Multi-choice opinion dynamics model based on Latané theory

In this paper Nowak–Szamrej–Latané model is reconsidered. This computerised model of opinion formation bases on Latané theory of social impact. We modify this model to allow for multi (more than two) opinions. With computer simulations we show that in the modified model the signatures of order/disorder phase transition are still observed. The transition may be observed in the average fraction of actors sharing the ith opinion, its variation and also average number of clusters of actors with the same opinion and the average size of the largest cluster of actors sharing the same opinion. Also an influence of model control parameters on simulation results is shortly reviewed. For a homogeneous society with identical actors’ supportiveness and persuasiveness the critical social temperature TC decreases with an increase of the number of available opinions K from TC = 6.1 (K = 2) via 4.7, 4.1 to TC = 3.6 for K = 3, 4, 5, respectively. The social temperature plays a role of a standard Boltzmann distribution parameter containing social impact as the equivalent of energy or one may think about it just as on a noise parameter.

The last classification includes system dynamics in terms of time evolution of the system, which again may occurring in discrete or in continuous time.Assumed scheme of system representation force choosing the most adequate numerical technique for computer simulation of the system, including solving set of differential equations [42] (continuous space of opinions, continuous geometry and continuous time) or cellular automata technique [43] (discrete space of opinions, discrete time and discrete geometry).
In this paper Nowak-Szamrej-Latané model is reconsidered [44].We propose multi-choice opinion dynamics model based on Latané [45][46][47] theory.With computer The mathematical model being the foundation of this work relies on Latané social impact theory [45][46][47] and its computerised version proposed by Nowak et al. [44].This approach for binary opinions and possible charismatic leader localised in the system centre has been thoroughly explored in Hołyst, Kacperski and Schweitzer papers [9,10] (see Ref. [49] for review).
Latané assumes that people are social animals and in their natural environment (society) they influence each other.These interactions do not have to be intentional.Under this assumption we understand all interactions among people.Persuasion, joke, sharing emotions and feelings-all of these can affect others.Latané describes these interactions as social impact.
The theory of social impact bases on three fundamental principles: i) social force, ii) psycho-social law and iii) multiplication/division of impact.

Social force
The social force principle [47] says that social impact I on i-th actors is a function of the product of strength S, immediacy J, and the number of sources N I = F(SJN ). ( The strength of influence is the intensity, power or importance of the source of influence.This concept may reflect socio-economical status of the one that affects on In subsequent rows the probabilities of changing opinion P i associated with sites i and for social temperature T = 0, 1 and 3 are presented.∀i : p i = s i = 0.5, α = 3 [48].
our opinion, his/her age, prestige or position in the society.
The immediacy determines the relationship between the source and the goal of influence.This may mean closeness in the social relationship, lack of communication barriers and ease of communication among actors.
Latané called this principle 'a bulb theory of social relations'.According to this analogy the social impact plays a role of illuminance.The illuminance depends on • the power of the bulb (physicists prefer to think about bulb's luminous flux)-equivalent of the strength of impact • the distance from sources (bulbs)-equivalent of the immediacy • and the number of bulbs-equivalent of the number of people.

Psycho-social law
The data of the famous Asch [50] and Milgram et al. [51] experiments may be fitted to formula proposed by Latané: where N is the number of people exerting the impact, S is a strength of impact and 0 < β < 1 is the scaling exponent.This means that each next actor j sharing the same opinion as actor i exerts the lower impact on the ith actor.This formula has been independently confirmed experimentally by Latané and Harkins [45].

Multiplication/division of impact
The lecture for single student influence his/her much more the same lecture given for hundred of students.In the latter case, the impact of lecture is roughly equally divided among all listeners [52].For this issue Latané proposes where the scaling exponent 0 < γ < 1. Latané and Nida [46] gathered results over hundred experiments to validate Eq. (3).

The limitations of the theory
The main limitation of the social impact theory lies in treating people as totally passive.The second trouble is the absence of dynamics in the model.These issues have been solved by Nowak et al. [44] in the computerised version of Latané model.

II. MODEL
Every actor at position i is characterised by his/her discrete opinion ξ i , his/her persuasiveness (0 ≤ p i ≤ 1) and his/her supportiveness (0 ≤ s i ≤ 1).Parameter p i describes the intensity of persuasion to change the opinion by actor i from a person with opinion different than ξ i , while s i describes the intensity of supporting people with the same views.

A. Two opinions (K = 2)
For two opinions one can assume integer values of ξ i ∈ {−1, +1}.For evaluation of social impact I i on actor at position i one can apply formula proposed in Ref. [49]: where J P (•), J S (•), q(•), g(•) stand for scaling functions and d i,j is Euclidean distance between sites i and j.The system dynamics may be governed by heat-bath-like dynamics [10], i.e.: where T is a noise parameter (social temperature [53]).For T = 0 the rule (5) may be reduced to fully deterministic rule [10] as I i (t) = 0 is practically impossible to occur.
B. Three and more opinions (K > 2) For multi-state space of opinions we do not assign numeric values to opinions where K is the number of available opinions.We rather prefer to think about various 'colours' of opinions, or about K orthogonal versors in K-dimensional vector space.Also we propose some modifications of Eq. ( 4).We propose to separate the social impact on actor i from actors j sharing opinion of actor i (ξ j = ξ i ) and all other actors having different K − 1 opinions (ξ j = ξ i ) where 1 ≤ k ≤ K enumerates the opinions.The factor of four in Eq. ( 7) guaranties exactly the same impact on actor i as calculated basing on Eq. ( 4) for K = 2.The calculated social impacts I i,k (t) influence the i-th actor opinion ξ i (t + 1) at the subsequent time step.For T = 0 this opinion is determined by those opinions which believers exert the largest social impact on i-th actor For finite values of social temperature T > 0 we apply the Boltzmann choice which yield probabilities of choosing by i-th actor in the next time step k-th opinion: The form of dependence (9) in statistics and economy is called logit function [54].We assume identity function for scaling functions J S (x) ≡ x, J P (x) ≡ x, q(x) ≡ x.The distance scaling function should be an increasing function of its argument.Here, we assume the distance scaling function as what ensures non-zero values g(0) = 1 of denominator for self-supportivenees in Eq. (7a).Newly evaluated opinions are applied synchronously to all actors.
The simulations are carried out on square lattice of linear size L = 40 with open boundary conditions.We assume identical values of supportivenees and persuasiveness for all actors ∀i : s i = p i = 0.5.We set exponent α = 3 in the distance scaling function (12).
The web application allowing for direct observation of the system evolution is available at http://www.zis.agh.edu.pl/app/MSc/Przemyslaw_Bancerowski/.The short manual for this application is available in Appendix A.

Influence of the model parameters on opinion dynamics
To understand better the system time evolution the maps of probabilities P i of opinion changes at sites i (for K = 2) are presented in Fig. 1.The snapshots of system states at t = 0, 1 and 10 are presented in the first row of Fig. 1.The corresponding to these states probabilities of opinion changing (flipping) for social temperatures T = 0, 1 and 3 are presented in the second, third and fourth row of the Fig. 1, respectively.For T = 0 (the second row) the system is fully deterministic and ∀i : P i ∈ {0, 1}.For long enough times of evolution the system reaches the nearly-steady state (with single spinsons1 going to change their minds) and clearly defined borders between groups (clusters) of spinsons with different opinions.The static picture of the system is also observed for T > 0, with non-zero probabilities of changing opinions P i for spinson i located at the clusters borders.
In Fig. 2 the time evolution of the spatial average of probabilities of opinion changes is presented.The spatial average over L 2 sites is marked through this paper by a bar (•).For long enough times the average probabilities of opinion changes P increases smoothly with increase of social temperature, reaching P ≈ 10% for T = 4.
As expected, an increase the social temperature T enhances the spinsons nonconformity, i.e. they are able to change their minds although social impact exerted on them by other members of the society with the same opinion.In the limit of infinite social temperature every actor chooses his/her opinion randomly, as lim T →∞ p i,k (t) = 1 and lim In Fig. 3 the maps of probabilities changes P i are presented again.The first row shows the snapshots from simulations indicating the spinsons opinions for t = 0 (first column) and t = 10 (second column).The subsequent rows correspond to probabilities of opinion changes for various values of exponent α in the distance scaling function g(x) [Eq.( 12)]-α = 2, 3, 6 in the second, third and fourth row, respectively.The random initial configu-ration of opinions leads to random maps of P i .However, ten time steps of system relaxation allows for an observation of both: the spatial clusterization of spinsons shearing the same opinion and high probabilities of opinion changing at the borders of these clusters.Moreover, for high values of exponent α differences among the minimal and the maximal values of P i are much smaller than for small values of α.
Quantitatively these differences may be observed in Fig. 4 for purely deterministic (T = 0) and nondeterministic (T = 1) cases.In principle, for T > 0 the higher value of the exponent α leads to the higher value of P which values saturate on the level P ≈ 1% after hundred simulation steps for T = 1 and α > 4.

Phase transition
In Fig. 5a the results on an average opinion for various values of the social temperature T are presented.Similarly to the Ising model some signatures of the phase transition in the system may be observed.For low social temperature (T < T C ) the system is in ordered phase with majority of one (initially dominant) opinion.However, for high enough temperature (T > T C ) the average opinion oscillates around ξ = 0.In Fig. 5b an example of time evolution of the spatial average opinion in the system for T → T + C is presented.Although the long-range interaction among actors is assumed, the time evolution ξ(t) is not different from 'magnetisation' evolution in the Ising model with characteristic 'magnetisation' switching between its positive and negative values above the Curie temperature.
In Fig. 6a we plot the temporal average for various temperatures T .The temporal average over τ times steps is marked through this paper by brackets ( • • • ).Here, τ = 5000 − 100, i.e. the first hundred of time steps is excluded from the averaging procedure.The ordered phase phase vanishes for T > T C ≈ 6.1.This critical value of T C coincidences nicely with a peak of average opinion dispersion as presented in Fig. 6b.The values of σ plays a role of static susceptibility χ in Ising-like systems.We confirm the earlier results indicating the phase transition in Nowak-Szamrej-Latané model for binary opinions [9].
In the next Section we show that the above mentioned results are generic also when multi-opinions are available in the system.

B. Three and more opinions
As we mentioned in Section II B for K > 2 we do not assign numerical values to opinions ξ i .Instead, we prefer to think about K 'colours' Ξ k=1,••• ,K of opinions (see Fig. 7 for snapshots from simulations presenting spatial distributions of opinions for T = 0, 6 and K = 3, 6).This assumption does not allow for dealing with ξ [Eq.( 14)] and σ( ξ) [Eq.(15)] in order to identify the critical social temperature T C .Thus for this purpose we propose to deal with a fraction nk of actors sharing the k-th opinion and its standard deviation σ(n k ).
In Figs.8a-8c and Figs.8d-8f we plot nk and σ(n k ) for K = 3, 4, 5, respectively.As we can see in Figs.8a-8c the majority n1 (t = 0) of holders of opinion Ξ 1 vanishes with increasing the social temperature T .For critical social temperature T ≥ T C all available opinions Ξ 1 , • • • , Ξ K in the system are equally occupied ( n1 = n2 = n3 ≈ 33% for K = 3 and n1 = • • • = n5 ≈ 20% for K = 5).Again, vanishing of initially major opinion at T = T C coincidences nicely with maximal values of σ(n k ) as presented in Figs.8d-8f.Similar critical behaviour may be observed in thermal evolution of the size of the larger cluster of actors sharing the same opinion Smax (see Figs. 8g-8i) and the total number C of clusters of actors sharing the same opinion (see Figs. 8j-8l).The increase of the number of clusters with increasing social temperature is also clearly visible in the Fig. 7.

IV. DISCUSSION AND CONCLUSIONS
In this paper we proposed multi-choice opinion dynamics model based on Latané theory.With computer simulation we show, that for multi-opinion version of the Nowak-Szamrej-Latané model of opinion dynamics even and probabilities of choosing these opinions given by Eqs. ( 9)- (10).Please note that these 'colours' are equally distanced to each other and none of them is better or worse than others.Thus our scale of opinions corresponds to the nominal level of measurement [56].Please note, that term responsible for actors interactions with other actors who share the same opinions [Eq.(7a)] is not dissimilar to the Potts model [57], where phase transition is also observed.As we do not assign numerical values ξ i to differentiate actors opinions we can observe the order/disorder phase transition in thermal dependence of nk , σ(n k ), Smax , C .The results of our simulations indicate that the critical temperature T C decreases with increasing the number of opinions K available in the system.We conclude, that for opinion Nowak-Szamrej-Latané modelwith multi-choice of opinions and long-rage interactions among actors-the phase transition from ordered to disordered phase is also observed.

FIG. 1 :
FIG.1:(Colour online) In the top line the snapshots from simulation of the system containing L 2 = 40 2 sites and K = 2 are presented.The subsequent columns correspond to time steps t = 0, 1, 10, respectively.In subsequent rows the probabilities of changing opinion P i associated with sites i and for social temperature T = 0, 1 and 3 are presented.∀i : p i = s i = 0.5, α = 3[48].

FIG. 2 :
FIG. 2: The time evolution of the average changing opinion probability P [%] for various values of social temperatures T [48].

FIG. 3 :
FIG. 3: (Colour online)The maps of probabilities of opinion changes P i [%] for social temperature T = 1 at the initial random distribution of opinions (t = 0) and after ten time steps of simulation (t = 10) and for various values of exponent α = 2, 3 and 6.In the first row the snapshots from simulations indicating the spinsons opinions for t = 0 (first column) and t = 10 (second column) are presented[48].

FIG. 4 :
FIG. 4: The time evolution of the average probability P [%] of opinion changes for various values of the distance function scaling exponents α [48].

6 FIG. 7 :FIG. 8 :
FIG. 7: (Colour online) The snapshots of opinions spatial distribution for social temperature T = 0 (the first column) and T = 6 > T C (the second column) for various numbers of available opinions K = 3 (the first row) and K = 6 (the second row).

TABLE I :
The values of critical social temperature T C for various number K of opinion available in the system deduced from Figs. 6 and 8.