Effect of initial concentration and spatial heterogeneity of active agent distribution on opinion dynamics

Physica A 390 (2011) 3876–3887

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Physica A

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Effect of initial concentration and spatial heterogeneity of active agentdistribution on opinion dynamics

Alexander S. Balankin ∗, Miguel Ángel Martínez Cruz, Alfredo Trejo MartínezGrupo Mecánica Fractal, Instituto Politécnico Nacional, México D.F., 07738, Mexico

a r t i c l e i n f o

Article history:Received 22 January 2011Received in revised form 3 May 2011Available online 12 June 2011

Keywords:SociophysicsMonte Carlo simulationFinite-size scalingCritical exponents

a b s t r a c t

We analyze the effect of spatial heterogeneity in the initial spin distribution on spindynamics in a three-state square lattice divided into spatial cells (districts). In the spiritof the statistical mechanics of social impact, we introduce a dominant influence rule (DIR),according to which, in a single update step, a chosen node adopts the state determinedby the influence of its discussion group formed by the node itself and its neighborswithin one or more coordination spheres. In contrast to models based on some formof majority rule (MR), a system governed by the DIR is easily trapped in a stable non-consensus state, if all nodes of the discussion group have the same weight of influence.To ensure that a consensus in the DIR system is necessarily reached, we need to put astochastic process in the update rule. Further, the stochastic DIRmodel is used as a startingpoint for understanding the effect of spatial heterogeneity of active agent (non-zero spin)distribution on the exit probabilities. Initially, the positive and negative spins (activeagents) are assigned to some nodes with non-uniform spatial distributions; while the restof the nodes remain in the state with spin zero (uncommitted voters). By varying therelative means and skewness of the initial spin distributions, we observe critical behaviorsof exit probabilities in finite size systems. The critical exponents are obtained by MonteCarlo simulations. The results of numerical simulations are discussed in the context of socialdynamics.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

It is well known that many features of the collective behavior of large social systems are independent of the attributesof individuals and details of social interactions [1–4]. In this way, stochastic spin models on networks are widely used tostudy the general features of a wide variety of social systems with the ultimate aim to explain the occurrence at a globallevel of complex phenomena like the formation of hierarchies and consensus (see Refs. [1–14] and references therein). Theuse of spin models to simulate the opinion dynamics is based on the social observation that people tend to cooperate whileexchanging their opinions, and these interactions cause an opinion shift towards consensus or compromise among somealternatives, reminiscent of the stable magnetic states of spin models on networks [15]. The aim of these simulations is notonly academic, as the spinmodels permit to reproduce and study the general features of real social systems. Elections providea precise global measurement of the state of the electorate opinions and so constitute an ideal playground for application ofthe statistical physics tools to model the opinion dynamics (see Refs. [1–14] and references therein). Analysis of electoratedynamics is one of the key problems for forecasting the results of voting [11]. In this way, it was found that the distributionsof the number of candidates according to the number of votes they received in Brazilian, Indian, and Mexican elections canbe reproduced with simple spin dynamic models [9,16–18]. The role of inflexible minorities in the breaking of democratic

∗ Corresponding author. Tel.: +52 5557296000.E-mail address: [email protected] (A.S. Balankin).

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opinion dynamics was studied in Ref. [10]. An interesting result was obtained in a three-candidate voter model: an initialdamage and suppression of one candidate may later lead to an enhancement of the same candidate [19].1

In general, a spin dynamic model consists of N agents that are fixed on network vertices. Each agent can either assumethe state with spin s = +1 or s = −1, or behave in the neutral state with s = 0 [15]. In the context of social dynamics, spinsof different signs can be associated with active agents of two different parties, while uncommitted voters are associatedwith si = 0.2 The system evolves according to a specific dynamic rule and generally tends to a stable magnetic state [15].

The social dynamic rules can be divided in two groups [23]. Those in which individuals form their beliefs based on theopinions of their neighbors in a social network of personal acquaintances (see, for example, Refs. [3,6,7,11]), and those inwhich, conversely, network connections form between individuals of similar beliefs [24,25].3 Various rules of neighbors’influence have been proposed to model the opinion formation (spin dynamics), among which are the voter model (VM)[26–28], the Sznajd model [6], the bounded confidence models [7,21,29,30], the social impact models [5], the majority rule(MR) models [3], and the ‘‘vacillating’’ VM [31,32], among others (see, for review Refs. [3,4]). Each of these models belongsto a well defined universality class determined by the system dimensionality (d), the conservation laws, the symmetry ofthe order parameter, the range of the interactions, and the coupling of the order parameter to conserved quantities [33].Each class of universality is characterized by a set of critical exponents governing the behavior of the spin model [33–35].

In the paradigmatic VM [27], at each time step, a randomly selected node adopts the opinion of a randomly-chosenneighbor.4 This step is repeated until the system reaches consensus. The VMnode-update dynamics conserves the ensembleaverage magnetization in a regular lattice. So, the probability that the system eventually ends with all positive (negative)spins equals the initial density of positive (negative) spins in all spatial dimensions. The time to consensus ts depends onthe system size N . In the VM on a two-dimensional lattice the time to consensus scales as ts ∝ N lnN , while ts ∝ N2 ind = 1 and ts ∝ N for d > 2 [4]. It has been shown that the two-dimensional VM represents a broad class of models forwhich phase ordering takes place without surface tension [28]. The effect of long-range interactions, such as the shortcutspresent in small-world networks on the ordering process of the VM was studied in Ref. [36]. The introduction of noisedrastically affects the VM dynamics [37]. In particular, VM with noise does not converge to a state of complete order in thethermodynamic limit [37]. The authors of Ref. [38] have introduced an additional rule in the VM termed as the latency:after a voter changes its opinion, it enters a waiting period of stochastic length where no further changes take place. Inthis case the average magnetization is not conserved, and the system is driven toward zero magnetization, independentlyof initial conditions [38]. The authors of Ref. [39] have proposed a variant of the voter model by introducing the socialdiversity in the evolution process. The VM update rule has also been used to study the dynamics of systems with morethan two opinion states [19,40,41]. The cyclically dominated three-state voter model on a square has been extended bytaking into consideration the variation of Potts energy during the nearest neighbor invasions [41], for study of the effect ofsurface tension on the self-organizing patterns maintained by the cyclic invasions. It was found that the model undergoesa continuous phase transition as this parameter is varied, from a regime in which opinions are arbitrarily diverse to one inwhich most individuals hold the same opinion [41]. Castellano et al. [42] have suggested a nonlinear variant of the VM, theq-voter model, in which q neighbors (with possible repetition) are consulted for a voter to change opinion. If the q neighborsagree, the voter takes their opinion; if they do not have a unanimous opinion, still a voter can flip its statewith probability δ.5A simple modification of VM was suggested by Sznajd–Weron and Sznajd [6]. It is based on the principle that if two friendsshare the same opinion, they may succeed in convincing their acquaintances of their opinion (‘‘united we stand, divided wefall’’).6 Fortunato [21] has proposed the extension of the Sznajd model with continuous opinions, based on the criterion ofbounded confidence.7 Masuda et al. [43] have introduced the heterogeneous (HVN) and partisan (PVM) voter models. In theHVM each agent has its own intrinsic rate to change state, reflective of the heterogeneity of real people, whereas in the PVMeach agent has an innate and fixed preference for one of two possible opinion states [43].8 Some other modifications of theclassic VM rule have been also suggested (see Ref. [4] and references therein) to model non-equilibrium systems.

1 While surprising at first glance, this phenomenon was observed during the electoral campaign in the 2006 Mexican presidential election [20].2 In models with continuous opinion, each individual has, at least initially, its own attitude/opinion, such that si are real numbers and a ‘‘confidence

bound’’ ε is introduced such that two agents are compatible if their opinions differ from each other by less than ε, i.e. |si − si+1| < ε [7,21]. Furthermore,the opinions can be represented by vectors with real-valued components [22].3 Holme andNewman [23] have suggested the combination of these two processeswithin a simplemodelwith a single parameter controlling the balance

between two procedures.4 Voter dynamics was first considered by Clifford and Sudbury [26] as a model for the competition of species and named the ‘‘voter model’’ by Holley

and Liggett [27] for the very natural interpretation of its rules in terms of opinion dynamics.5 This model illustrates how apparently innocuous changes in the microscopic dynamics can lead to different types of collective phenomena, and in

particular to different paths to reach consensus [42].6 In the original Sznajd’s algorithm, if a randomly chosen pair of neighboring agents are in the same state si = si+1 , their neighbors i−1 and i+2 adopt the

same state,whereas if si = −si+1 , each agent of the pair ‘‘imposes’’ its opinion to theneighbor of the other agent of the pair, so si−1 = si+1 = −si = −si+2 [6].With these rules there are two possibilities of a final state: Complete consensus (perfect ferromagnetic) or a perfect splitting of the community in twofactions, with exactly half of the agents sharing either opinion (perfect antiferromagnetic). However, in most successive studies on the Sznajd model thesecond rule has usually been neglected, such that a consensus becomes the only possible final.7 It was found that in the model with continuous opinion and the original Sznajd update rules the bi-polarization is very likely to occur at low values of

confidence bound ε < εC ∼= 0.5 [21].8 Both, the partisan and heterogeneous VM rules move a finite size system toward a consensus, but much slower than the classic VM [43].

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In particular, in the ‘‘vacillating’’ VM [31] a sampled node checks the state of two of its neighbors and flips if either isdifferent from it. This leads to a bias toward the zero magnetization state. Because of the bias, the average magnetization isnot conserved and theprobability to reach a consensus is essentially independent of the initial composition of thepopulation.A finite system can achieve the consensus only via macroscopic fluctuations allowing the system to escape the bias-inducedpotential well.9

Another class of irreversible systems is provided by the spinmodels evolving according to aMR [1–3,9,12,44–47]. TheMRimplies that at each update the selected groups of nodes adopt the states of the majority inside the group. MR was actuallyfirst employed to model a continuous phase transition within a framework of geometric statistical problems [48]. In thecontext of opinion formation, the majority rule was first used in a model describing hierarchical voting in a society [1]. Incontrast to the classical VM, the MR models do not conserve the average magnetization. Since the average magnetizationis not conserved by the MR dynamics, the exit probability has a nontrivial dependence on the initial magnetization inthe thermodynamic limit and a minority can actually win the contest [4]. The MR model 1 with a fixed group size G wasanalytically solved in the mean-field limit [44]. In the mean-field limit, where a group consists of randomly selected agents,the time to consensus scales with the number of agents as ts ∝ lnN , while on finite-dimensional lattices, where a groupis a contiguous cluster, the consensus time fluctuates strongly between realizations and grows as a dimension-dependentpower of N [44,45]. If the noise parameter ξ is introduced to put a stochastic process into the majority rule, a minorityopinion can be selected with probability ξ and a phase transition occurs at the critical noise parameter ξC [49].10 Theauthors of Ref. [50] have developed themajority-vote model with noise in a network of social interactions for a systemwithtwo classes of individuals obeying different dynamics: individuals of A-class being influenced by neighbors of both classes,while the individuals of type B are influenced only by neighbors of that class. Nonetheless the dynamics of this model ismore reach that of the classic MR model, it belongs to the same universality class as the two-dimensional Ising model [50].Othermodifications ofMR include the introduction of a the latency [38] and a probability to favor a particular opinion,whichcould vary among different individuals and/or social groups [51], the variable group sizes [52], voters liking to win [53], theagentsmoving in space [54,55], and extension to three [56] andmore opinions [57], among others (see, for review, Refs. [3,4,58,59]). In themajority–minority model [60] in a single update step, a fixed-size group is defined and all agents in the groupadopt the state of the local majority with probability p or that of the local minority with probability 1− p. It was shown thatthe spin dynamics drastically changes due to a phase transition at p = pC .11

Besides, opinion dynamics is deeply dependent on the statistical topology of the social network [4,16,17,24,61–66]. It wasshown that the existence of community structures in the social network can drastically affect opinion formation [12,66].In the context of the VM, the effect of communities has been studied in the extreme situations where some voters are‘‘zealots’’ who never change opinion [10,61]. This attribute prevents consensus from being reached when zealots withdifferent opinions exist. In the MRmodels considered in Refs. [12,66], the community is characterized by a different degreeof agent connectivity inside and outside the community. This leads to more rich collective behaviors.

In this work, we are especially interested to model the empirical observation that the ‘‘electoral strength’’ of a politicalparty depends not only on the number of active members, but also on their distribution among the electoral districts (seeRefs. [67,68]). Accordingly, we study the effects of initial concentration and spatial heterogeneity in the initial distributionof active agents (non-zero spins) on the ‘‘electoral strength’’ of political parties.

This paper is organized as follows. In Section 2, we introduce the model of neighbors influence. The details of oursimulational procedure are specified in Section 3. Section 4 is devoted to the results and scaling analysis of the Monte Carlosimulations; in Section 4.4 the relevance of our findings is discussed in the context of social dynamics. A summary of findingsis given in Section 5.

2. Dominant influence rule

In the spirit of the statistical mechanics of social impact (see Ref. [5]), to study the ‘‘electoral strength’’ behavior weintroduce a dominant influence rule (DIR), according to which, in a single update step, a chosen node adopts the statedetermined by the influence of itself and some other nodes. That is, in each update step, the state of the node chosen byrandom sampling without replacement is updated as si(t − 1) → si(t), such that the state of each node of the lattice withN nodes is updated once per the cycle of t = N update steps.

In this work, we used three-state spin systems on lattices with N nodes. Each node i ∈ [1,N] can be in one of three spinstates of si = 1, −1, associated with active agents of two different parties, or si = 0, associated with uncommitted voters.According to the DIR, node i sampled at step t adopts the state with

si(t) = S

N−

m=1

dimξmi(t)sm(t − 1)

, (1)

9 Consequently, the time to consensus exponentially increases with the system size [31].10 The choice of transition rate satisfying the detailed balance condition is not unique. Though, it was found that the critical behavior of themajority votermodel with noise is independent of transition rates and belongs to the same universality class as the Ising model because of up–down symmetry [46,49].11 For group size G = 3, pC = 2/3 in all spatial dimensions [60].

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Fig. 1. Distributions of positive (black), negative (white) and zero (gray) spins in the lattice of size 50 × 50: (a) Initial distribution with 50% negativeand 50% positive spins, (b) the trapped two-state stable configuration of the system (a) after 4976 updates according to the DIR with ξmi(t) = const;(c) initial random distribution of 10% negative and 10% positive spins, while the rest of nodes are in the zero-spin state, (d) the trapped three-state stableconfiguration of the system (a) after 5523 updates according to the DIR with ξmi(t) = const .

where S(X) = sign(X), if X = 0, or S(X) = 0, if X = 0; dim denotes the weight of node m influence on node i; ξmi(t) is astochastic variable with the given probability distribution independently for each i,m, and t . In the context of social impacttheory the influence of different members of the discussion group on the sampled voter (node) is generally different andstochastically changed in time (see Ref. [12]). So, dimξmi(t) represents the stochastic impact of nodem on node i at the updatestep t [12].

Following to the spirit of the MR arguments, we assume that the state of the selected node after update is determined bythe members of its discussion group before the update.12 Generally, the discussion group of node i is defined as the group ofnodes including the i-node itself (dii = 1) and its g neighbor nodes with non-zero influence, i.e. with |dim| > 0,13 whereasdim = 0, if nodem-node does not belong to the discussion group of the i-node.

In practice, there are two ways for discussion group definition. Specifically, the discussion group of each node can bedefined deterministically as the node itself with all its neighbors within some coordination spheres, e.g. the von Neumannneighborhood (the selected node plus its g = 2d nearest neighbors). Alternatively, the neighbors with nonzero influenceon the selected node can be chosen by a random sampling at each update, as, for example, it was employed in Ref. [45] withthe MR model. So, in some sense, the discussion group is reminiscent of the group of G agents obeying the MR.

The primary operational difference between the DIR and theMR is that in the end, all G spins of the sampled (discussion)group adopt the same state, whereas in the DIRmodel, only the state of the sampled node is updated in a single update step.This distinction in the update rule has fundamental consequences. In particular, if G is odd, a consensus is the only possiblefinal state in the MR model [45]. In contrast to this, if for all members of the discussion group dim = 1 and ξmi(t) = const ,as this is in the MR model, the DIR system is easily trapped in a stable non-consensus configuration (see Fig. 1a, b) in whicheach node has the number of discussion groupmembers in the same spin state equal or larger than the number of the group

12 The model can be easily extended for the case of continuous opinion, such that si are real numbers between −1 and 1. In this case si(t) =

N−1∑Nm=1 dimξim(t)sm(t − 1), where 0 ≤ ξim(t) ≤ 1.

13 Generally, the weight of influence of the member of discussion group on the i-node can be positive or negative and the equality dim = dmi does notnecessarily hold.

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Fig. 2. Spatial distributions of positive (black) and negative (white) spins in the lattice of size 50×50 divided into 25 districts (gray nodes have spin zero):(a) Positive and negative spins are distributed according to normal law (nsj = 10, σs = 2); (b) snapshot of the system (a) after 2 cycles; (c) positive spinsare distributed according to normal law (n1j = 10, σ1 = 2), whereas negative spins obey a log–logistic distribution (2) with n−1j = 10 and q = 0.6;(d) snapshot of the system (c) after 2 cycles.

members in the opposite spin state.14 Furthermore, in the three-state spin systems obeying the DIR some nodes of stablenon-consensus configurations remain in the neutral state with equal numbers of group neighbors in the opposite states (seeFig. 1c, d). Therefore, to ensure that a consensus in the DIR model is necessarily reached, we need to put a stochastic processξmi(t) into the update rule (1).15

We also note that, in some sense, the DIR dynamics can be viewed as a generalization of the ‘‘vacillating’’ VM dynamics.In fact, if the discussion group is formed by the sampled node with its two nearest neighbors (g = 2), the dynamics oftwo-state DIR system coincides with the dynamics of the ‘‘vacillating’’ VM suggested in Ref. [31]: a sampled node changes itsstate only if both neighbors are in the same opposite state, or otherwise the state of sampled node is conserved. This update rule,however, differs from DIR (1) for the three-state spin systems. Furthermore, if the size of discussion group G = 1 + g > 3,the DIR dynamics differs from the ‘‘vacillating’’ VM dynamics even conceptually.

3. Details of numerical simulations

In this work, we study the effect of the initial concentrations and spatial distributions of active agents of two competingparties (non-zero spins) on their ‘‘electoral strength’’, defined as its probability to win P−1 = 1 − P1.16 In order to accountthe spatial heterogeneity of active agent distributions, square lattices of different sizes 400 ≤ N ≤ 40,000 are dividedinto k = N/100 square segments (districts) with 100 nodes each (see Fig. 2). Initially, we assign non-zero spins ton = n1 + n−1 < N nodes distributed non-uniformly among districts, while the rest of nodes remain in the state with

14 Notice that in the kinetic Ising model with zero-temperature Glauber kinetics static states with non-zero magnetization can also occur, that consistof perfectly flat interfaces in d = 2, or states where all interfaces have zero net curvature for d = 3 (see Ref. [69]). However, these striped configurationsdiffer from the static configurations in the DIR model without noise (see Fig. 1b, d).15 Alternatively, a discussion group can be defined as a sampled agent together with g agents chosen at random among gn > g neighbors.16 Another interesting topic to study with the DIR is the effect of uncommitted voters on the consensus probabilities and on stable non-consensusconfigurations, such as shown in Fig. 2b, d. However these topics are not aborded in this work.

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s = 0 (see Fig. 2a, c). Notice that the non-zero spins can be successfully distributed among districts according to specificnon-uniform distributions fs(nsj), where s = ±1 and j ∈ [1, k], only when n = n1 + n−1 << N , because n1j + n−1j ≤ 100,where nsj are the numbers of positive (s = 1) and negative (s = −1) spins in the district j.17 Therefore, in all simulationsperformed in this work the initial number of positive spins is fixed according to relation

n1 = 0.1k−j

n1j = 0.1 N = 10 k,

while the ratio x = n−1/n1 can be varied in the range of 0.1 ≤ x < 1.Accordingly, two series of Monte Carlo simulations were performed on square lattices with dim = dii = 1 for m-nodes

within two first coordination spheres of i-node (g = 8 neighbors), while dim = dmi = 0 otherwise.18 Additionally, to havea more intimate reference to the MR model, we have performed Monte Carlo simulations of the MR model considered inRef. [45], but with discussion groups of size G = g + 1 = 9 on square lattices divided in the districts with 100 nodes. Theperiodic boundary conditions were employed in all simulations. To ensure that a consensus is necessarily reached, in thiswork, following Ref. [12], we have used random functions ξim(t) with the uniform distributions on [0; 1].

In the first series of numerical simulations, the positive and negative spins are initially distributed among districtsaccording to normal laws. However, the mean and standard deviation of positive spin distributions are fixed as n1j =

0.1N/k = 10 and σ1 = 0.02N/k = 2, respectively, whereas the mean and standard deviation of negative spin distributionsare varied in the ranges of 1 ≤ n−1j ≤ 10 and 1 ≤ σ−1 ≤ 6, respectively.

In the second series of simulations, the positive spins are also normally distributed among districts with the meann1j = 10 and the standard deviation σ1 = 2, whereas the negative spins are distributed among districts according to apositively skewed distribution. Specifically, to generate the initial distribution of negative spins among the districts, in thiswork we used the log–logistic distribution19

f−1(n−1j) =α

m

n−1j/m

α−11 +

n−1j/m

α2 , (2)

where m is the scale parameter and α is the shape parameter.20 The log–logistic distribution is positively skewed. Thereare different parameters to characterize the distribution skewness. In this work, to characterize the skewness of initialdistributions we use the ratio of the median to the mean.21 For log–logistic distribution (2) the median (m = α) to mean(n−1j = n−1/k = 10) ratio is equal to

q =m

n−1j=

α

πsinπ

α

(3)

for α > 1 [70]. In practice, however, it is very difficult to achieve a good log–logistic distribution of integer numbers0 ≤ n−1j ≤ 100 between a finite number of districts 4 ≤ k = N/100 ≤ 100, especially when k ≤ 25. So, to characterizethe heterogeneity of active agent distribution, we used the actual ratio 0.1 ≤ q ≤ 1 of the output distribution, rather thanthe shape parameter α or the ratio q of the input log–logistic distribution.

In all our numerical simulations we used random sampling without replacement. Specifically, in each cycle, districts areselected by random sampling without replacement until all districts are sampled. Once the district is selected, all nodesof this district are updated in random order without replacement. Here, it should be pointed out that we verified that thedetails of randomsampling determines the absolute values of the time to consensus only and do not affect neither qualitativebehavior nor the scaling exponents reported in the next section.

4. Results and discussion

We first noted that, in all our simulations of the DIR model with random impact functions ξim(t), the two-spin state ofs = 1 and s = −1 is achieved after two cycles (see Fig. 2). However, further, there are two fundamentally different scenariosof system evolution, similar to those studied in Ref. [69] using the Ising model with Glauber dynamics and in Ref. [45] using

17 Notice that nsj represents the concentrations (in percent) of active agents in the district j, while 100ns/N is the concentration of s-spins in the lattice.18 In fact, we found that the DIR models with discussion groups of different size and random ξim(t) lead to qualitatively similar results (see also Ref. [45]).Though, we noted that some features studied in this work are somewhat more pronounced in the DIR model with the group size of G = q + 1 = 9, thanin the DIR model with the von Neumann neighborhood (G = 5).19 Initially, we perform numerical simulations with positively skewed Pareto, gamma, and log–logistic distributions. However, we found that theconsensus time and the exit probability are controlled by the skewness of distribution rather than by a specific type of distribution.20 The use of the log–logistic distribution is inspired by the empirical observation from six federal elections in México in period 1991–2006. Mexico isdivided in 300 electoral districts with one deputy from each. The relative electoral strength of political party in a district is characterized by the percentageof votes to the party in this district. It was found that for ‘‘emergent’’ parties the statistical distribution of relative electoral strength among 300 districtsare always best fitted with a log–logistic distribution, whereas the relative strengths of leading parties obey a normal distribution [68].21 We have tried to use different measures of the skewness of the output initial distribution of negative spins and found that the most consistent results(presented in Fig. 6) are obtained with the use of the median to mean ratio.

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Fig. 3. Dynamic of systems without (a–c) and with (d–f) coherent states: (a, d) Snapshots after 120 cycles; (b, e) concentrations ps = ns/N of positive(curves 1) and negative (curves 2) spins versus time t (square and full circle—initial concentrations); (c, f) log–log plots of consensus time Tc (in number ofcycles) versus lattice size N for simulations when only a state of one sampled node is updated according to the DIR rule (1) with random ξmi(t) (curves 1)and when 9 spins of a selected majority group simultaneously adopt the same state according to the MR with random ξmi(t) (curves 2); symbols—data ofMonte Carlo simulations, straight lines—power law fittings.

theMRmodel. In the first scenario, the evolution of finite clusters of spins of the same sign is governedbydiffusive coarsening(see Fig. 3a, b). In another scenario, the opposite-spin domains organize into coherent geometries–stripes (see Fig. 3d) witha very slow kinetics of evolution (see Fig. 3e).

We found that stating from the initial state with x = q = 1, the probability for the system to reach a coherent statein the DIR model simulations is PDIR

coh (1, 1) = 37.5% ± 4%. This value is close to, but somewhat larger than the probabilityP IGcoh(1, 1) = 1/3 expected for the Ising model with Glauber dynamics [69] and the probability PMR

coh(1, 1) = 30% ± 5%observed in numerical simulations with the two-state MR model [45]. As the ratio of initial concentrations of non-zerospins is moved away from zero, PDIR

coh (x), as well as P IGcoh(1, 1) and PMR

coh(x), quickly decay to zero, reflecting the fact that if onephase is initially below the percolation threshold, there is a very small possibility for minority phase droplets to merge andform a stripe that spans the system (see Ref. [45]). We also observed a quick decrease of PDIR

coh (q) as the ratio q decreases.This observation is easy to understand, because the probability to form a system-spanning cluster decreases due to the localclustering of negative spins.

The approach to consensus is very different for realizations with and without coherent states (see Fig. 3b, e). So, in thiswork, the Monte Carlo realizations with and without stripe formations are treated separately.

4.1. Time to consensus

The relevant property of spin models on networks is the time to consensus [1–4]. Following to Ref. [45], in this workthe consensus time Tc is measured in the number of cycles and so, Tc = ts/N , where ts is the number of update steps to

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consensus. In Monte Carlo simulations with the DIR model we found that for both scenarios of evolution (see Fig. 3) themean time to consensus scales as

Tc ∝ Nν, (4)

but with different values of the scaling exponent ν (see Fig. 3c, f). Specifically, for realizations without coherent states wefound the value22

ν = 1.00 ± 0.05, (5)

which differs from the value ν = 1.24 ± 0.04 obtained for this scenario in our simulations of the MR model (G = 9)with noise associated with the use of random functions ξmi(t)23 (compare curves 1 and 2 in Fig. 3c). Thus, the operationaldifference in the update rule causes a change of the universality class of spin dynamics. This is not surprising because theMRand DIR produce different kinds of spin dynamics (see Section 2). As far as we know, the value (5) differs from the universalvalues of ν for the known universality classes of spin dynamics in d = 2. So, the DIR model does not belong to any knownuniversality class of lattice spin models.

In contrast to this, for realizations with spins organized into coherent single-opinion bands the scaling exponent

νc = 1.70 ± 0.05 (6)

found in theMonte Carlo simulations of the DIRmodel is numerically close to the corresponding exponent νc = 1.72±0.05obtained in our numerical simulations with the simultaneous update of all spins of a sampled group, nonetheless that thetime to consensus in the DIR model is more than 10 times larger than Tc in the simulations of the MR model with the sameG = 924 (see curves 1 and 2 in Fig. 3f).

While the numerical coincidence of νc for realizations with the long-living coherent single-opinion bands in the MR andDIR models can be accidental, we can not exclude that this coincidence may reflect some similarities of interface evolutionin the MR and the DIR models. In fact, in this scenario only the nodes belonging to the interface can change their states.These nodes always have at least 3 neighbors with the same spin and at least 3 neighbors with the opposite spin. Therefore,a specific position of a node on the interface determines the state of two neighbors only. So, the dynamics of interfacebetween two coherent states is the same as in the one-dimensional ‘‘vacillating’’ VM: a sampled node changes its state onlyif both of these two ‘‘active’’ neighbors are in the same opposite state. Otherwise the state of sampled node is conserved.Moreover, this dynamics is somewhat similar to the interface dynamics in theMRmodelwithG = 3. Namely, if both ‘‘active’’neighbors are in the same state, after update all members of the discussion group in the DIR model are in the same state, asthis is after the MR update. However, if the ‘‘active’’ neighbors are in opposite states, in the MR model all members of thesampled discussion group adopt the same state of themajority, whereas there is no change of interface after the DIR update.Nonetheless, our data suggest that this difference in the update dynamics does not change the scaling exponent νc , becausethe system achieves a consensus due to a larger fluctuation (see Fig. 3e), rather than by the interface diffusion. This alsoexplains the difference between the values of νc obtained in Ref. [45] by the numerical simulations (νc = 1.7) and throughthe theoretical arguments based on the analysis of interface diffusion (νc = 1.5).

Further, we noted that once the opposite-spin domains are organized into stripes (see Fig. 3d), both alternatives havealmost equal probabilities to win (see Fig. 3e). Thus, in this scenario, the exit probability P−1(x, q) and the consensus timeTc are almost independent on the x and q, whereas the probability Pcoh(x, q) vanishes as x or q decreases.25

In the casewhen the DIRmodel evolution is governed by diffusive coarsening (see Fig. 3a–c), the exit probability P−1(x, q)and the consensus time Tc both decrease as x or q decreases. So, in the next two sub-sectionswe are focused on this scenario.Besides, to link the DIR model with the elections in a two-party society, the nodes with non-zero spins below are referredto as the active agents.

4.2. Effect of the initial concentrations on the exit probabilities

When active agents of both parties are initially distributed among districts according to normal laws, we found that theprobability to end in the state with all spins equal to −1 decreases as the initial relative concentration of negative spinsdecreases (see Fig. 4a). We found that the results of Monte Carlo simulations in lattices of size N = 400 are best fitted withthe power law relationship

P−1 = 0.5x3, (7)

22 The same value (5) was also found in simulations with the two-state DIR (S(0) = si(t − 1) in Eq. (1)) in which, initially, agents of both parties arerandomly distributed (n1 + n−1 = N).23 It should be pointed out that the value ν = 1.24 ± 0.04 coincides with the result obtained in Ref. [45] for the MR model (G = 3) without nose (that isequivalent to ξ(t) = 1). So, neither the discussion group size, nor the random noise associated with ξim(t) affect the scaling behavior (4, 5).24 Notice that the same value of νc was obtained by the numerical simulations ofMRmodel with G = 3 (see Ref. [45]), whereas the general considerationsof interface evolution according to the MR dynamics suggest a simple scaling relation νc = (d + 1)/d = 1.5 [45].25 We does not studied the functional form of Pcoh(x, q). Though, we noted that Pcoh(x, q) = 0, if x < xC or q < qC .

3884 A.S. Balankin et al. / Physica A 390 (2011) 3876–3887

Fig. 4. Results of Monte Carlo simulations the three-state DIRmodel when initially the positive and the negative spins are distributed according to normallaws such that n−1 ≤ n1 = 0.1N: (a) Graphs of P−1 versus x for lattices of size N = 400 (1), 900 (2), and 40,000 (3); for visual clarity the data for latticesof other sizes are not presented; (b) log–log plots of P−1 versus (x − xC ) for lattices of size: N = 400 (1), 900 (2) 2500 (3), 3600 (4), 6400 (5), 10,000(6), and 40,000 (7); symbols—results of Monte Carlo simulations, straight lines—power law fittings with Eqs. (7)–(10); (c) log–log plot of 1 − xC versusN; circles—results of Monte Carlo simulations, straight line—power law fitting by Eq. (10) with NX = 141; (d) data collapse in coordinates P−1 versusX∗

= (x − xC )/(1 − xC ), symbols—the same as in the panel (b), straight line—power law fittings by Eq. (8) with β = 3.

whereas in lattices of larger sizes N ≥ NC > 900, the exit probability behaves, as in the case of standard second-ordertransitions, according to relationship

P−1 =12

x − xC1 − xC

β

, if x ≥ xC , (8)

and it becomes zero when the ratio x is less or equal to the critical value xC (see Fig. 4b). We establish that the scalingexponent

β = 3.0 ± 0.1 (9)

is independent on the lattice size N and the standard deviations of initial distributions fs(nsj)26(see Fig. 4b).

The critical ratio xC is found to increase with the lattice size N as

xC = 1 −

NX

N, (10)

such that for lattices of size less than the critical size NX = 212= 441, the minority always has a chance to win (xC = 0),

whereas in the limit of infinite lattice size xC (N → ∞) = 1 (see Fig. 4c). So, while the minority may win in a finite system,the probability for this event quickly vanishes asN → ∞. That is, the exit probability P−1(N, x) approaches a step function asN → ∞ (see Fig. 4a) with a characteristic width that scales asN−1/2, reflecting the fact that the exit probability is controlledby the global bias. Fig. 4d shows the data collapse for lattices of different size.

It should be pointed out that while in the MR model the probability that the initially minority opinion wins alsoapproaches a step function as N → ∞, in a finite size system obeying the MR the exit probability P−1(x) does not exhibita critical behavior. The critical behavior of exit probability (8) also differs from an exponential decrease of the probabilitythat the initially minority phase wins in the Ising model with Glauber dynamics [69]. To have a more intimate referenceto the two-state models, we also perform simulations with the two-state DIR model (see footnote 22), such that, initially,n1 +n−1 = N , while agents of both parties are randomly distributed on a square networkwithout districting.We found (seeFig. 5) that in this case the exit probability also obeys the scaling behavior (8) with the universal exponent (9) and the criticalrelative concentration (10),27 while the critical size N (2)

X = 102= 100 is smaller than in the case when the uncommitted

26 Additional simulations with lattice of size N = 10,000 divided in 25 square districts of size 200 nodes each suggest that neither the time to consensusno the exponent (9) are dependent on the size of square districts.27 This finding, together with the finding that both two- and three-state DIR models are characterized by the same time-to-consensus exponent (5) (seefootnote 22) suggest that these models belong to the same universality class which is different from the known universality classes of spin dynamics.

A.S. Balankin et al. / Physica A 390 (2011) 3876–3887 3885

Fig. 5. Comparison between the results of Monte Carlo simulations for the three-state (open circles) and the two-state (full rhombus) DIR model on thesquare lattice of size N = 2500 (log–log plot of P−1 versus X∗). Straight line—data fitting by Eq. (8) with β = 3. Notice that for the three-state DIR modelxC (N = 2500, y ≥ 0.8) = 0.422 ± 0.006, while for the two-state DIR model xC (N = 2500, y = 0) = 0.800 ± 0.005.

voters are presented. Hence, when the lattice size is fixed, the critical concentration (10) decreases as the percentage (y) ofinitially uncommitted voters increases. This is easy to understand. In fact, in the DIR model with a noise all uncommittedvoters disappear after two cycles (see Fig. 2). Once the two-spin state is achieved, further DIR dynamics of models with andwithout the neutral state s = 0 is essentially the same (see footnote 22). However, the relative concentration x(2)(N, y, x)after the two-spin state is achieved generally differs from the initial relative concentration x. In fact, for a fixed x the values ofx(2) are distributed within the interval x(2)

m ≤ x(2)≤ x(2)

M , where x(2)m and x(2)

M are functions of the percentage of uncommittedvoters (y), the relative concentration x, and the lattice size N , such that x(2)

m ≤ x ≤ x(2)M . Accordingly, when the uncommitted

voters are presented, the critical relative concentration xC (y,N) can be obtained from the following equation

x(2)M (N, y, x = xC ) = x(2s)

C (N, y = 0),

where x(2s)C (N, y = 0) = 1 −

N (2)

X /N is the critical relative concentration for the two-state DIR model. So, xC (N, y > 0) <

x(2)C (N, y = 0) (see the caption of Fig. 5).

4.3. Effect of spatial heterogeneity in the initial distributions of active agents on the exit probabilities

In the case when initially the active agents of both parties have the same concentration, i.e., x = 1, but different types ofdistribution among districts (see Fig. 2c) we found that P−1(q) decreases with q decreasing. Specifically, in simulations withlattices of size N = 400 and 900 we found that the probability to end in the state with all negative spins behaves as

P−1 = 0.5q1.75, (11)

whereas in lattices of larger sizes N ≥ 1600 the exit probability P−1(q) displays the critical behavior

P−1 =12

q − qC1 − qC

γ

(12)

with the scaling exponent

γ = 1.75 ± 0.15, (13)

which is independent on the lattice size (see Fig. 6a). The critical ratio qC is found to increase with the lattice size N as

qC = 1 −

NQ

N, (14)

such that in the limit of infinite lattice size the exit probability P−1(x) approaches a step function, similarly to the caseof different initial concentrations of non-zero spins. (see Fig. 6b). So, the effect of distribution skewness becomes morepronounced as the system size increases. We note that the critical system size for the effect of spatial distribution NQ =

332= 1089 is somewhat larger than NX = 212

= 441 in the case considered in the Section 4.2. The data collapse for latticesof different size is shown in the insert of Fig. 6b.

3886 A.S. Balankin et al. / Physica A 390 (2011) 3876–3887

Fig. 6. Log–log plots of: (a) P−1 versus (q − qC ) for lattices of size N = 400 (1), 900 (2), 1600 (3), 2500 (4), 3600 (5), 4900 (6), 10,000 (7), and 40,000(8); symbols—results of Monte Carlo simulations, straight lines—power law fitting by Eqs. (11)–(14); (b) 1 − qC versus N; circles—results of Monte Carlosimulations, straight line—fitting by Eq. (14) with NQ = 1089, insert shows the data collapse in coordinates P−1 versus q∗

= (q − qC )/(1 − qC ).

4.4. Relevance in the context of social dynamics

The critical dependence of exit probability on the skewness of the active agent distribution illustrates the well knownfact that the ‘‘electoral strength’’ of political party depends not only on the number of active members, but also on theirdistribution among the electoral districts (see Refs. [67,68]). Actual electoral districts commonly have different sizes, e.g. inMéxico, the distribution of district sizes is an exponential. So, a more adequate measure of relative electoral strength ofpolitical party in a district is the percentage of votes to this party in this district (pj(T −1), d = 1, 2, . . . , k) in past elections(T − 1). In this way, the active agent distribution between districts of fixed size can be qualitatively associated with thedistribution of relative electoral strength of political party in different districts between districts of different size.

The existence of the critical value of the skewness of the distribution of relative strength of political partywas first pointedout in the empirical study of deputy elections inMéxico in period 1991–2006 [68]. Empirically, it was found that if pj(T −1),is distributed according to log–logistic distribution (2) with α ≤ 2.3 (q < 0.72), the total number of votes for this party inthe next deputy elections (T ) is much less than in the past election (T − 1).28

5. Conclusions

In summary, we introduce a dominant influence rule (DIR) to model the spin dynamics in non-equilibrium system. Wefound that the three-state DIR model on a two-dimensional lattice belongs neither to the VM universality classes, nor to theuniversality classes of the MR models. The three-state DIR model displays critical behavior of exit probabilities in a finitesize system. The critical scaling exponents β = 3.0 ± 0.1 and γ = 1.75 ± 0.15 are found to be independent on the latticesize and the initial concentrations of non-zero spins. In the context of social dynamics, the DIR model accounts the effect ofspatial heterogeneity of active agent distribution on the dynamics of opinion formation. Specifically, our findings illustratethe well known fact that the ‘‘electoral strength’’ of a political party depends not only on the number of active members, butalso on the distribution of their relative electoral strength in different districts among the districts.

28 In fact, in Mexican federal elections all parties with q < 0.72 have lost their registrations after the next elections in which they have obtained less than2% of the votes in all districts [68].

A.S. Balankin et al. / Physica A 390 (2011) 3876–3887 3887

Acknowledgment

This work was partially supported by the Government of Mexico City under Grant PICCT08-64.

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