Non-equilibrium dynamics in a three state opinion formation model with stochastic extreme switches Kathakali Biswas1 2and Parongama Sen2

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Non-equilibrium dynamics in a three state opinion formation model with stochastic

extreme switches

Kathakali Biswas1, 2 and Parongama Sen2

1Department of Physics, Victoria Institution (College), 78B Acharya Prafulla Chandra Road, Kolkata 700009, India.

2Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India.

We investigate the non-equilibrium dynamics of a three state kinetic exchange model of opinion

formation, where switches between extreme states are possible, depending on the value of a param-

eter q. The mean ﬁeld dynamical equations are derived and analysed for any q. The fate of the

system under the evolutionary rules used in S. Biswas et al, Physica A 391, 3257 (2012) shows that

it is dependent on the value of qand the initial state in general. For q= 1, which allows the extreme

switches maximally, a quasi-conservation in the dynamics is obtained which renders it equivalent to

the voter model. For general qvalues, a “frozen” disordered ﬁxed point is obtained which acts as an

attractor for all initially disordered states. For other initial states, the order parameter grows with

time tas exp[α(q)t] where α=1−q

3−qfor q6= 1 and follows a power law behaviour for q= 1. Numeri-

cal simulations using a fully connected agent based model provide additional results like the system

size dependence of the exit probability and consensus times that further accentuate the diﬀerent

behaviour of the model for q= 1 and q6= 1. The results are compared with the non-equilibrium

phenomena in other well known dynamical systems.

I. INTRODUCTION

One of the main motivations in studying non-

equilibrium phenomena is to check what kind of steady

states can be reached using diﬀerent initial conditions.

In the well known studies of Ising-Glauber model at zero

temperature, on lattices or networks, several studies have

been made to show that the steady states may not be

the equilibrium steady states [1–14]. Exit probability, a

quantity related to the type of ﬁnal state reached from an

initially biased state, has also been studied extensively in

recent times in spin and opinion formation models [15–

26]. In systems with more than two states, several other

interesting features like two stage ordering process has

been noted [26]. In addition, how a system evolves to a

stable state starting from an unstable ﬁxed point is also

a matter of interest [27].

Opinion dynamics models relevant to social phenom-

ena have received extensive attention recently [28–31].

These models typically show a rich non-equilibrium be-

haviour. Usually, the opinion of an agent is updated

following the interaction with other individuals; some-

times the inﬂuence of media is also incorporated. In the

numerous models studied so far, the interaction and the

choice of the interacting agent(s) play crucial roles. The

simplest models involve binary opinions typically rep-

resented by 0,1 or ±1. The Voter model [32, 33], in

which an agent just copies the opinion of another ran-

domly picked up agent, is one of the simplest and earliest

opinion dynamics models. Later, models involving more

complexities have been constructed [29, 30]. The binary

models obviously cannot capture all the intricacies of the

real world. Hence, models with three or more opinion

states as well as continuous values of opinions have been

considered in the recent past. The voter model can be

generalised with more number of states easily [34] while

other multistate models which involve the eﬀect of more

neighbours have also been considered [35, 36]. In compar-

ison to the simple binary state models, here the opinions

are not merely ﬂipped but can change in more than one

possible way. We focus our attention on the so called

kinetic exchange models where pairwise interactions are

considered at each step [37]. However, these models gen-

erally have some restrictions. In particular, in the ki-

netic exchange models most recently studied with three

discrete opinion states quantiﬁed by -1,0,1 (assumed to

represent e.g., left, central and right ideologies), a tran-

sition from 1 to -1 or vice versa (i.e., an extreme switch

of opinion) is not allowed to the best of our knowledge

[38–42]. Also, in many other similar three-state mod-

els such a restriction is imposed [43–49]. However, hu-

man behaviour being complex and unpredictable such

switches cannot be completely ruled out. In fact, there

are real world examples where even political cadres or

leaders shift their allegiance to parties with totally op-

posite principles [50, 51]. The reasons may be associated

with immediate gains and selﬁsh interests, lack of strong

ideological beliefs etc. We have considered a model for

opinion dynamics where extreme switches are allowed to

happen and see how the dynamics are aﬀected by this.

It may be added here that for the multistate voter model

or Potts type models, such extreme switches can take

place, however, in the relevant studies, the eﬀect of such

switches has not been the issue of interest speciﬁcally

[34–36].

In this article, we have considered a kinetic exchange

model of opinion dynamics with three states, with the

possibility of switching between extreme opinions. In the

mean ﬁeld approach, the equations for the time deriva-

tives are set up for the three population densities of diﬀer-

ent opinions and solved numerically. We have introduced

a parameter qwhich governs the probability with which

switches between extreme opinions can occur and stud-

ied its eﬀect on the time evolution. qvaries between zero

and unity, the zero case is already considered where no

arXiv:2210.02043v3 [cond-mat.stat-mech] 27 Feb 2023

such switch is allowed [38]. Parallely, numerical simula-

tions have been conducted using a fully connected agent

based model. The model and quantities of interest are

discussed in the next section followed by the results pre-

sented in section III and ﬁnally in the concluding section,

the results are discussed and compared to existing results

in similar models.

II. MEAN FIELD KINETIC EXCHANGE

MODEL

We have considered a kinetic exchange model (KEM)

for opinion formation which incorporates three opinion

values are quantiﬁed by 0,±1. The possible correspon-

dence with left, central and right ideologies has already

been mentioned. The three opinion values may even

mimic a 2-party voting system, where the the ±1 opin-

ions represent support for the two parties while people

with zero opinion (the neutral population) are those who

refrain from voting for either of them. The opinion of an

individual is updated by taking into account her present

opinion and an interaction with a randomly chosen indi-

vidual in the fully connected model. The opinion of the

ith individual is denoted by oi(t). The time evolution of

oi, after an interaction with the kth individual, chosen

randomly, is given by

oi(t+ 1) = oi(t) + µok(t),(1)

where µcan be interpreted as an interaction parameter.

The opinions are bounded in the sense |oi| ≤ 1 at all

times and therefore oiis taken as 1 (-1) if it is more

(less) then 1 (-1). There is no self-interaction so i6=k

in general. This evolutionary rule was introduced in [38].

Here time is assumed to be discrete but one can easily use

a continuous time model as will be done in this paper.

In several previous works [26, 38–42], µ, the interac-

tion parameter, has been chosen randomly, allowing also

negative values albeit being bounded; |µ| ≤ 1. Such a

bound allows a transition between opinion values with a

diﬀerence of maximum ±1 only. In the present work, the

interaction parameter µis allowed to take two discrete

values. The values are µ= 1 and µ= 2 which occur with

probabilities 1 −qand qrespectively. Hence, for exam-

ple, if an agent with opinion +1 interacts with another

with opinion −1 and µ= 2, her opinion can change to

-1, the other extreme value. The possibilities of all the

interactions and resulting opinions are shown in Fig.1 for

the extreme values q= 0 and q= 1. Note that in the

present work, only positive values of µare allowed.

The densities of the three populations with opinion

0,±1 are denoted by f0, f±1with f0+f+1 +f−1= 1.

The ensemble averaged order parameter obtained from

the time dependent equations for the densities is given

as hO(t)i=f+1 −f−1with −1≤ hO(t)i ≤ 1.

FIG. 1: The updated opinions of the ith individual following

an interaction with another individual (denoted by k) for all

possible opinion values at time tare shown for q= 0 (left

panel), which implies µ= 1 and q= 1 (right panel) for which

µ= 2.

Usually, to study the opinion dynamics models, one

starts with a random disordered conﬁguration such that

the average opinion is 0. Given that there are three

states, one can choose this state with diﬀerent combina-

tions of f0

is, keeping f+1 =f−1. A conventional choice is

f0=f±1= 1/3.

One can also study the eﬀect of an initial bias in the

distribution of opinions in the starting conﬁguration of

the system. The homogeneous conﬁguration being one

with all the densities equal to 1/3, one can consider a

deviation from this such that the net opinion is nonzero

by choosing f0= 1/3, f+1 = 1/3+∆/2 and f−1=

1/3−∆/2. Here −2/3≤∆≤2/3. Apart from this case,

one can take other initial conﬁgurations which have a net

nonzero opinion. We have discussed such cases as well to

show the initial conﬁguration dependence.

We present in this paper the rate equations derived an-

alytically using mean ﬁeld theory for the three densities,

and study their behaviour as functions of time. The ﬁxed

point analysis of the equations present some interesting

and non-intuitive results. We also obtain the exit prob-

ability. Here the exit probability Eis considered as a

function of ∆, i.e., E(∆) is the probability that the ﬁnal

conﬁguration has f+1 = 1 starting from f+1 = 1/3+∆/2

and f−1= 1/3−∆/2. The saturation value of hOiis

related to E(∆) by

hOisat = 2E(∆) −1 (2)

from which the exit probability can be estimated.

We have also conducted numerical simulations by con-

sidering an agent based model where each agent can in-

teract with any other agent. Here, the order parameter

for a given conﬁguration is deﬁned as ¯

O(t) = |Poi(t)|

where Nis the system size with h¯

Oidenoting the conﬁg-

uration average. To calculate the exit probability E(∆),

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Non-equilibriumdynamicsinathreestateopinionformationmodelwithstochasticextremeswitchesKathakaliBiswas1,2andParongamaSen21DepartmentofPhysics,VictoriaInstitution(College),78BAcharyaPrafullaChandraRoad,Kolkata700009,India.2DepartmentofPhysics,UniversityofCalcutta,92AcharyaPrafullaChandraRoad,Kolkata70...

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Non-equilibrium dynamics in a three state opinion formation model with stochastic extreme switches Kathakali Biswas1 2and Parongama Sen2

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