Equilibrium and dynamics of a three-state opinion model Irene Ferri Albert D az-Guilera and Matteo Palassini Departament de F sica de la Mat eria Condensada and Institute of Complex Systems UBICS

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Equilibrium and dynamics of a three-state opinion model
Irene Ferri, Albert D´ıaz-Guilera, and Matteo Palassini∗
Departament de F´ısica de la Mat`eria Condensada and Institute of Complex Systems (UBICS)
Universitat de Barcelona, 08028 Barcelona, Spain
(Dated: October 7, 2022)
Abstract: We introduce a three-state model to study the effects of a neutral party on opinion
spreading, in which the tendency of agents to agree with their neighbors can be tuned to favor either
the neutral party or two oppositely polarized parties, and can be disrupted by social agitation mim-
icked as temperature. We study the equilibrium phase diagram and the non-equilibrium stochastic
dynamics of the model with various analytical approaches and with Monte Carlo simulations on dif-
ferent substrates: the fully-connected (FC) graph, the one-dimensional (1D) chain, and Erd¨os-R´enyi
(ER) random graphs. We show that, in the mean-field approximation, the phase boundary between
the disordered and polarized phases is characterized by a tricritical point. On the FC graph, in the
absence of social agitation, kinetic barriers prevent the system from reaching optimal consensus. On
the 1D chain, the main result is that the dynamics is governed by the growth of opinion clusters.
Finally, for the ER ensemble a phase transition analogous to that of the FC graph takes place, but
now the system is able to reach optimal consensus at low temperatures, except when the average
connectivity is low, in which case dynamical traps arise from local frozen configurations.
I. INTRODUCTION
Within the field of complex systems, social questions
are perhaps the most elusive, as the agents involved (hu-
mans) exhibit a sophisticated individual behavior, not
easily reducible to a few analyzable parameters. Never-
theless, many models have been proposed to capture dif-
ferent aspects of societal interaction, such as bipartidism
[1–4], gerrymandering [5–7], or echo chambers formation
[8–10].
The consensus problem on a given social question, such
as which kind of energy is the most suitable for subsis-
tence or which political party should govern, has been
addressed using a variety of agent-based models, both
discrete, such as the voter [9, 11, 12], Ising [13] or Potts
models [14], and continuous, such as the Deffuant model
[15]. Continuous models often predict the formation of
opinion clusters [16, 17], thereby reinforcing a discrete
description of the opinion space.
A common goal in many of these works is to under-
stand the transition between an initial disordered state,
in which opinions are random and uncorrelated, to a state
in which agents exhibit some kind of local or global con-
sensus. The simplest case occurs when there are only
two opinions, as in polarized situations with a clear bi-
partidist scenario. In other situations, however, it is
more realistic to consider at least one intermediate or
neutral state representing, for example, centrists or un-
decided voters. Several three-state models have been
proposed, using various approaches for introducing the
neutral state. Some of these models prevent agents that
hold extreme opinions from interacting directly, forcing
them to pass through the neutral opinion. For instance,
Vazquez and Redner [18] study a stochastic kinetic model
∗palassini@ub.edu
in the mean-field approximation, and find that the final
configuration depends strongly on the initial proportion
of agents in each state. Along similar lines, the authors of
Ref.[19] incorporate temperature and find a phase tran-
sition analogous to that of the Ising model.
In this paper we propose a three-state Hamiltonian
agent-based model for opinion spreading in which agents
interact in a pairwise manner that tends to promote con-
sensus, with a tunable neutrality parameter that controls
the relative preference for the neutral state over the po-
larized states. Agents can change their opinion according
to a stochastic dynamics in which the effects of social ag-
itation are taken into account by mimicking them as a
temperature.
The model can be mapped to a special case of
the Blume-Emery-Griffiths (BEG) model [20] from con-
densed matter physics. Other variants of the BEG model
have been applied to sociophysics before [21, 22], but our
model allows to study directly the role of the neutral
state in the dynamics of the opinion formation.
The geometry of social structures is crucial in opin-
ion formation and other contemporary questions such as
pandemic spreading, economics [23–25] and smart cities
design [26, 27]. In order to understand the role of the net-
work geometry in the dynamics of opinion formation, we
embed our model on different types of networks: the fully
connected (FC) graph, the one-dimensional (1D) chain,
and Erd¨os-R´enyi (ER) random graphs. We study with
different analytical approaches the equilibrium phase di-
agram of the model, and use Monte Carlo (MC) simu-
lations at zero and non-zero temperature to investigate
the stochastic evolution of the population starting from
random configurations.
The paper is organized as follows. In Section II we
introduce the model, placing it in a social context and
discussing its general features. In Section III we deter-
mine the equilibrium phase diagram in the mean-field ap-
proximation, which is characterized by a phase boundary
arXiv:2210.03054v1 [cond-mat.stat-mech] 6 Oct 2022
2
with a tricritical point. In Section IV we analyze the zero-
temperature dynamics on the FC graph and identify the
basins of attraction of the absorbing states. We predict,
and confirm via MC simulations, that random configu-
rations always evolve towards an all-neutral state, and a
finite temperature is necessary to overcome the barriers
towards equilibrium. In Section V we derive an exact
solution for the 1D chain, and obtain the scaling of the
magnetization from a domain-growth argument, which
is validated by our MC results. In Section VI, devoted
to ER graphs, we determine the transition temperature
using the annealed mean-field approximation, and show
that for low connectivities the system gets stuck in dy-
namical traps, while for high connectivities it is always
able to reach the optimal configuration. Finally, in Sec-
tion VII we present our conclusions. The Appendices
contain details of the analytical calculations and numer-
ical methods.
II. THE MODEL
We propose an agent-based model with a discrete opin-
ion space. Agents live on the Nnodes of an undi-
rected graph, and their opinions are represented by two-
dimensional vectors Si(i= 1, . . . , N) that can take three
orientations:
•Si= (1,0); positive opinion / rightist
•Si= (0, α); neutral opinion / centrist
•Si= (−1,0); negative opinion / leftist
where αis the neutrality parameter. We assume that
the agents prefer to agree with their neighbors so as to
minimize, in the absence of social agitation, the following
cost function, or Hamiltonian:
H=−JX
hi,ji
Si·Sj,(1)
where J > 0 and the sum runs over all the undirected
edges of the graph. In the optimal configuration, i.e. the
ground state of the Hamiltonian, agents reach consen-
sus on the neutral opinion if α > 1, or on one of the
two polarized opinions if α < 1. The possible values of
the interaction energy between two agents, −JSi·Sjare
shown in Fig. 1. The quantity J(α2−1) thus measures
the reward of the neutral opinion over the polarized ones.
We assume that the system is in contact with a thermal
bath at temperature T, which should be understood as
a coarse-graining of all the sources of noise (e.g. social
agitation) that affect individual opinions.
An equivalent way to express the above Hamiltonian is
to replace the state vectors with scalar variables taking
the values σi∈ {1,0,−1}, which gives
H=−JX
hi,jiσiσj+α2(σ2
i−1)(σ2
j−1).(2)
FIG. 1. Contribution of a pair of interacting agents to the
energy of the system.
In this form, the model can be seen as a special case
of the BEG model [20], which in its general form is de-
scribed by the Hamiltonian
HBEG =−JX
hi,ji
σiσj−KX
hi,ji
σ2
iσ2
j+X
i
∆iσ2
i+C . (3)
In fact, we recover Eq.(2) by setting
K=Jα2,∆i=kiK, C =−NzK / 2,(4)
where kiis degree of node i, i.e. the number of nodes
connected to it, and z=Piki/N is the average degree
of the graph. Note that the standard BEG model has
a unique value ∆i= ∆ for all i. Therefore, for graphs
with constant degree ki=z, we can obtain the phase
diagram of our model by projecting that of the BEG
model, which is defined in the three-dimensional param-
eter space (K/J, ∆/J,T/J), onto the two-dimensional
semiplane (α, T/J) defined by Eq.(4).
We assume that the system evolves stochastically via
either one of two common discrete-time Markov pro-
cesses, the Metropolis and the Glauber dynamics, both
described in Appendix A. For T > 0, both dynamics con-
verge, given sufficient time, to a stationary state in which
the probability of a configuration σ={σi}N
i=1 is given
by the Boltzmann distribution p(σ)∝exp[−H(σ)/T ].
At T= 0, depending on the graph and initial condi-
tions, they can either converge to the ground state of
the Hamiltonian, or get trapped forever in metastable
configurations, as we will discuss later. We perform MC
simulations with both dynamics starting, unless other-
wise specified, from a random configuration in which the
agents independently take one of the three opinion states
with uniform probability (i.e. sampling from the infinite-
temperature Boltzmann distribution), and then quench-
ing the system instantaneously to the desired tempera-
ture Tand letting it evolve.
The order parameters of the model are the Ising-like
magnetization m=Pihσii/N, namely the difference be-
tween the fractions of rightists and leftists, and the frac-
tion of neutral agents, n0= 1 −Pihσ2
ii/N. Here h. . .i
denotes the expectation with respect to the Boltzmann
distribution.
In the rest of the paper we will use dimensionless units
for the energy and temperature such that J= 1. We
can anticipate some general features of the equilibrium
phase diagram in the plane (α, T ). For α < 1, the ground
state is ferromagnetic, namely the agents achieve a global
3
consensus either in the positive or in the negative state
(m=±1, n0= 0). For α > 1, all agents assume the neu-
tral opinion (m= 0, n0= 1) in the ground state. Hence,
moving along the zero-temperature axis we encounter a
discontinuous phase transition at α= 1.
For α= 0, the model can be thought of as an Ising
model with vacancies. Hence, in the thermodynamic
limit it will generically display a continuous phase tran-
sition in the Ising universality class at a critical temper-
ature T0
c>0, between a low-temperature polarized (fer-
romagnetic) phase, characterized by m6= 0, and a high-
temperature disordered (paramagnetic) phase, in which
m= 0.
We thus generally expect a phase boundary in the
plane (α, T ) connecting the two points (α= 0, T =T0
c)
and (α= 1, T = 0). Since the transition is discontinuous
at one end of the boundary and continuous at the other,
we also expect a tricritical point (αtc, Ttc) at some point
along the boundary, separating a line of continuous tran-
sitions at T=Tc(α) for 0 ≤α≤αtc from a line of discon-
tinuous transitions at T=Td(α) for αtc < α ≤1. Such a
phase boundary was indeed observed for a different pro-
jection of the BEG model, the Blume-Capel model, on
various types of random graphs, including some cases in
which the phase boundary is reentrant [28]. Exceptions
to this scenario are represented by 1D systems with short-
range interaction, in which T0
c= 0, since no long-range
order can survive at finite temperature, and by graphs
with a degree distribution falling more slowly than k−3
for large k, in which the system is known to remain fer-
romagnetic at all temperatures for α= 0 [28] (we expect
this to remain true for all α < 1).
III. MEAN-FIELD PHASE DIAGRAM
The mean-field approximation provides a useful first
understanding of the equilibrium behavior of the model.
In this approximation, the free-energy per spin is given
by the minimum of a free-energy function L(m, n, β) with
respect to mand n= 1−n0, where β= 1/T is the inverse
temperature. As shown in Appendix A, the stationarity
conditions ∂L/∂m =∂L/∂n = 0 give two coupled self-
consistent equations (SCEs),
m=2eβzα2(n−1) sinh(βzm)
1+2eβzα2(n−1) cosh(βzm),(5)
n=2eβzα2(n−1) cosh(βzm)
1+2eβzα2(n−1) cosh(βzm).(6)
By expanding them for small m, we find a line of con-
tinous transitions between the disordered phase and the
polarized phase, at an inverse critical temperature βc(α)
given by
α2=1
βc(α)z−1ln [2(βc(α)z−1)] .(7)
In particular, we have βc(0)−1= 2z/3, which is below
the critical temperature Tc=zof the Ising model in
the mean-field approximation. This is because, even if
at α= 0 the neutral state does not contribute to the
energy, it brings an additional entropy that destabilizes
the polarized phase. The line of critical points, shown in
Fig. 2, ends at a tricritical point (αtc, βtc) determined by
the condition
2 ln[2(βtcz−1)] = 3 −βtcz(8)
By solving this numerically we obtain β−1
tc = 0.532573 z,
and substituting this value into Eq.(7) gives αtc =
0.800354.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.2 0.4 0.6 0.8 1 1.2
Tricritical point
T/z
α
Continuous transition
Discontinuous transition
Metastability limit
FIG. 2. Mean-field phase diagram. The line of continuous
transitions T=βc(α)−1for α < αtc is given by Eq.(7). For
α > αtc , the points are obtained numerically as explained in
the text, the lines being only a guide to the eye.
For α > αtc, the phase diagram displays a line of
discontinuous transitions at inverse temperature βd(α),
which we locate by finding numerically the values of m,
n,βthat satisfy simultaneously Eqs.(5) and (6), together
with the condition of equality between the free energies
of the ferromagnetic and paramagnetic phases. The re-
sulting phase boundary is shown in Fig. 2. Also shown
in the figure is the limit of metastability of the ferromag-
netic phase, namely the value of Tabove which the SCEs
no longer admit a solution with m6= 0.
The equilibrium values hmiand hn0i= 1 − hni, ob-
tained by solving numerically Eqs.(5) and (6), are shown
in Fig. 3 as a function of temperature. For α < αtc,
the magnetization vanishes continuously at the critical
temperature, and is described by the usual mean-field
critical and tricritical exponents as T→Tc(α)−, namely
hmi ∼ (Tc(α)−T)1/2for α < αtc and hmi ∼ (Ttc −T)1/4
for α=αtc. The fraction of neutral agents hn0iincreases
with Tup to the critical point Tc(α), then it decreases
monotonically towards hn0i= 1/3 in the T→ ∞ limit,
in which the three states become equiprobable. We note
that at T=Tc(α) we have hn0i= 1 −Tc(α)/z.
For αtc < α < 1, hmidecreases monotonically with T
and jumps to zero at T=Td(α), while hn0ihas a strongly
non-monotonic temperature dependence: starting from
4
hn0i= 0 at T= 0, it increases slowly with T, then it
jumps to a large value at the discontinuous transition,
before decreasing monotonically towards 1/3.
Finally, for α≥1, we have hmi= 0 at all tempera-
tures, and hn0idecreases monotonically with T, starting
from hn0i= 1 at T= 0, since in the ground state all
agents are in the neutral state.
From a social point of view, we see that agents agree
on one of the polarized opinions when coupling domi-
nates over temperature, while at intermediate levels of
upheaval above the discontinuous transition they have a
neutral preference.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
a)
hmi
T/z
α= 0.10
0.50
0.75
αtc
0.85
0.90
0.95
0.99
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
b)
hn0i
T/z
α= 1.2
1−T/z
FIG. 3. Mean-field results for the equilibrium values of the
magnetization (a) and the fraction of neutral sites (b), as
a function of the temperature and for different values of α.
The thick red line corresponds to α=αtc , separating the
continuous transition from the discontinuous transition. In
b) the thick green curve corresponds to α= 1.2, and the
other curves are as in a).
The mean-field approximation is exact for the FC
graph in the N→ ∞ limit, formally setting z= 1 (see
Section IV). In addition, the qualitative features of the
phase diagram and of the behavior of hmiand hn0ide-
scribed above will hold in a large class of graphs. In par-
ticular, for d-dimensional regular lattices we expect the
same qualitative phase diagram when d≥2 (no finite-
temperature phase transition can exist for d= 1), with
mean-field critical exponents for d > 4 and non-mean-
field exponents of the Ising universality class for d= 2,3.
For random graphs, the mean-field approximation is
not exact and various other theoretical approaches have
been developed. For the Ising model, it was found, using
the replica method, that if the degree distribution falls
off as k−5for large kor faster (which includes the case
of random-regular and ER graphs), a continuous transi-
tion with mean-field exponents takes place. On the other
hand, for γ≤5 different scenarios depending on γare
observed [28]. In the Blume-Capel model, a tricritical
point was found using an annealed mean-field approxi-
mation (see Section C) [29].
IV. FULLY-CONNECTED GRAPH
Next, we discuss the stochastic dynamics of the model
on the FC graph, in which each agent interacts with all
other agents. In this case, in order for the Hamiltonian
to be extensive (i.e. proportional to N), we must replace
the coupling constant Jin Eq.(1) by J/N. Since the
degree is z=N−1, for large Nwe have zJ/N =J,
which is equivalent to setting z= 1 in the mean-field
solution. The Hamiltonian can then be written, up to
terms of order 1/N and recalling that J= 1 in our units,
as
H=−N
2(m2+α2n2
0).(9)
The dynamics can then be represented as the evolu-
tion of a point (m, n0) inside the triangle of vertices
(1,0),(−1,0),(0,1) shown in Fig. 4. We will refer to any
such point as the macrostate of the system, to distinguish
it from the microscopic configuration of the Nagents.
A. Zero-temperature dynamics
If we sample the initial configuration of the Nagents
independently and uniformly at random among the
three states, the probability distribution of the initial
macrostate is
p(m, n0) = 3−NN!
(Nn+)! (Nn−)!(Nn0)! ,(10)
where n±= (1 −n0±m)/2 is the fraction of agents in
the positive and negative state, respectively. For large
N, one has p(m, n0)∼exp[Nf(m, n0)], where f(m, n0)
is a large-deviation function that has a maximum at m=
0, n0= 1/3, which corresponds to equiprobable opinions,
and is shown by the red point in Fig. 4. The probability
of any macrostate different than the equiprobable one
decreases exponentially with N.
It is nevertheless interesting to analyze the fate of the
system prepared in an arbitrary macrostate (m, n0), even
摘要：

Equilibriumanddynamicsofathree-stateopinionmodelIreneFerri,AlbertDaz-Guilera,andMatteoPalassiniDepartamentdeFsicadelaMateriaCondensadaandInstituteofComplexSystems(UBICS)UniversitatdeBarcelona,08028Barcelona,Spain(Dated:October7,2022)Abstract:Weintroduceathree-statemodeltostudythee ectsofaneutr...

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