The impact of social noise on the majority-rule model across various network topologies
Roni Muslima,∗, Didi Ahmad Mulyaa,b, Zulkaida Akbara, Rinto Anugraha NQZc
aResearch Center for Quantum Physics, National Research and Innovation Agency (BRIN), South Tangerang, 15314, Indonesia
bDepartment of Industrial Engineering, University of Technology Yogyakarta, Yogyakarta, 55285, Indonesia
cDepartment of Physics, Universitas Gadjah Mada, Yogyakarta, 55281, Indonesia
Abstract
We explore the impact of social noise, characterized by nonconformist behavior, on the phase transition within the framework of the majority-rule model. The order-disorder transition can reflect the consensus-polarization state in a social context. This study covers various network topologies, including complete graphs, two-dimensional (2-D) square lattices, three-dimensional (3-D) square lattices, and heterogeneous or complex networks such as Watts-Strogatz (W-S), Barab´asi-Albert (B-A), and Erd˝os-R´enyi (E-R) networks, as well as their combinations. Social behavior is represented by the parameter p, which indicates the probability of agents exhibiting nonconformist behavior. Our results show that the model exhibits a continuous phase transition across all networks. Through finite-size scaling analysis and evaluation of critical exponents, our results suggest that the model falls into the same universality class as the Ising model.
Keywords: Majority-rule model, continuous phase transition, universality class, complex network.
1. Introduction
1
Scientists apply principles from physics to social science to
2
understand pervasive social phenomena in society [1, 2, 3, 4].
3
For instance, discrete opinion models, inspired by real-world
4
dynamics, capture how individuals tend to align with major-
5
ity viewpoints [5, 6, 7], how social pressures influence opinion
6
shifts [8], and how social validation drives conformity to pre-
7
vailing trends [9, 10]. Another example is continuous opinion
8
frameworks, which suggest that trust between individuals can
9
influence their opinions [11, 12]. While most opinion-dynamic
10
models eventually lead to homogeneity, with everyone adopt-
11
ing the same opinion, the coexistence of minority and majority
12
opinions can persist. To capture these complexities, scientists
13
introduce features like social noise, which challenges prevail-
14
ing opinions. The presence of such noise can lead to intriguing
15
new phenomena worth exploring through a physics lens.
16
Social noise occurs when individuals or groups diverge
17
from prevailing societal norms or expectations. This devia-
18
tion often means departing from established conventions, tra-
19
ditions, or cultural practices, a behavior commonly known as
20
nonconformity[13, 14, 15, 16, 17]. The study of nonconformity
21
behavior in opinion dynamics models that induce the emer-
22
gence of new phases has been extensively explored across var-
23
ious scenarios [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
24
30, 31]. The emergence of these novel phases sparks interest in
25
both social sciences and physics. The phase transitions induced
26
∗Corresponding author
Email addresses:[email protected](Roni Muslim), [email protected](Didi Ahmad Mulya),
[email protected](Zulkaida Akbar),[email protected] (Rinto Anugraha NQZ )
by nonconformity behavior are analogous to the ferromagnetic-
27
antiferromagnetic or ferromagnetic-paramagnetic phase transi-
28
tions observed in magnetic systems. In this context, these phase
29
transitions can represent a shift from consensus to discord in
30
socio-political discourse.
31
As the Ising model in physics describes, a phase transition
32
from an ordered to a disordered phase occurs at a critical tem-
33
perature. At low temperatures, spin orientations align, forming
34
an ordered arrangement. Conversely, spin orientations become
35
random above the critical temperature, leading to a disordered
36
state. This concept also applies to opinion-dynamic models.
37
During interactions and discussions, people typically agree and
38
converge on a common viewpoint, potentially reaching a con-
39
sensus. Conversely, nonconformity behavior could result in po-
40
larization or a stalemate situation.
41
In physics, as the Ising model exemplifies, systems with dif-
42
ferent underlying interactions or lattice structures can exhibit
43
the same critical behavior near the critical point. This implies
44
that observables such as critical exponents (e.g.,β,γ,ν) and
45
scaling functions are independent of specific details of the sys-
46
tem and instead depend only on the dimensionality of space
47
and the symmetries of the order parameter [32]. This concept is
48
called universality. This universality allows physicists to study
49
critical phenomena in simpler models or theoretical frameworks
50
and then apply their findings to understand complex systems
51
and sociophysics.
52
In the social interaction model with nonconformity behavior,
53
the network topology significantly influences the occurrence
54
and type of phase transitions. Applying the same interaction
55
model to different network structures can yield different re-
56
sults. Conversely, even with different network structures, vary-
57
ing degrees of connectivity, and distinct microscopic interac-
58
July 10, 2024
Preprint not peer reviewed
tion mechanisms—meaning that interactions between individu-
59
als and their nearest neighbors differ across networks—they can
60
exhibit the same critical phenomena, for example, by analyzing
61
the model’s universality. This phenomenon will be discussed in
62
this paper.
63
This paper investigates how social noises, like independence
64
and anticonformity, influence the transition from order to dis-
65
order in the majority-rule model. We study this model on dif-
66
ferent types of networks to better understand the transition and
67
the behavior of the models, specifically focusing on the uni-
68
versality class of the observed phase transitions. We look at
69
various networks, homogeneous networks such as the complete
70
graph, a two-dimensional (2-D) lattice, and a three-dimensional
71
(3-D) lattice, as well as heterogeneous networks such as the
72
Watts-Strogatz (W-S) network [33], Barabasi-Albert (B-A) net-
73
work [34], and Erdos-Renyi (E-R) network [35]. These het-
74
erogeneous networks refer to networks where nodes or edges
75
possess diverse properties. These networks contrast with ho-
76
mogeneous networks, where all nodes and edges are essentially
77
similar. Heterogeneous networks are prevalent in various real-
78
world systems, including social networks, biological networks,
79
transportation networks, and technological networks [36].
80
Our study demonstrates that the model defined on all net-
81
works undergoes a continuous phase transition. We also ana-
82
lyzed the universality class of the model by estimating the crit-
83
ical exponents associated with finite-size scaling analysis. Our
84
results indicate that the model defined on the complete graph
85
and heterogeneous networks falls into the same universality
86
class as the mean-field Ising model. Additionally, the model
87
on the 2-D lattice falls into the same universality class as the 2-
88
D Ising model, and the model on the 3-D lattice also falls into
89
the same universality class as the 3-D Ising model. Detailed
90
results are discussed in Section 3.
91
2. Model and methods
92
The majority-rule model, also called the Galam majority-rule
93
model, is an opinion dynamics model that illustrates how the
94
majority opinion in a small group (such as in a discussion)
95
within a population always dominates or wins [5, 37]. The
96
majority-rule operates as follows. A small group of agents or
97
individuals is randomly selected. Within this small group, all
98
members interact with each other and adopt the majority opin-
99
ion. This dynamic makes sense when considering groups with
100
an odd number of members. In social psychology, when agents
101
follow the majority opinion, they exhibit conformity behavior,
102
often called “conformist agents.” Conformity means adjusting
103
one’s attitudes, beliefs, and behavior to fit the established norms
104
of a group [38]. In contrast to conformity, another important so-
105
cial behavior is nonconformity, which can be divided into two
106
categories: anticonformity and independence. Independent be-
107
havior acts according to its own will without the influence of
108
others. In contrast, anticonformity behavior evaluates the will
109
of others and adopts the opposite stance[14, 15, 16, 17].
110
In this paper, we explore these types of social behaviors and
111
introduce a probability parameter denoted as p, which rep-
112
resents the likelihood of agents adopting either independence
113
(model with independence) or anticonformity (model with an-
114
ticonformity). Simply put, with a probability ofp, agents opt to
115
act independently or display anticonformity. Conversely, with a
116
probability of 1−p, agents conform by aligning with the major-
117
ity opinion. Each agent has two possible opinions, for instance,
118
an “up” opinion or+1, and a “down” opinion or−1. To elabo-
119
rate further, we outline the model algorithm as follows:
120
1. The initial state of the system is prepared in a disordered
121
state, where the number of agents with positive and nega-
122
tive opinions is equal.
123
2. Microscopic interaction of agents within the model:
124
(a) Model with independence: A group of agents is ran-
125
domly selected from the population, and with a prob-
126
ability ofp, each group member acts independently.
127
Then, with the same probability of 1/2, each mem-
128
ber of the group changes its opinion to the opposite
129
one.
130
(b) Model with anticonformity: A group of agents is ran-
131
domly selected from the population, and with a prob-
132
ability ofp, all agents act in an anticonformist man-
133
ner. If all agents within the group share the same
134
opinion, they all change their opinions to the oppo-
135
site one.
136
3. Alternatively, with a probability of 1−p, all agents choose
137
to conform by following the majority opinion.
138
We examine the model on several homogeneous networks,
139
such as the complete graph, 2-D lattice, and 3-D lattice, and on
140
several heterogeneous networks, such as the B-A, W-S, and E-
141
R networks. All agent opinions are embedded in the network
142
nodes, while the links or edges between nodes signify social
143
connections. The complete graph depicts a network structure
144
where every node is linked to every other node. In the com-
145
plete graph, all agents are neighbors and can interact with each
146
other with equal probability. Each agent has four nearest neigh-
147
bors in the two-dimensional square lattice, while in the three-
148
dimensional square lattice, each agent has six nearest neigh-
149
bors. In the heterogeneous networks, we examined networks
150
where the minimum degree of connectivity for each node is
151
two, and we selected three agents to adhere to the majority-rule
152
model algorithm, as mentioned earlier.
153
For the model on the complete graph, we can conveniently
154
perform analytical treatment to compute the order parameter
155
(magnetization) of the model using the following formula:
156
m=N+−N−
N++N−=2r−1, (1) whereN+ andN− represent the total number of agents with
157
opinion+1 and−1, respectively, andr=N+/N++N−denotes
158
the fraction of opinion+1, or the probability of finding an agent
159
with+1 within the population. In the numerical simulation,
160
we also compute the susceptibilityχ and Binder cumulantU,
161
which are defined as [39]:
162
χ=N
⟨m2⟩ − ⟨m⟩2
, (2)
U=1−1 3
⟨m4⟩
⟨m2⟩2. (3)
Preprint not peer reviewed
2The parametersm,χ, andU are calculated once the simulation
163
reaches an equilibrium state.
164
We use finite-size scaling analysis to calculate the critical ex-
165
ponents corresponding to the order parameterm, susceptibility
166
χ, and Binder cumulantU. The finite-size scaling relations can
167
be written as [40]:
168
p−pc=c N−1/ν, (4) m=φm(x)N−β/ν, (5)
U=φU(x), (6)
χ=φχ(x)Nγ/ν, (7) whereφ represents the dimensionless scaling function that fits
169
the data near the critical point pc. The critical exponents β,
170
γ, andν are important in the vicinity of the critical point pc.
171
We can identify the critical point pc, where the system shifts
172
between ordered and disordered phases, by pinpointing the in-
173
tersection of the Binder cumulant curveUand the probability
174
curve p. This scaling analysis is important for understanding
175
the universality class of the systems.
176
3. Result and Discussion
177
3.1. Time evolution of the complete graph
178
The evolution of the fractionrover time in the model can be
179
obtained from the discrete-time master equation [7]. For any
180
givenN, the time evolutiontis expressed as:
181
r=r0+1
N[ρ+(r)−ρ−(r)], (8) whereρ+(r)andρ−(r)represent the probabilities of the frac-
182
tionrincreasing and decreasing, respectively, at each time step
183
∆t=1/N. For comparing the analytical and numerical simula-
184
tions of the model on a complete graph, it is more appropriate
185
to write the evolution ofrfor the limitN≫1, so that Eq. (8)
186
can be written as:
187
dr
dt =ρ+(r)−ρ−(r), (9) which serves as the rate equation governing the fraction of
188
opinion-upr.
189
The form ofρ+(r)andρ−(r)in Eq. (9) varies depending on
190
the model. This study considers a scenario where three agents
191
are randomly selected from the population to interact based on
192
the outlined algorithm. We use the mean-field approach, as-
193
suming that the concentration of the global state is equal to the
194
concentration of the local state, so the system’s state can be
195
represented by a single parameter—for example, the fraction of
196
opinionr.
197
For the model with independence, three selected agents be-
198
have independently with a probability of p. All three agents
199
change their opinion oppositely with a probability of 1/2, tran-
200
sitioning from−1 to+1 if initially−1, and from+1 to−1 if
201
initially+1. Hence, the probability density of the three agents
202
changing their opinions from +1 to−1 is 3p/2(N+/N), and
203
from−1 to+1 is 3p/2(N−/N). When the three agents do not
204
exhibit independent behavior, with a probability of 1−p, they
205
adopt the majority opinion. The opinion+1 increases when the
206
three agents have a configuration of two opinions+1 and one
207
opinion−1, such as+ +−,+−+, and−+ +. Similarly, the
208
opinion+1 decreases when the three agents have a configura-
209
tion of two opinions−1 and one opinion+1, such as− −+,
210
−+−, and+− −. Based on these configurations, the proba-
211
bility density of the three agents adopting the majority opinion
212
+1 is:
213
3(1−p)N+
N
N+−1 N−1
N− N−2,
and the probability density of adopting the majority opinion−1
214
is:
215
3(1−p)N− N
N−−1 N−1
N+ N−2.
Therefore, for anyN, the total probabilities of the opinion+1
216
is increasing or decreasing, denoted byρ+orρ−respectively,
217
can be written as follows:
218
ρ+=3 p
2 N−
N + (1−p)N+
N N+−1
N−1 N− N−2
, (10)
ρ−=3 p
2 N+
N + (1−p)N− N
N−−1 N−1
N+
N−2
. (11)
And forN≫1, the equations (10) and (11) can be written as
219
follows (see Appendix A.1 for the general formulation):
220
ρ+(r) =3hp
2(1−r) + (1−p) (1−r)r2i
, (12)
ρ−(r) =3hp
2r+ (1−p)r(1−r)2i
. (13)
For the model with anticonformity, three agents randomly se-
221
lected will adopt anticonformist behavior with a probability of
222
p. These agents will change their opinions from−1 to+1 or
223
vice versa when they share the same opinion. The opinion+1
224
will increase or decrease when all three agents have the same
225
opinion−1 (i.e.,− − −) or+1 (i.e.,+ + +), with total proba-
226
bilities of
227
3p·N−
N ·N−−1
N−1 ·N−−2 N−2 , and
228
3p·N+
N ·N+−1
N−1 ·N+−2 N−2 , respectively.
229
When the three agents do not adopt anticonformist behav-
230
ior, they adopt the majority opinion with a probability of 1−p,
231
following the same probability configuration as in the indepen-
232
dence model. Thus, the probability density of the opinion+1
233
increasing or decreasing for anyNcan be written as follows:
234
ρ+(r) =3
pN− N
N−−1 N−1
N−−2
N−2 + (1−p)N+ N
N+−1 N−1
N− N−2
, (14) ρ−(r) =3
pN+
N N+−1
N−1 N+−2
N−2 + (1−p)N− N
N−−1 N−1
N+ N−2
. (15)
Preprint not peer reviewed
3ForN≫1, the equations (14) and (15) can be written as follows
235
(see Appendix A.2 for the general formulation):
236
ρ+(r) =3h
p(1−r)3+ (1−p) (1−r)r2i
, (16)
ρ−(r) =3h
pr3+ (1−p)r(1−r)2i
. (17)
Eqs. (12)-(13) and (16)-(17) are essential for analyzing the sys-
237
tem’s state on the complete graph, especially in identifying
238
order-disorder phase transitions. These equations allow us to
239
understand the dynamics of opinion changes and the conditions
240
under which the system transitions from an ordered to a disor-
241
dered state at a certain critical point.
242
We can solve Eq. (9) to find an explicit expression for the
243
fraction opinion-up r at time t for the models with indepen-
244
dence and anticonformity. By substituting Eqs. (12) and (13),
245
and integrating it, the fraction opinionrfor the model with in-
246
dependence can be written as:
247
r(t,p,r0) =1 2
"
1±
1−3p 1−p+2e−3(1−3p)(t+A)
1/2#
, (18)
whereA=ln[(1−2r0)2/(2(1−p)(r02+r0) +p]/[3(1−3p)]is
248
a parameter that satisfies the initial condition ofr(t)att=0.
249
Similarly, for the model with anticonformity, we obtain:
250
r(t,p,r0) =1 2
"
1±
1−4p 1−4e−3(1−4p)(t+A)
1/2#
, (19)
whereA=ln
(1−2r0)2/(r02−r0+p)
/(1−4p). These equa-
251
tions provide analytic expressions forrat timetfor both mod-
252
els, where p represents the probability of agents adopting in-
253
dependence or anticonformity, andr0denotes the initial opin-
254
ion fraction. For instance, when p=0, both Eqs. (18) and
255
(19) converge to the same form, indicating the evolution of
256
rtowards complete consensus states or completely disordered
257
states, depending on the initial fractionr0. Critical points oc-
258
cur at p=1/3 for the model with independence and p=1/4
259
for the model with anticonformity, where r→1/2, signifying
260
complete disorder.
261
Figure 1 compares Eq. (18) (red) and Eq. (19) (blue) with
262
numerical simulations for a large population (N=104) and dif-
263
ferent values ofp, demonstrating close alignment between an-
264
alytical and numerical results. Atp=0 and|r0|>0.5, all ini-
265
tial fractions evolve towards complete consensus withr=1 (all
266
agents have the same opinion +1) forr0>0.5 andr=0 (all
267
agents have the same opinion−1) forr0<0.5. Additionally,
268
at 0<p<pc,r evolves towards two stable values, while at
269
p=pc,rconverges to 1/2, representing complete disorder.
270
3.2. Phase diagram and critical exponents of the complete
271
graph
272
To investigate the order-disorder phase transition of the
273
model, we consider the stationary condition of Eq. (9), where
274
dr/dt =0 or ρ+=ρ−. For the model with independence,
275
0 2 4 6 8 10 0.0
0.2 0.4 0.6 0.8 1.0
Frac.opinionhri p= 0.00
0 5 10 15 20 0.0
0.2 0.4 0.6 0.8
1.0 p= 0.25
0 5 10 15 20 0.0
0.2 0.4 0.6 0.8
1.0 p=pc= 1/3
0 2 4 6 8 10 Time sweept 0.0
0.2 0.4 0.6 0.8 1.0
Frac.opinionhri p= 0.00
0 2 4 6 8 10 Time sweept 0.0
0.2 0.4 0.6 0.8
1.0 p= 0.10
0 5 10 15 20 Time sweept 0.0
0.2 0.4 0.6 0.8
1.0 p=pc= 1/4
Figure 1: The comparison between analytical calculation (dashed lines) and numerical simulation (points) for both models with independence (red) and an- ticonformity (blue) across various probability valuesp, based on Eqs. (18) and (19), respectively, is shown. At p=0, all data points forrconverge either to complete consensus withr=1 (forr0>1/2) orr=0 (forr0<1/2). For 0<p<pc, all data points evolve towards two stable valuesrst, while atp=pc, all data converge tor→1/2 (representing a disordered state). The population size isN=104, and each data point averages over 300 independent realizations.
this yields three stationary solutions: r1 =1/2 and r2,3 =
276
1 2
1±
r1−3p 1−p
. Consequently, the order parametermis:
277
m2,3=± s
1−3p
1−p . (20)
The critical point occurs at pc=1/3, wherem2,3=0. Sim-
278
ilarly, for the anticonformity model, the stationary condition
279
for the opinion fraction dr/dt=0 yields three stationary states:
280
r1=1/2 andr2,3=1 2
1±√ 1−4p
. Consequently, the order
281
parametermis:
282
m2,3=±p
1−4p, (21)
and hence, the critical point for the anticonformity model oc-
283
curs atpc=1/4. Both Eqs. (20) and (21) can be expressed as
284
power law in terms ofp, wherem∼(p−pc)β, withβ =1/2,
285
typical of the critical exponent of the mean-field Ising model
286
[41].
287
As previously discussed, the topology of the complete graph
288
can be approximated analytically using a mean-field approach.
289
To validate these analytical results, Monte Carlo simulations
290
were performed with a large population size ofN=106. The re-
291
sults, illustrated in Fig. 2 [panel (a)], demonstrate a close match
292
between the analytical calculations and the Monte Carlo simu-
293
lations. These data indicate that the model undergoes a contin-
294
uous phase transition with a critical pointpc=1/3 for the in-
295
dependence model andpc=1/4 for the anticonformity model.
296
Another method to analyze the order-disorder phase tran-
297
sition is the effective potential, obtained through integration
298
of the effective force. Traditionally, the effective potential is
299
derived from the effective force using the formulaV(r)eff=
300
Preprint not peer reviewed
40.0 0.1 0.2 0.3 0.4 0.5 Probabilityp 0.0
0.2 0.4 0.6 0.8 1.0
Orderparam.m (a)
−1.0−0.5 0.0 0.5 1.0 Order param. m 0
2 4 6 8 10
PotentialV(10−2) (b)
p= 0.1
p= 0.2 pc= 1/3
p= 0.4
Indep. model
−1.0−0.5 0.0 0.5 1.0 Order param. m 0
1 2 3 4 5 6
PotentialV(10−2) (c)
p= 0.1 p= 0.2 pc= 1/4
p= 0.3
Antic. model
DataIndep.
DataAntic.
Eq.(20) Eq.(21)
Figure 2: The phase diagram [panel (a)] of the model on the complete graph for models with independence and anticonformity is shown. Analytical results from Eqs. (20) and (21) are compared with the Monte Carlo simulation data and agree very well. Both models undergo continuous phase transitions, revealing critical points atpc=1/3 for the model with independence andpc=1/4 for the model with anticonformity. Panels (b) and (c) illustrate the effective po- tential described by Eqs. (22) and (23), demonstrating bistable forp<pcand monostable behaviors forp>pc, indicating the continuous phase transition is occurred.
−R f(r)effdr. Here, f(r)eff=ρ+(r)−ρ−(r) represents the
301
force that drives opinion change during the dynamics process.
302
For the independence model on the complete graph, the effec-
303
tive potential (in terms ofm) can be written as follows:
304
Vindep.= 3
32(1−p)−1 1−3p−(1−p)m22
. (22)
And for the model with anticonformity, the effective potential
305
can be written as follows:
306
Vantic.= 3
32 1−4p−m22
. (23)
Plots of Eqs. (22) and (23) are shown in panels (b) and (c) of
307
Fig. 2. For both potentials, there are bistable states forp<pc, a
308
bistable-monostable transition at p=pc, revealing the model’s
309
critical point, and the system is monostable forp>pc, indicat-
310
ing a continuous phase transition at pc.
311
The critical point of the model can also be analyzed using
312
Landau’s theory. According to this theory, the potential can
313
be expanded in terms of the magnetization masV =∑iVimi,
314
whereVigenerally depends on thermodynamic parameters [42].
315
In this model,Vi can be influenced by noise parameters, such
316
as the probabilities of independence and anticonformity. The
317
Landau potentialV exhibits symmetry under the inversion of
318
the order parameter, m→ −m. Consequently, only the even
319
terms of the potential are considered. Therefore, the Landau
320
potential takes the form:
321
V=V2m2+V4m4+··· (24) Understanding the termsV2andV4is sufficient for analyzing
322
the model’s phase transition using the potentialV. The critical
323
point can be determined by settingV2=0, and the nature of the
324
phase transition is characterized byV4(pc), whereV4(pc)≥0
325
denotes a continuous phase transition, whileV4(pc)<0 indi-
326
cates a discontinuous phase transition. By comparing Eq. (24)
327
with Eqs. (22) and (23), we can determineV2andV4for both
328
the independence and anticonformity models. For the inde-
329
pendence model, we obtainV2(p) =3(1−3p)/8 andV4(p) =
330
9(1−p)/4. Meanwhile, for the anticonformity model, we ob-
331
tainV2(p) =−3(1−4p)/8 andV4(p) =9/4. Hence, the criti-
332
cal pointspcalign with those obtained from the previous anal-
333
ysis: pc=1/3 for the independence model and pc=1/4 for
334
the anticonformity model. Furthermore,V4(pc)≥0 for both
335
models confirms their continuous phase transition.
336
The model’s critical points and critical exponents can be esti-
337
mated numerically using finite-size scaling relations [Eqs. (4)-
338
(7)]. By varying the population size N from 2000 to 10000,
339
we compute the magnetizationm, susceptibilityχ, and Binder
340
cumulantU as shown in Fig. 3. Each data point is averaged
341
over 105independent realizations to ensure accurate results. In
342
Fig. 3, the inset graphs display standard plots, while the main
343
graphs present the scaling plots of the model. The critical point
344
is determined using the Binder method by observing the cross-
345
ing of lines between the Binder cumulantUand the probability
346
of anticonformity p. In this instance, the critical point is es-
347
timated to bepc≈0.251 [inset graph of panel (a)], consistent
348
with the analytical result in Eq. (21), namelypc=1/4.
349
The plots in Fig. 3 show the dynamics of the scaled param-
350
eters for the model with anticonformity. The best critical ex-
351
ponents obtained from fitting the data for various values ofN
352
areβ ≈0.5, ν≈2, andγ ≈1.0. It is important to note that
353
althoughβ=1/2 andγ=1 are the same with the usual critical
354
exponents for the mean-field Ising model,ν=2.0 does not fit
355
in this pattern. However, a direct connection exists betweenν
356
and the critical dimensiondc=4 of the mean-field Ising model,
357
expressed asν=dcν′=2, whereν′=1/2 is an effective ex-
358
ponent. This result is also observed in several discrete dynamic
359
models [21, 24, 28, 43, 44, 45]. These critical exponents sug-
360
gest that the model belongs to the mean-field Ising universality
361
class. Notably, identical critical exponents are obtained for the
362
independence model, indicating a similarity between the inde-
363
pendence and anticonformity models. Furthermore, these mod-
364
els resemble well-known models like the Sznajd [28] and kinet-
365
ics exchange models [46, 47]. This finite-size scaling analysis
366
provides robust numerical evidence supporting the model’s crit-
367
ical point and exponents, validating the analytical findings and
368
classifying the model within the mean-field Ising universality
369
class.
370
3.3. Critical exponents of the model on the 2-D lattice
371
We explored various population sizes denoted as N =L2,
372
whereLtakes values of 32,45,64,100,150, and 200, to inves-
373
tigate the model’s critical point and critical exponents on the
374
2-D lattice. The numerical results concerning the order param-
375
eter m, susceptibility χ, and Binder cumulantU are depicted
376
in Fig. 4. The critical point, marking the instance of a contin-
377
uous phase transition in the model, is identified at pc≈0.106
378
[as observed in the inset panel (a) of Fig. 4]. By employing
379
finite-size scaling relations described in Eqs. (4)-(7), we deter-
380
mined the critical exponents that give the best description of
381
the data. These critical exponents areβ ≈0.125,γ≈1.75,and
382
ν≈1.00. These values suggest similarities with the 2-D Sz-
383
najd model [30, 48] and align with the universality class of the
384
two-dimensional Ising model [41].
385
Preprint not peer reviewed
5−4 −2 0 2 (p−pc)N1/ν 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
U
(a)
N= 2000 N= 4000 N= 6000 N= 8000 N= 10000
−4 −2 0 2 4 (p−pc)N1/ν 0
1 2 3 4 5
mNβ/ν
(b)
−4 −2 0 2 4 (p−pc)N1/ν 0.0
0.1 0.2 0.3 0.4
χN−γ/ν(10−1)
(c)
0.2 0.3
p 0.0 0.2 0.4 0.6
U
0.0 0.2 0.4 p 0.0 0.2 0.4 0.6 0.8 1.0
m
0.0 0.2 0.4 p 0 10 20 30
χ
Figure 3: The M-C simulation results of the model on the complete graph for the order parameterm, susceptibilityχ, and Binder cumulantUacross various population sizesN. The critical point is identified from the crossing lines of the Binder cumulantUversus probabilitypwhich is found atpc≈0.250 [inset graph of the panel (a)], validating the analytical result in Eq. (21). The best critical exponents facilitating the collapse of all data near the critical pointpc areβ≈0.5,ν≈2.0, andγ≈1.0. These values suggest that the model belongs to the mean-field Ising model class.
We extended our analysis to incorporate the model with anti-
386
conformity and found that it also undergoes a continuous phase
387
transition. The critical point for the model with anticonformity
388
ispc≈0.062, as illustrated in the inset panel (a) of Fig. 5. No-
389
tably, our investigations yielded similar critical exponents for
390
this model: β≈0.125,γ≈1.75,andν≈1.00. These shared
391
critical exponents suggest that both the model with indepen-
392
dence and the model with anticonformity exhibit analogous be-
393
havior, indicating their belonging to the same universality class.
394
Our results align these models with the two-dimensional Ising
395
universality class. The critical exponents satisfy the hyperscal-
396
ing relationνd=2β+γ, whered=2, the spatial dimension of
397
the model.
398
3.4. Critical exponents of the model on the 3-D lattice
399
We investigated the model using different population sizes
400
N=L3, where the linear dimensionsLvaried from 15 to 35.
401
Similar to the 2-D lattice model, we used periodic boundary
402
conditions. In this network, each agent has six nearest neigh-
403
bors and interacts based on the aforementioned algorithm. The
404
numerical results for the Binder cumulantU, order parameter
405
m, and the susceptibility χ of the model with independence
406
are shown in Fig. 6. Each data point averages over 106inde-
407
pendent realizations. Our findings indicate that the model un-
408
dergoes a continuous phase transition, with the critical point
409
−4 −2 0 2 4 (p−pc)N1/ν 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
U
(a)
−4 −2 0 2 4 (p−pc)N1/ν 0.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
mNβ/ν
(b)
−4 −2 0 2 4 (p−pc)N1/ν 0.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4
χN−γ/ν(10−1)
(c)
L= 32 L= 45 L= 64 L= 100 L= 150 L= 200 0.0 0.1 0.2
p 0.0 0.2 0.4 0.6 0.8
U
0.0 0.1 0.2 p 0.0 0.2 0.4 0.6 0.8 1.0
m
0.1 0.2 p 0 2 4 6 8 10 12 14
χ(102)
Figure 4: The M-C simulation results of the model on the 2-D lattice for the model with independence for the order parameterm, susceptibilityχ, and Binder cumulantUacross various population sizesN=L2. The model under- goes a continuous phase transition with a critical point atpc≈0.106. The best critical exponents of the model areβ≈0.125,ν≈1.0, andγ≈1.75. These results suggest that the model falls into the same universality class as the 2-D Ising model.
occurring atpc≈0.311. Using finite-size scaling analysis near
410
the critical pointpc, we determined the critical exponents of the
411
model to beν≈0.630,β ≈0.326, andγ≈1.237. These crit-
412
ical exponent values are consistent with those of the 3-D Ising
413
model, suggesting that this model belongs to the same univer-
414
sality class as the 3-D Ising model [41].
415
Our analysis of the model with anticonformity reveals a
416
second-order phase transition with a critical point atpc≈0.268,
417
as shown in Fig. 7. In this model, we obtained similar critical
418
exponents to those of the model with independence, yielding
419
values ofν≈0.630,β ≈0.326, andγ≈1.237. These results
420
suggest that both the models with independence and anticonfor-
421
mity are identical and belong to the same universality class. No-
422
tably, these critical exponents remain consistent across various
423
datasets for different system sizesN, indicating their universal-
424
ity. It is important to note that the critical exponents of both
425
models satisfy the hyperscaling relationνd =2β+γ, where
426
d=3, the spatial dimension of the model.
427
3.5. Critical exponents of the model on the heterogeneous net-
428
works
429
Compared to the homogeneous networks mentioned earlier,
430
heterogeneous networks such as Watts-Strogatz (W-S), Albert-
431
Barab´asi (A-B), and Erd˝os-R´enyi (E-R) networks better reflect
432
real social networks [34, 36]. These three types of networks
433
Preprint not peer reviewed
6−4 −2 0 2 4 (p−pc)N1/ν 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
U
(a)
−4 −2 0 2 4 (p−pc)N1/ν 0.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
mNβ/ν
(b)
−4 −2 0 2 4 (p−pc)N1/ν 0.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4
χN−γ/ν(10−1)
(c)
L= 32 L= 45 L= 64 L= 100 L= 150 L= 200 0.0 0.1 0.2
p 0.0 0.2 0.4 0.6 0.8
U
0.0 0.1 0.2 p 0.0 0.2 0.4 0.6 0.8 1.0
m
0.1 0.2 p 0 1 2 3
χ(102)
Figure 5: The M-C simulation results of the model on the 2-D lattice for the model with anticonformity for the order parameterm, susceptibilityχ, and Binder cumulantUacross various population sizesN=L2. The model un- dergoes a continuous phase transition with a critical point atpc≈0.062. The best critical exponents of the model areβ≈0.125,ν≈1.0, andγ≈1.75.
These results suggest that the model falls into the same universality class as the 2-D Ising model.
have been extensively studied across various research areas and
434
applied to understand diverse social phenomena, such as epi-
435
demic processes [49], the analysis of the structure and charac-
436
teristics of scientific collaboration networks [50], including in
437
fields like medicine [51].
438
In the heterogeneous networks, the three chosen agents at
439
random interacted according to the model’s algorithm. We as-
440
signed varying node degrees in all networks, ensuring that each
441
agent has at least two nearest neighbors. The population size
442
wasN=104, with each data point representing the average of
443
105independent realizations. Our numerical results for the or-
444
der parametermare shown in Fig. 8. One can see that the model
445
undergoes a continuous phase transition in all three networks,
446
each with different critical points. Atp=0, the system exhibits
447
complete order with|m|=1 (complete consensus with all mem-
448
bers having the same opinion). The value of |m|decreases as
449
pincreases and approaches zero near the critical point pc. For
450
p<pc, the system is in a state of consensus with a majority-
451
minority opinion existing, and for p>pc, the system is in a
452
state of polarization. Interestingly, for all three networks, the
453
critical point for the model with anticonformity is smaller com-
454
pared to the model with independence on the same network,
455
consistent with the results on homogeneous networks.
456
We analyzed the critical exponents of the model using finite-
457
size scaling in Eqs. (4)-(7). The numerical results for the model
458
−8−6−4−2 0 2 4 6 (p−pc)N1/ν 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
U
(a)
−8−6−4−2 0 2 4 6 8 (p−pc)N1/ν 0.0
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
mNβ/ν
(b)
L= 15 L= 20 L= 25 L= 30 L= 35
−8−6−4−2 0 2 4 6 (p−pc)N1/ν 0
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
χN−γ/ν(10−1)
(c)
0.2 0.3 0.4 p 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
U 0.00.10.20.30.4
p 0.0 0.2 0.4 0.6 0.8 1.0
m
0.2 0.4 p 0 1 2 3
χ(102)
Figure 6: The M-C simulation results of the model on the 3-D lattice for the model with independence for the order parameterm, susceptibilityχ, and Binder cumulantUacross various population sizesN=L3. The model un- dergoes a continuous phase transition with a critical point atpc≈0.311. The best critical exponents of the model areβ≈0.326,ν≈0.630, andγ≈1.237 . These results suggest that the model falls into the same universality class as the 3-D Ising model.
with independence on the B-A network are shown in Fig. 9.
459
The model exhibits a continuous phase transition with a criti-
460
cal point at pc≈0.245 [see the inset graph in panel (a)]. The
461
best fit for the critical exponents across differentN values are
462
ν≈2.0,γ≈1.0, andβ ≈0.5. These results suggest that the
463
model belongs to the same universality class as the model on
464
a complete graph and aligns with the universality class of the
465
mean-field Ising model.
466
We also analyzed the model on a combination of the B-A
467
and W-S networks, where several nodes between the two net-
468
works are interconnected, as shown in Fig. 10. This combi-
469
nation of networks can represent two communities or groups
470
where each member of the community can communicate with
471
others. Within these networks, we also examined the interac-
472
tions among three agents, who interact with each other and fol-
473
low the algorithm above. We ensured that each node in the
474
combined network has at least two directly connected neigh-
475
bors.
476
The numerical results form,χ, andUfor the model with in-
477
dependence on this combined network are shown in Fig. 11.
478
It can be seen that the model undergoes a continuous phase
479
transition with a critical point atpc≈0.280. Interestingly, the
480
finite-size scaling analysis results show that the model has crit-
481
ical exponents ofν≈2.0,γ≈1.0, andβ≈0.5 across different
482
N. This indicates that although the topology of the combined
483
