passage Processes (New York: Cambridge University Press)chapter 2[8] Wolfram S, 2002 A New Kind of Science (Champaign, IL: Wolfram Media)[9] Glauber R J, 1963 J. Math. Phys. 4 294[10] Mobilia M and Redner S, 2003 Phys. Rev. E 68 046106doi:10.1088/1742-5468/2007/10/L10001 8

Dynamics of vacillating voters

We introduce the vacillating voter model in which each voter consults two neighbors to decide its state, and changes opinion if it disagrees with either neighbor. This irresolution leads to a global bias toward zero magnetization. In spatial dimension d > 1, anti-coarsening arises in which the linear dimension L of minority domains grows as t(1/(d+ 1)). One consequence is that the time to reach consensus scales exponentially with the number of voters

Lambiotte, Renaud

DIAL UCLouvain

RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHEAuthor(s) - Auteur(s) :Publication date - Date de publication :Permanent link - Permalien :Rights / License - Licence de droit d’auteur :Bibliothèque Universitaire Moretus PlantinInstitutional Repository - Research PortalDépôt Institutionnel - Portail de la Rechercheresearchportal.unamur.beDynamics of vacillating votersLambiotte, R.; Redner, S.Published in:Journal of Statistical Mechanics: Theory and ExperimentDOI:10.1088/1742-5468/2007/10/L10001Publication date:2007Document VersionPublisher's PDF, also known as Version of recordLink to publicationCitation for pulished version (HARVARD):Lambiotte, R & Redner, S 2007, 'Dynamics of vacillating voters' Journal of Statistical Mechanics: Theory andExp riment, no. 10. https://doi.org/10.1088/1742-5468/2007/10/L10001General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 21. May. 2019Dynamics of vacillating votersThis article has been downloaded from IOPscience. Please scroll down to see the full text article.J. Stat. Mech. (2007) L10001(http://iopscience.iop.org/1742-5468/2007/10/L10001)Download details:IP Address: 138.48.20.157The article was downloaded on 30/12/2012 at 07:21Please note that terms and conditions apply.View the table of contents for this issue, or go to the journal homepage for moreHome Search Collections Journals About Contact us My IOPscienceJ.Stat.Mech.(2007)L10001ournal of Statistical Mechanics:An IOP and SISSA journalJ Theory and ExperimentLETTERDynamics of vacillating votersR Lambiotte1,2 and S Redner31 GRAPES, Universite´ de Lie`ge, Sart-Tilman, B-4000 Lie`ge, Belgium2 INMA, Universite´ catholique de Louvain, 4 avenue Georges Lemaitre,B-1348 Louvain-la-Neuve, Belgium3 Center for Polymer Studies and Physics Department, Boston University,Boston, MA 02215, USAE-mail: Renaud.Lambiotte@uclouvain.be and redner@bu.eduReceived 5 October 2007Accepted 8 October 2007Published 23 October 2007Online at stacks.iop.org/JSTAT/2007/L10001doi:10.1088/1742-5468/2007/10/L10001Abstract. We introduce the vacillating voter model in which each voterconsults two neighbors to decide its state, and changes opinion if it disagreeswith either neighbor. This irresolution leads to a global bias toward zeromagnetization. In spatial dimension d > 1, anti-coarsening arises in which thelinear dimension L of minority domains grows as t1/(d+1). One consequence isthat the time to reach consensus scales exponentially with the number of voters.Keywords: coarsening processes (theory), critical phenomena of socio-economicsystems, scaling in socio-economic systems, probability theoryArXiv ePrint: 0710.0914c©2007 IOP Publishing Ltd and SISSA 1742-5468/07/L10001+8$30.00J.Stat.Mech.(2007)L10001Dynamics of vacillating votersThe voter model [1] gives an appealing, albeit idealized, description for the opiniondynamics of a socially interacting population. In this model, each node of a graphis occupied by a voter that has one of two opinions, ↑ or ↓. The population evolvesby: (i) picking a random voter; (ii) the selected voter adopting the state of a randomlychosen neighbor; (iii) repeating these steps ad inﬁnitum or until a ﬁnite system necessarilyreaches consensus. Descriptively, each voter has no self conﬁdence and follows one of itsneighbors. With this dynamics, a voter chooses a state with a probability equal to thefraction of neighbors in that state, a feature that renders the voter model soluble in alldimensions [1, 2].In this work, we investigate a variation that we term the vacillating voter model.By vacillating, we mean that a voter very much lacks conﬁdence in its state. In anupdate, if a voter happens to select a random neighbor of the same persuasion, the voteris still not convinced that this state is right. Thus the voter selects another randomneighbor and adopts this state. This vacillation causes a voter to change state with alarger probability than the fraction of disagreeing neighbors, and leads to a bias towardthe zero-magnetization state in which there are equal densities of voters of each type.Thus vacillation inhibits consensus, but due to a diﬀerent mechanism than that in theprototypical Axelrod model [3], the bounded compromise model [4] and its variants [5]. Forthese latter models, consensus is hindered because of the absence of interaction whenevertwo agents become suﬃciently incompatible. For vacillating voters, it is individualuncertainty that forestalls consensus. The vacillating voter model also diﬀers from modelsthat incorporate ‘contrarians’ [6] because voters still try to imitate their neighbors.The update steps in the vacillating voter model are as follows.(1) Pick a random voter.(2) The voter picks a random neighbor. If the neighbor disagrees with the voter, thevoter changes state.(3) If the neighbor and the voter agree, the voter picks another random neighbor andadopts its state.(4) Repeat steps 1 and 2 ad inﬁnitum or until consensus is reached.For example, the probability that a vacillating voter on the square lattice ﬂips is0, 12, 56, and 1, respectively, when the number of anti-aligned neighbors is 0, 1, 2, and ≥ 3(ﬁgure 1). In contrast, for the classic voter model, the ﬂip probability is k4, where k is thenumber of neighbors of the opposite opinion. We now explore the consequences of thisvacillation on voter dynamics.Consider ﬁrst the mean-ﬁeld limit. Here the density x of ↑ voters obeys the rateequationx˙ = −x[1− x2] + (1− x)[1− (1− x)2]= x(1− x)(1− 2x). (1)The ﬁrst term on the right accounts for the loss of ↑ voters in which a ↑ voter is ﬁrstpicked (factor x), and then the neighborhood cannot consist of two ↑ voters (factor 1−x2).Similarly, in the second (gain) term, a ↓ voter is ﬁrst picked, and then the neighborhoodmust contain at least one ↑ voter. The factorized form shows that there are unstable ﬁxeddoi:10.1088/1742-5468/2007/10/L10001 2J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 1. Illustration of an update for the vacillating voter on the square lattice(left and middle). For the conﬁguration on the right, the central voter ﬂips withprobability 5/6 because out of the six ways of selecting two neighbors, only onechoice leads to both neighbors agreeable (dashed).points at x = 0, 1 and a stable ﬁxed point at x = 1/2. Thus a population is driven to thezero-magnetization state.However, because consensus is the only absorbing state of the stochastic dynamics, aﬁnite population ultimately reaches consensus. To characterize the evolution to this state,we ﬁrst study the exit probability En, deﬁned as the probability that a population of Nvoters ultimately reaches ↑ consensus when there are initially n ↑ voters. Then En obeysthe backward equation [7]En = wn→n+1En+1 + wn→n−1En−1 + wn→nEn, (2)where wn→m is the probability for the transition from the state with n ↑ voters to m ↑ votersin an update. This equation expresses the probability to exit from n as the probabilityto take one step (the factors w) times the probability to exit from the point reached afterone step. In the large-n limit, we write x = n/N , and the transition probabilities becomewn→n+1 = (1− x)[1− (1− x)2]wn→n−1 = x(1− x2)wn→n = x3 + (1− x)3.Substituting these in (2), writing En±1 → E(x± δx), and expanding to second orderin δx, gives3x(1− x)2N∂2E∂x2+ x(1− x)(1− 2x)∂E∂x= 0, (3)with solutionE(x) =∫ x−1/2−1/2e2Ny2/3 dy/∫ 1/2−1/2e2Ny2/3 dy. (4)Notice that E(x) approaches the constant value 1/2 for increasing N (ﬁgure 2), reﬂectingthe bias towards the zero-magnetization state. Almost all initial states are driven to thepotential well at x = 1/2, so that the exit probability becomes independent of the initialdensity of ↑ voters.doi:10.1088/1742-5468/2007/10/L10001 3J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 2. Exit probability E(x) versus the density of ↑ voters x for the caseN = 16, N = 25 and N = 100.Similarly, we study the time to reach consensus as a function of the initial compositionof voters. Let tn denote the time to reach consensus (either all ↑ or all ↓) when startingwith n ↑ voters in a population of N voters. Similar to (2), tn obeys the backwardequation [7]tn = δt + wn→n+1tn+1 + wn→n−1tn−1 + wn→ntn, (5)where δt = 1/N is the time elapsed in an update. In the large-n limit, this equationbecomes3x(1− x)2N∂2t∂x2+ x(1− x)(1− 2x) ∂t∂x= −1. (6)The formal solution is again elementary, but the result can no longer be expressed inclosed form. The main result is that the consensus time scales as eaN , with a a constantof order 1. In contrast to the classical voter model, the global bias drives the system intoa potential well that must be surmounted to reach consensus. Thus the consensus time isanomalously long.In one dimension, a voter changes its opinion if at least one of its neighbors is indisagreement. For example, a ↑ voter ﬂips with rate 1 if the neighborhood conﬁgurationsare ↑↑↓, ↓↑↑, and ↓↑↓. As an amusing side-note, this dynamics is equivalent to rule178 of the one-dimensional cellular automaton [8], except that this rule is implementedasynchronously in the vacillating voter model. In the framework of the Ising–Glaubermodel [9], the ﬂip rate of a voter at site i, whose states are now represented by σi = ±1,isw({σ} → {σ′}i) = − [σi(σi+1 + σi−1) + σi−1σi+1 − 3]4, (7)with {σ} denoting the state of all voters and {σ′}i the state where the ith voter ﬂips.The ﬁrst two terms correspond to conventional Glauber kinetics, but as mentionedparenthetically in [9], the presence of the σi−1σi+1 term couples the rate equation forthe mean spin to three-body terms and the model is not exactly soluble.doi:10.1088/1742-5468/2007/10/L10001 4J.Stat.Mech.(2007)L10001Dynamics of vacillating votersThe mean spin, sj ≡ 〈σj〉 =∑{σ} σjP ({σ}; t), evolves according to∂sj∂t=∑{σ}σj[∑iw({σ′}i → {σ})P ({σ′}i; t)− w({σ} → {σ′}i)P ({σ}; t)], (8)which reduces to, after straightforward but tedious steps,∂sj∂t=12(sj+1 + sj−1 + 〈σj−1σjσj+1〉 − 3sj) . (9)In a similar spirit, the rate equation for the nearest-neighbor correlation function,〈σjσj+1〉, is∂〈σjσj+1〉∂t=12[〈σj−1(σj + σj+1)〉+ 〈(σj + σj+1)σj+2〉] + 1− 3〈σjσj+1〉. (10)We can simplify equation (10) by considering domain walls—nearest-neighbor anti-aligned voters—whose density is given by ρ = (1− 〈σiσi+1〉)/2. According to the ﬂip ratein equation (7), an isolated domain wall diﬀuses freely, just as in the pure voter model.However, when two domain walls are adjacent, they annihilate with probability 1/3 orone hops away from the other with probability 2/3. This process is isomorphic to single-species annihilation, A + A → 0, but with a reduced reaction rate compared to freelydiﬀusing reactants because of the nearest-neighbor repulsion. The domain wall densitystill asymptotically decays as t−1/2 with an amplitude that depends on the magnitude ofthe repulsion.Because domain walls are widely separated at long times, the second-neighborcorrelation function is〈σjσj+2〉 = +prob (0 or 2 walls between j and j + 2)− prob(1 wall between j and j + 2) ≈ 1− 2ρ.Using the approximation of widely separated domain walls, 〈σjσj+2〉 ≈ 〈σjσj+1〉 ≡ m2,and the rate equation for the nearest-neighbor correlation function m2 becomes ∂m2/∂t =1−m2, with solutionm2(t) = 1 + [m(0)2 − 1]e−t. (11)Here we chose the uncorrelated initial condition, so that m2(0) = m(0)2, where m(0) ≡〈sj(0)〉 is the average magnetization at t = 0.Let us now return to the rate equation (9) for the mean spin. For a spatiallyhomogeneous system, the 〈sj〉 are all identical and the magnetization is m ≡ 〈sj〉. Also,we follow [10] and decouple the three-spin correlation function as 〈σj−1σjσj+1〉 ≈ mm2.Then by averaging over all sites, the rate equation, equation (9), becomes∂m∂t=12(mm2 −m) = m2e−t(m(0)2 − 1), (12)whose solution, for the initial condition m(0), ism(t) = m(0)e1/2(1−e−t)(m(0)2−1). (13)Thus we obtain a non-trivial relation between the ﬁnal magnetization m(∞) and m(0):m(∞) = m(0)e1/2(m(0)2−1). (14)doi:10.1088/1742-5468/2007/10/L10001 5J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 3. Exit probability E(x) as a function of the initial density of ↑ votersx for a one-dimensional system composed of 25, 36 and 1000 voters respectively.The voter model result, E(x) = x, that follows from magnetization conservationis shown for comparison.Since the density of ↑ voters is x = (1+m)/2, while m(∞) = 2E(x)−1, the exit probabilityE(x) becomesE(x) = 12[(2x− 1)e2x(x−1) + 1]. (15)This result is in excellent agreement with our simulation results (ﬁgure 3). For smallsystems (N = 25 and 36), we directly measure the probability E(n) that the populationultimately reaches a ↑ consensus when there are initially n ↑ voters by averaging over 5000realizations of the dynamics. We also veriﬁed equation (15) for large systems (N = 1000nodes) by a diﬀerent approach that avoids the need to measure E(n) directly by simulatinguntil ultimate consensus. Instead, we run the dynamics up to 1000 time steps and measurethe magnetization at this time. We then average over 200 realizations of the process toobtain m(∞) and ﬁnally obtain E(x) from E(x) = (1+m(∞))/2. We again ﬁnd excellentagreement with our prediction (15).The vacillating voter model in greater than one dimension has the new qualitativefeature that small minority domains tend to grow. This anti-coarsening is a manifestationof the bias toward the zero-magnetization state. To appreciate how this anti-coarseningarises, consider a circular two-dimensional island domain of ↑ voters of linear dimension Land area A in a sea of ↓ voters. For large L, each voter at the interface has the same localenvironment, so that there is no environmental bias. However, there are slightly more ↓voters just outside the circle than ↑ voters just inside. In a time of the order of δt ∼ Leach interface voter is updated once, on average, so that the island area increases by anamount δA that is of the order of the diﬀerence in the number of ↑ and ↓ voters at theinterface. Thus δA/δt ∼ 1/L, which gives L ∼ t1/3. In d dimensions, this same reasoninggives L ∼ t1/(d+1). We probed for this anti-coarsening by simulating the evolution of aninitial small square domain of ↑ voters in a ↓ background in two dimensions (ﬁgure 4).Although such domains do not remain contiguous, the data suggest that the number, oroccupied area, of ↑ voters grows as tα, with α around 0.73, in reasonable agreement withour expectation α = 2/3.doi:10.1088/1742-5468/2007/10/L10001 6J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 4. Double logarithmic plot of the number of ↑ voters versus time on thesquare lattice starting from a 4×4 square of ↑ voters in a background of ↓ voters.Figure 5. Exit probability E(x) as a function of the initial density of ↑ voters x fora square lattice of 16, 25, 36 and 49 voters, respectively, with periodic boundaryconditions.A system with non-zero initial magnetization is therefore again drawn to the attractorwhere the density x of ↑ voters equals 1/2 before ﬁnal consensus is eventually reached.It is only for x initially very close to 0 or 1 that the system achieves consensus withoutﬁrst being drawn to this attractor. Thus the exit probability E(x) should be nearlyindependent of x for almost all x, just as in the mean-ﬁeld limit. Simulations of thevacillating voter model on the square lattice (ﬁgure 5) conﬁrm that E(x) approaches1/2 for a progressively wider range of x as L increases. Simulations also show that thecorrelation function C1 ≡ 〈σi,jσi,j+1〉 does not approach 1 in the long-time limit, as inone dimension or in the pure voter model in two dimensions. Rather, C1 reaches thestationary value 0.31, so that domains of opposite opinions coexist (ﬁgure 6), and only arare macroscopic ﬂuctuation allows consensus to be reached.doi:10.1088/1742-5468/2007/10/L10001 7J.Stat.Mech.(2007)L10001Dynamics of vacillating votersFigure 6. Snapshots of the vacillating (left) and pure (right) voter model on a50 × 50 lattice starting with a random zero-magnetization state after 100 timesteps. The correlation function C1 equals 0.31 (left) and 0.59 (right) respectively.In summary, when vacillation is incorporated into the voter model, consensus isinhibited but not prevented. In the mean-ﬁeld limit, the vacillation drives a populationaway from consensus and toward the zero-magnetization state. A ﬁnite system ultimatelyachieves consensus only via a macroscopic ﬂuctuation that allows the system to escape thisbias-induced potential well. Because of the bias, the probability to reach ↑ consensus isessentially independent of the initial composition of the population. In one dimension, thesystem coarsens, albeit more slowly than in the pure voter model because of the repulsionof neighboring domain walls, and the probability to reach the ﬁnal state of ↑ consensus hasa non-trivial initial state dependence. In two and higher dimensions, the domains slowlyanti-coarsen to drive the system to the zero-magnetization state. The overall behavior isqualitatively similar to that of the mean-ﬁeld vacillating voter model, and very diﬀerentfrom the pure voter model.We gratefully acknowledge the support of the European Commission Project CREENFP6-2003-NEST-Path-012864 (RL), the ARC ‘Large Graphs and Networks’ (RL) andNSF grant DMR0535503 (SR), and the hospitality of the Ettore Majorana Center wherethis project was initiated.References[1] Liggett T M, 1999 Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes (New York:Springer)[2] Krapivsky P L, 1992 Phys. Rev. A 45 1067[3] Axelrod R, 1977 J. Conﬂ. Res. 41 203Axtell R, Axelrod R, Epstein J and Cohen M D, 1996 Comput. Math. Organ. Theory 1 123[4] Weisbuch G, Deﬀuant G, Amblard F and Nadal J P, 2002 Complexity 7 55Hegselmann R and Krause U, J. Artif. Soc. Social Simul. 5 (3)Ben-Naim E, Krapivsky P L and Redner S, 2003 Physica D 183 190[5] Vazquez F and Redner S, 2004 J. Phys. A: Math. Gen. 37 8479[6] Galam S, 2004 Physica A 333 453[7] See e.g. Redner S, 2001 A Guide to First-Passage Processes (New York: Cambridge University Press)chapter 2[8] Wolfram S, 2002 A New Kind of Science (Champaign, IL: Wolfram Media)[9] Glauber R J, 1963 J. Math. Phys. 4 294[10] Mobilia M and Redner S, 2003 Phys. Rev. E 68 046106doi:10.1088/1742-5468/2007/10/L10001 8

Dynamics of Vacillating Voters

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