aa r X i v : . [ phy s i c s . s o c - ph ] M a r Dynamics of Confident Voting
D. Volovik and S. Redner Center for Polymer Studies and Department of Physics, Boston University, Boston,Massachusetts 02215, USA
Abstract.
We introduce the confident voter model, in which each voter can be in one of twoopinions and can additionally have two levels of commitment to an opinion — confidentand unsure. Upon interacting with an agent of a different opinion, a confident voterbecomes less committed, or unsure, but does not change opinion. However, an unsureagent changes opinion by interacting with an agent of a different opinion. In the mean-field limit, a population of size N is quickly driven to a mixed state and remains closeto this state before consensus is eventually achieved in a time of the order of ln N . Intwo dimensions, the distribution of consensus times is characterized by two distincttimes — one that scales linearly with N and another that appears to scale as N / .The longer time arises from configurations that fall into long-lived states that consistof two (or more) single-opinion stripes before consensus is reached. These stripe statesarise from an effective surface tension between domains of different opinions.PACS numbers: 02.50.-r, 05.40.-a ynamics of Confident Voting
1. INTRODUCTION
The voter model [1] describes the evolution toward consensus in a population of N agents, each of which can be in one of two possible opinion states. In an update event, arandomly-selected voter adopts the state of a randomly-selected neighbor. As a result ofrepeated update events, a finite population necessarily reaches consensus in a time T N that scales as a power law in N (with a logarithmic correction in two dimensions) [1, 2].Because of its simplicity and its natural connection to opinion dynamics, the votermodel has been extensively investigated (see, e.g., [3, 4]). The connection withsocial phenomena has also motivated efforts to extend the voter model to incorporatevarious aspects of social reality, such as, among others, stubbornness/contrarianism [5–8], multiple states [9, 10], internal dissonance [11], individual heterogeneity [12],environmental heterogeneity [13–16], vacillation [17], and non-linear interactions [18, 19].These studies have uncovered many new phenomena that are still being activelyexplored.Our investigation was initially motivated by recent social experiments ofCentola [20], who studied the spread of a specific behavior in a controlled online networkwhere reinforcement played a crucial role. Reinforcement means that an individualadopts a particular state only after receiving multiple prompts to adopt this behaviorfrom socially-connected neighbors. These experiments found that social reinforcementplayed a decisive role in determining how a new behavior is adopted [20].Previous research that has a connection with this type of reinforcement mechanisminclude the q-voter model [19], where multiple same-opinion neighbors initiate change,the naming game, and the AB model [21]. An example that is perhaps most closelyconnected to reinforcement arises in the noise-reduced voter model [22], where a voterkeeps a running total of inputs towards changing opinions, but actually changes opinionsonly when this counter reaches a predefined threshold. A similar notion of reinforcementarises in a model of fad and innovation dynamics [23] and in a model of contagionspread [24]. The use of multiple discrete opinions is not the only option for incorporatingvarying opinion strength. Previous models have used a continuous range of opinionsquantifying the tendency for an agent to change its opinion. [25] For example, in thebounded confidence model, an agent can possesses an opinion in a continuous range, withthe spatial distance between points representing the difference in those opinions. [26]In this paper, we study how reinforcement affects the dynamics of the voter model.In our confident voter model , we assume that agents possess some modicum of intrinsicconfidence in their beliefs and, unlike the classic voter model, need multiple promptsbefore changing their opinion state. We investigate a simple realization of this confidentvoting in which each opinion state is further demarcated into two substates of differentconfidence levels. The basic variables are thus the opinion of each voter and theconfidence level with which this opinion is held. For concreteness, we label the twoopinion states as plus (P) and minus (M). Thus the possible states of an agent are P and P for confident and unsure plus agents, respectively, and correspondingly M and M ynamics of Confident Voting marginal version, an unsure agent that changes opinion stillremains unsure. Such an agent is often labelled a “flip-flopper”, a routinely-invokedmoniker by American politicians to characterize political opponents. Figuratively, anagent who switches opinion remains ambivalent about the new opinion state and canswitch back. In the extremal version, an unsure agent becomes confident after an opinionchange. Such an agent “sees the light” and therefore becomes fully committed to thenew opinion state. This behavior is typified by Paul the Apostle, who switched frombeing dedicated to finding and persecuting early Christians to embracing Christianityafter experiencing a vision of Jesus. confident PP PP MM (a) MM (b) unsure Figure 1.
Illustration of the states and possible transitions in the: (a) marginal,and (b) extremal versions of the confident voter model. Dashed arrows indicatepossible confidence level changes (biased toward higher confidence), while solidarrows indicate possible opinion change events.
2. MEAN-FIELD DESCRIPTION
The basic variables are the densities of the four types of agents. We use P , P , M , M to denote both the agent types and their densities. In the mean-field description, a pairof agents is randomly selected, and the state of one the two agents, chosen equiprobably,changes according to the voter-like dynamics illustrated in Fig. 1. We now outline thetime evolution for the two variations of the confident voter model. For writing the rate equations, we first enumerate the possible outcomes when a pair ofagents interact: M P → M M or P P ; M P → M P or M P ; P P → P P or P P ; M M → M M or M M ; M P → M P or P P ; M P → M P or M M . ynamics of Confident Voting M P ) leadsto no net density change, as in the classic voter model. However, when two confidentagents of different opinions meet ( M P ), one of the agents becomes unsure. The nexttwo lines account for interactions between agents of the same opinion but differentconfidence levels. We assume that an unsure agent exerts no influence on a confidentagent by virtue of the latter being confident, while a confident agent is persuasive andconverts an unsure agent to confident. Finally, the last line accounts for an unsure agentchanging opinion upon interacting with a confident agent of a different opinion.The corresponding rate equations are:˙ P = − ( M + M ) P + P P ≡ − M P + P P , (1)˙ P = M P − P P + ( M P − M P ) , with parallel equations for M and M that are obtained by interchanging M ↔ P inEq. (1). The rate equation for the total density of plus agents is˙ P = M P − M P , and from the complementary equation for ˙ M , it is evident that the total density ofagents is conserved, ˙ P + ˙ M = 0. For the extremal version, we again enumerate the possible outcomes when a pair ofagents interact. These are: M P → P P or M M ; M P → M P or M P ; P P → P P or P P ; M M → M M or M M ; M P → M P or P P ; M P → M P or M M .The point of departure, compared to the marginal version, is that a voter is now confidentin its new opinion state upon changing opinion. The rate equations corresponding tothese steps are:˙ P = − M P + M P + P P , (2)˙ P = M P − M P − P P + ( M P − M P ) , with parallel equations for M and M . The rate equation for the total density of plusagents is the same as that for the marginal version, so that again the total density ofagents is manifestly conserved. For both variants of the confident voter model, the time evolution is dominated by thepresence of a saddle point that corresponds not to consensus, but a balance betweenplus and minus agents. For nearly-symmetric initial conditions, the densities of thedifferent species are initially attracted to this unstable fixed point, but eventually flowto a stable fixed point that corresponds to consensus. However when the initial condition ynamics of Confident Voting
It is instructive to first study the initial conditions M (0) = P (0) = 1 / M (0) = P (0) = 0. The rate equations (1) for the marginal version ofconfident voting now reduce to ˙ P = − ˙ P = − P , with solution P ( t ) = P (0) / [1 + P (0) t ] , (3) P ( t ) = 12 − P ( t ) . Thus in an initially symmetric system, confident voters are slowly eliminated becausethere is no mechanism for their replenishment, and all that remains asymptotically areequal densities of unsure voters.For the extremal version of confident voting, the rate equations (2) reduce to˙ P = − ˙ P = P + 12 P −
14 (4)= − ( P − λ + )( P − λ − ) , with λ ± = ( − ± √ ≈ . , − . P < λ + and negative for P > λ + , the fixedpoint at λ + is stable. Thus P ( t ) approaches λ + exponentially in time. We solve for P by a partial fraction expansion to give P ( t ) − λ + P ( t ) − λ − = P (0) − λ + P (0) − λ − e − ( λ + − λ − ) t , (5)which indeed gives an exponential approach to the final state of P = − P = λ + .Thus all four voting states are represented in the long-time limit. If the initial condition is slightly non-symmetric, thennumerical integrations of the rate equations clearly show that the evolution of thedensities turns out to be controlled by two distinct time scales — a fast time scalethat is O (1) and a longer time scale that is O (ln N ), where N is the population size.To incorporate N in the rate equations, we interpret these equations as describingthe dynamics of voters that live on a complete graph of N ≫ − N in the rate equations. Similarly, an initial small deviation ǫ = N from the symmetricinitial conditions in the rate equations (i.e., P (0) = + ǫ and M (0) = − ǫ , with ǫ = N ), should be interpreted as the departure from a symmetric state by a singleparticle on a complete graph of N sites.In the marginal model (Fig. 2(a)), the system begins to approach the point M = P = algebraically in time, as discussed above. For a slightly asymmetricinitial condition, the densities remain close to this unstable fixed point for a time thatnumerical integration shows is of order ln N . Ultimately, the system is driven to the ynamics of Confident Voting P M M P M M Figure 2.
Evolution of the densities for the: (left) marginal and (right) extremalmodels with the near-symmetric initial condition P = 0 . M = 0 . P = M = 0. fixed point that corresponds to the initial majority opinion. For the extremal model,qualitatively similar behavior occurs, except that in the initial stages of evolution thesystem is quickly driven towards the fixed point at P = M = λ + and P = − λ + . Thisfixed point is a saddle node, with one stable and two unstable directions (Fig. 2(b)).Thus for nearly-symmetric initial conditions, the densities remain close to this fixedpoint for a time of the order ln N , after which the densities are suddenly driven to oneof the two stable fixed points, either M = 1 or P = 1. P P M M (a) M P (b) M P Figure 3.
Composition tetrahedron for the: (a) marginal and (b) extremalmodels. Shown in (a) are the consensus fixed points (dots), the unstable fixedline (thick), and the symmetry line P = M , P = M (dashed arrow) thatterminates in a symmetric fixed point (circle). Shown in (b) are the unstable(circle) and stable (dots) fixed points. For both cases, two representative flowsthat start from nearly symmetric initial conditions are shown. The full state space is the composition tetrahedron, which consists of theintersection of the set { P , P , M , M | P i , M i ≤ } with the normalization constraintplane P + P + M + M = 1 (Fig. 3). Each corner corresponds to a pure system that isentirely comprised of the labeled species. For the marginal version, there are only two ynamics of Confident Voting P = 1 and M = 1, corresponding to consensus of either confidentplus voters or confident minus voters. There is also a fixed line, defined by P + M = 1,where the population consists only of unsure agents. This fixed line is locally unstableexcept at the point P = M = . Thus if the system starts along the symmetry linedefined by P = M and P = M , the system flows to the final state of P = M = .However, near-symmetric initial states execute a sharp U-turn and eventually flow toone of the consensus fixed points P = 1 or M = 1, as illustrated in Fig. 3.For the extremal version, qualitatively similar dynamics arises, except that insteadof a fixed line, there is an unstable fixed point at P = M = λ + and P = M = − λ + .Nearly symmetric initial states first flow to this unstable fixed point and remain in thevicinity of this point for a time scale that is of order ln N , after which the densitiesquickly flow to the consensus fixed points, either M = 1 or P = 1.
3. CONFIDENT VOTING ON LATTICES
We now investigate confident voting dynamics when voters are situated on the sites of afinite-dimensional lattice of linear dimension L (with N = L d ), with periodic boundaryconditions. For the classic lattice voter model, it was found that the consensus time T N asymptotically scales as N in one dimension d = 1, as N ln N for d = 2, and as N for d > d = 2 and the lack ofdimension dependence for d > d c = 2 for the classicvoter model.The confident voter model has quite different dynamics because the magnetization isnot conserved, except in the symmetric limit P = M and P = M , whereas the averagemagnetization is conserved in the classic voter model [1, 2]. Here the magnetization isdefined as the difference in the densities of plus and minus voters of any kind. Theabsence of this conservation law leads to an effective surface tension between domainsof plus and minus voters [22]. Consequently confident voting is closer in character tothe kinetic Ising model with single-spin flip dynamics at low temperatures rather thanto the classic voter model. In the simplest case of one dimension, the agents organize at long times into domainsthat are in a single state and the evolution is determined by the motion of the interfacebetween two dissimilar domains. Thus we consider the evolution of a single interfacebetween two semi-infinite domains — for example, one in state P and the other in state M . By enumerating all possible ways that the voters at the interface can evolve (Fig. 4),we find that the domain wall moves one site to the left or to the right equiprobably afterfour time steps. Thus isolated interfaces between domains undergo a random walk, butwith the domain wall hopping at one-fourth the rate of a symmetric nearest-neighborrandom walk. ynamics of Confident Voting P M P MP M M P PMM P P M M P P P M M P P M P P M M P M M M P
Figure 4.
First three evolution steps of an interface between a P and M domain. Voters that change their state are shown green. After one more step,a sharp domain wall that is translated by ± Similarly, we determine the fate of two adjacent diffusing domain walls by studyingthe evolution of a single voter in state M in a sea of P voters. By again enumeratingthe possible ways these two adjoining interfaces evolve, we find that the domain wallsannihilate with probability 1 / /
2. Additionally, we verified that the distribution of survival times for a single confidentvoter in a sea of opposite-opinion voters scales as S ( t ) ≡ t − / , as in the classic votermodel. We also studied the analogous single-defect initial condition for unsure voters. Inall such cases, the long-time behavior is essentially the same as in the classic voter model,albeit with an overall slower time scale. Finally, we confirmed that the time to reachconsensus starting from an arbitrary initial state scales quadratically with N . Thus theone-dimensional confident voter model at long times exhibits the same evolution as theclassic voter model, but with a rescaled time. In our simulations of confident voting in two dimensions, we typically start a populationwith exactly one-half of the voters in the confident plus state and one-half in theconfident minus state, with their locations randomly distributed. Periodic boundaryconditions are always employed. For both the marginal and the extremal versions ofconfident voting, T N appears to grow algebraically in N , with an exponent that isvisually close to (Fig. 5). However, the local two-point slopes in the plot of ln T N versus ln N are slowly and non-monotonically varying with N so that it is difficult tomake a precise estimate of the exponent value.We argue that this slow approach to asymptotic behavior arises because there aretwo different routes by which consensus is achieved. For random initial conditions,most realizations reach consensus by domain coarsening, a process that ends with theformation of a large single-opinion droplet that engulfs the system. However, for asubstantial fraction of realizations (roughly 38% for the extremal model and 42% for ynamics of Confident Voting NT N MarginalExtremal 1 1.2 1.4 1.6 1.8 0 0.05 0.1 0.15 0.21/Log(N)MarginalExtremal
Figure 5. (left) Average consensus time T N on the square lattice as a functionof N . For both models, the initial number of confident plus and minus votersare equal and randomly-distributed in space. The number of realizations forthe largest system size is 40 , the marginal model), voters first segregate into alternating stripe-like enclaves of plusand minus voters (Fig. 6). This feature is akin to what occurs in the two-dimensionalIsing model with zero-temperature Glauber dynamics, where roughly one-third of allrealizations fall into a stripe state (which happens to be infinitely long lived at zerotemperature [27–29]). A similar condensation into stripe states also occurs in themajority vote model [30], the AB model, the naming game [21], and now the confidentvoter model. It is striking that this symmetry breaking occurs in a wide range of non-equilibrium systems for which the underlying dynamics is symmetric in x and y . It isan open challenge to understand why this symmetry breaking occurs. Figure 6.
Typical configurations of the extremal version of the confident votermodel on a 30 ×
30 square lattice that reach either a stripe state (left) or anisland state (right). Black and white pixels correspond to unsure plus and minusagents; these form a sharp interface between domains of confident agents.
The existence of these two distinct modes of evolution is reflected in the probabilitydistribution of consensus times P ( T N ) (Fig. 7). Starting from the random butsymmetrical initial condition, the distribution P ( T N ) first has a sharp peak at acharacteristic time that scales linearly with N , and then a distinct exponential tail ynamics of Confident Voting N / . The shorter time scale corresponds tothe subset of realizations that reach consensus by conventional coarsening. For theserealizations, the length scale ℓ of the coarsening grows as √ t . When this coarsening scalereaches L , consensus is achieved. The consensus time is thus given by ℓ = L = √ t ; since N ∝ L , we have T N ≃ N . The longer time scale stems from the subset of realizationsthat fall into a stripe state before consensus is eventually reached. -7 -6 -5 -4 -3 -2 T N P(T N ) MarginalExtremal Figure 7.
Consensus time distribution for a 64 ×
64 system on a doublelogarithmic scale. The initial condition is the same as in Fig. 5 and the dataare based on 750,000 realizations.
To help understand the quantitative nature of the approach to consensus via thetwo different routes of coarsening and stripe states, we studied the confident voter modelwith the initial conditions of: (i) a large circular single-opinion island and (ii) a stripestate (Fig. 8). For the former, the initial condition is a circular region of radius r thatcontains agents in state M , surrounded by agents in state P . For the latter, agents instate P occupy the top half of the system, while the bottom half is occupied of agentsin state M . For these two initial conditions, the consensus time T N grows as N and as N / , respectively (Fig. 8). In the latter case, the approach to asymptotic behavior isboth non-monotonic and extremely slow (Fig. 8); we do not understand the mechanismresponsible for these anomalies. These limiting behaviors account for the two time scalesthat arise in the distribution of consensus times for a system with a random, symmetricinitial condition.Although the confident voter model has an appreciable probability of falling intoa stripe state, such a state is not stable because the interface between the domainscan diffuse. When the two interfaces of a stripe diffuse by a distance that is of theorder of their separation, one stripe is cut in two and resulting droplet geometry quicklyevolves to consensus. We estimate the time for two such interfaces to meet by followingessentially the same argument as that developed for the majority vote model [30]. For aflat interface, every site on the interface can change its opinion. Such an opinion changemoves the local position of the interface by ±
1. For a smooth interface of length L , therewill therefore be of the order of L ± √ L opinion change events of plus to minus and vice ynamics of Confident Voting NT N IslandNStripeN
Figure 8. (left) Average consensus time for the extremal model on the squarelattice as a function of the population size for: (i) island and (ii) stripe initialconfigurations. For the stripe initial condition, the data for the largest systemsize is based on 13,000 realizations. The stripe-state data has been verticallydisplaced for clarity. (right) The local two-point exponent for T N for an initialstripe state. The error bars indicate the statistical uncertainty. versa. Thus the net change in the number of agents of a given opinion is of the orderof ±√ L . Consequently, the average position of the interface moves by √ L/L = 1 / √ L .Correspondingly the diffusion coefficient D L of the interface scales as 1 /L . The time fortwo such interfaces that are separated by a distance of the order of L to meet thereforescales as L /D L ∼ L ∼ N / .In a d -dimensional system, the analog of two-stripe state is a two-slab state witha ( d − N ( d +1) /d as the time scale for two initially flat interfaces to meet.According to this approach, the consensus time scales linearly with N in the limit of d → ∞ , a limit that one normally associates with the mean-field limit. However, therate equation approach gives a consensus time that grows as ln N . We do not know howto resolve this dichotomy.
4. Summary
We introduced the notion of individual confidence in the context of the voter model.Our model is based on recent social experiments that point to the importance of multiplereinforcing inputs as an important influence for adopting a new opinion or behavior [20].We studied two variants of confident voting in which an agent who has just switchedopinion will be either have confidence in the new opinion — the extremal model — orbe unsure of the new opinion — the marginal model. In the mean-field limit, a nearlysymmetric system quickly evolves to an intermediate metastable state before finallyreaching a consensus in one of the confident opinion states. This intermediate state isreached in a time of the order of one, while the time to reach consensus scales as ln N .On a two-dimensional lattice, a substantial fraction of all realizations of a randominitial condition reach a long-lived stripe state before ultimate consensus is reached. This EFERENCES N , corresponds to realizations that reach consensus by domain coarsening. Thelonger time corresponds to realizations that get stuck in a metastable stripe state beforeultimately reaching consensus.An unexpected feature of confident voting is that the behavior in two dimensions,where the consensus time T N varies as a power law in N , is drastically different thanthat of the mean-field limit, where T N varies logarithmically with N . In contrast, in theclassic voter model, T N ∼ N ln N in two dimensions, whereas the mean-field behavior is T N ∼ N . This dichotomy suggests that confident voting on the complete graph does notcorrespond to the limiting behavior of confident voting on a high-dimensional hypercubiclattice. Moreover the argument that T N on a d -dimensional hypercubic lattice scales as N ( d +1) /d suggests that the upper critical dimension for confident voting is infinite. Acknowledgments
We gratefully acknowledge financial support from NSF Grant No. DMR-0906504.
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