# Phase diagram in a one-dimensional civil disorder model

Ignacio Ormazábal<EMAIL_ADDRESS>Departamento de Física, Universidad de
Concepción, Concepción, Chile  Felipe Urbina  Centro de Investigación DAiTA
Lab, Facultad de Estudios Interdisciplinarios, Universidad Mayor, Santiago,
Chile  Félix A. Borotto  Hernán F. Astudillo  Departamento de Física,
Universidad de Concepción, Concepción, Chile

###### Abstract

Epstein’s model for a civil disorder is an agent-based model that simulates a
social protest process where the central authority uses the police force to
dissuade it. The interactions of police officers and citizens produce dynamics
that do not yet have any analysis from the sociophysics approach. We present
numerical simulations to characterize the properties of the one-dimensional
civil disorder model on stationary-state. To do this, we consider interactions
on a Moore neighborhood and a random neighborhood with two different visions.
We introduce a Potts-like energy function and construct the phase diagram
using the agent state concentration. We find order-disorder phases and reveal
the principle of minimum grievance as the underlying principle of the model’s
dynamics. Besides, we identify when the system can reach stable or an
instability conditions based on the agents’ interactions. Finally, we
identified the most relevant role of the police based on their capacity to
dissuade a protest and their effect on facilitating a stable scenario.

##  I Introduction

In recent years, the sociophysics or statistical physics of social dynamics
has described different social phenomena as collective effects of the
interaction between individuals [1, 2, 3]. In particular, the study of opinion
dynamics has generated various models describing consensus, agreement, or
uniformity using tools from statistical physics [4, 5, 6, 7]. Recent efforts
aim at describing these models considering a diversity of individual traits of
a population and size of group discussion [8], or their multi-state
variations, such as, for example, the majority voting model [9], the multi-
state voter model [10], multichoice opinion dynamics models [11], or the
multi-state noisy q-voter model [12].

In parallel, social scientists have used agent-based models to reproduce
emerging social phenomena [13, 14], such as the Schelling model of urban
segregation [15] and the Axelrod model of cultural dissemination [16]. These
models have attracted the attention of physicists, who have described the
Schelling model as interacting physical particles [17] and as an Ising-like
model [18]. Furthermore, they have characterized the static and dynamic
properties in one and two dimensions [19] and their different behaviors using
a phase diagram [20]. They have recently used a similar energy function to
characterize the Schelling and Sakoda models [21, 22, 23]. On the other hand,
physicists described the Axelrod model in two dimensions showing order-
disorder phase transitions [24]. Then, they described the one-dimensional
Axelrod model as a starting point for its description in more complex
topologies [25]. In addition, they described the role of dimensionality on the
order-disorder phase transitions [26] and the stability model using Lyapunov
functions [27, 28]. In this context, Epstein presented an agent-based model to
describe the social dynamics of protests and rebellions through recognizable
macroscopic phenomena [29]. This model simulates a social protest process
where the central authority uses police force to dissuade a protest.

For a generalized rebellion, Epstein presents five study cases and reports
statistical regularities observed in the punctuated equilibrium dynamics,
opening new questions about how the civil disorder dynamics work. Different
scientists have modified this model to describe other social conflicts such as
workers’ protests [30], the spread of criminal activity [31], or civil war
cases between ethnic groups [32]. In addition, some variations include
legitimacy with endogenous feedback [33] or the influence of the distribution
of money on the dynamics [34]. Despite these modifications, nobody
characterized this agent-based model using concepts and tools of statistical
physics in its original form.

This paper aims to characterize the one-dimensional civil disorder model on
stationary-state as a first approach to studying this model in other
dimensions or topologies. To do this, we perform numerical simulations of the
Epstein model with and without police officers and use two visions to define
interactions in a Moore neighborhood and a random neighborhood. We introduced
two macroscopic quantities and built the phase diagram to identify different
behaviors. On the one side, we define a Potts-like energy function to deduce a
guiding principle to understand civil disorder dynamics. For the Schelling and
Sakoda model [22, 23], this energy-like function allows identifying a
minimization principle to understand the spatial segregation patterns as
efficient or inefficient. In the case of the dynamics of a factory workers’
protest [35], a function similar to the Ising model’s free energy allows
identifying the steady-state of the system and describing two phases based on
the Principle of Minimum Dissatisfaction. On the other side, we used the
concentration of agents to identify and characterize phase changes. The
concentration allows describing phase transitions in the q-voter model with
two types of stochasticity [36] and the multi-state noisy q-voter model [12].
Besides, this macroscopic quantity is helpful to building phase diagrams. In
the Schelling and Axelrod model [20, 24], this strategy allows identifying
domain boundaries of the different qualitative behaviors. Hence, in this work,
the agents’ concentration and the phase diagram are crucial elements to
characterize the original Epstein model and allow us to describe order-
disorder transitions not reported in other works.

When considering the system without police officers, we identify transitions
from two orders with a majority phase, a disordered phase, and a consensus
phase. Furthermore, we identify when the system can reach stability or an
instability conditions based on the agent’s interactions. Besides, we reveal
the Principle of Minimum Grievance, the underlying principle of the model’s
dynamics. On the other hand, in the system with police officers, we study the
effects of police officers’ concentrations in different scenarios generated by
the kind of neighborhoods and vision. We find the same order-disorder
transition, but now we observe six ordered phases with a majority, one
disordered phase, and the consensus phase. With the global quantities that we
introduced, we can determine the role of police officers to dissuade a social
protest. Finally, we identify stability and instability conditions of the
system dynamics, and we show the energetic cost of using the police force to
facilitate a stable scenario. These results from the perspective of
sociophysics yield new qualitative elements and contribute to the future to
study the dynamics of this model in other dimensions and topologies to
approach the complexity of the dynamics of social protest.

The paper is organized as follows: In Sec. II, we introduce Epstein’s model
for a civil disorder and the global quantities used to describe the model’s
behavior to reach the stationary state. The simulations for the model without
and with police officers and their respective phase diagrams are presented in
Sec. III. The discussion of our results and concluding remarks are in Sec. IV.

##  II The model and global quantities

###  II.1 The Epstein Model

The civil disorder model has two agents: citizens and police officers.
Citizens can be active when they participate in social protest, passive when
they do not participate, or jailed when the police officers catch them.
Citizens can switch from one state to another depending on their neighborhood,
local parameters, and the global parameters of the system. On the other hand,
a police officer agent represents the central authority’s force. They are
responsible for deterring a protest by capturing the active agents in their
neighborhood. The neighborhood for all agents can be a von Neumann
neighborhood used by Epstein [29] or a Moore neighborhood as in other works
[31, 34].

The system’s dynamics emerge by relating the legitimacy of the authority and
the grievance of the population, i.e., it depends on the relationship between
the global parameters and the agents’ parameters. The global parameters are
the same for all agents: legitimacy $L$, a state change threshold $T$, the
maximum jail term $J_{\text{max}}$, and the vision $v$. The original model’s
vision determines the neighborhood’s size, similarly to the rule radius in
cellular automata [37] and range in other opinion models [38, 39]. On the
other hand, the local agent parameters are hardship $H$ and risk aversion $R$.
Both parameters are random values between zero and one uniformly distributed
among all agents.

Figure 1: Schematic visualization of agents state changes in the Epstein’s
model for a civil disorder. For the system without police officers, the
citizen agents can switch between two states depending on their local and
global parameters. For the system with police officers, the agents can change
between three states depending on their local and global parameters and
neighborhood conditions. The switch from the active to jailed state is a
product of the interaction with the police officers’ agents.

The rules that determine the agents’ actions are as follows:

(1) State change rule: each agent will decide whether or not to join the
protest, evaluating the equation $G-N_{R}>T$, where $G=H(1-L)$ symbolizes the
grievance and $N_{R}=RP$ the net risk. The arrest probability equation
$P=1-\exp[-k(C/A)_{v}]$ depends on the active agents and police officers ratio
in the neighborhood defined by the vision. Hence when any agents evaluate if
they switch their state, consider all active agents and police officers in
their neighborhood. Therefore, for a fixed number of police officers, the
agent’s arrest probability falls the more active agents there are. Notice that
$A$ is always at least one because the agent always counts himself as active
when computing $P$. The value of $k$ is $2.3$ for ensuring plausible $P$
values, as reported by Epstein [29]. In the complete form of the state switch
equation,

$H(1-L)-R(1-\exp[-k(C/A)_{v}])>T,$ (1)

we notice that the first element on the left depends on a combination of local
and global values, and the other depends on the neighborhood conditions. In
this way, when the difference of the agents’ state variables exceeds the
threshold, they switch from passive to active; otherwise, they remain passive
agents.

(2) Capture rule: police officers randomly capture an active agent from their
neighborhood. If there are no active agents, they do nothing. A jailed agent
stops participating in the dynamics according to the jail parameter assigned
value randomly, with values between zero and the maximum determined at the
beginning of the simulation. We used $30$ time steps as a maximum jail term,
the same value used by Epstein [29]. When jailed agents finish their
sentences, they return to the model dynamics as passive agents.

(3) Movement rule: each agent will move to an empty space at random within
their neighborhood.

After setting global and local parameters of the model, we placed all the
agents in random positions in the lattice to start the simulations. At each
time step, all agents evaluate the dynamics rules asynchronously [29]. We show
a schematic visualization of the changes of the agents’ state changes due to
the interaction rules in Fig. 1.

###  II.2 Global quantities

In order to characterize the model, we labeled each agent with the variable
$\alpha$, which can take values between one and four to represent an active
agent, passive, jailed, or police officer. Then, we have defined the following
quantities.

(a) Concentration of agents. To see the predominant state in the system and
study its macroscopic behavior as a function of the global parameters, we
define:

$C_{\alpha}=\frac{N_{\alpha}}{N},$ (2)

where $N_{\alpha}$ denotes the number of agents in the state $\alpha$ and $N$
the number of agents in the system. As usually used in opinion dynamics models
[36, 12], $\sum C_{\alpha}=1$ and we distinguish the following phases:

(i) The disordered phase, when all agent states are of a similar concentration
in the system.

(ii) The ordered phase, when one agent state is majority over the others.

(iii) The consensus phase is when the system reaches a particular ordered
phase where all agents have the same state.

It is essential to note what we define order from the opinion dynamics
perspective to describe order-disorder transitions. Thus, by order, we refer
to a macroscopic pattern in which we could find a majority opinion state. We
do not refer to the common idea related to public order as the absence of
criminal or political violence in society. Moreover, these definitions are
convenient because they allow us to identify a macroscopic state with the
agents’ state and the system’s dynamics.

(b) Energy. Now, we introduce a global quantity that allows us to analyze and
interpret the system based on the macroscopic states that emerge from the
agents’ states of the system. Hence, we introduce a Potts-like energy function
[40],

$E[\alpha]=-\dfrac{1}{2vN}\sum_{i=1}^{N}\sum_{j\in
V_{i}}J_{ij}\delta(\alpha_{i},\alpha_{j}),$ (3)

where the symbol $\sum_{j\in V_{i}}$ means the sum over all neighbors $j$ of
the agent $i$ with the same state $\alpha$. Here $J_{ij}$ is called the
interaction strength. However, we will take $J_{ij}=1$ (for all $i$ and $j$),
due to the characteristics original model. In other words, in this work, we do
not consider different interaction strength values.
$\delta\left(\alpha_{i},\alpha_{j}\right)$ is a Kronecker delta, i.e.,
$\delta\left(\alpha_{i},\alpha_{j}\right)=1$ if $\alpha_{i}=\alpha_{j}$ and
zero for all $\alpha_{i}\neq\alpha_{j}$.

Note that the energy functions introduced in the Schelling model [21, 22] are
like the Ising model because the models have two possible states. For the case
of the Sakoda model [23] and this model, it is more natural to use the Potts
energy because these models have more than two states.

Besides, we can observe that the energy definition shows the absolute minimum
or ground state when all the agents in the system are in the same state. On
the other hand, the energy may reach the maximum energy when the system takes
a chessboard aspect. This behavior is convenient because it allows us to
establish analogies or interpretations of the dynamics.

Figure 2: Agents’ concentration variation on time. Figure (a) corresponds to a
system without police officers and Fig. (b) a system with police officers. In
both cases, the system reaches a stationary state quickly.  Figure 3:
Concentration variations for different threshold fixed values in a system
without police officers. With low values to legitimacy, the active agents are
predominant, but as legitimacy increases, the passive agents dominate. There
are points when active and passive agents have a similar concentration
(segmented lines) and when all the agents in the system are passives (solid
lines). As the threshold increases, we can see a translation of these points
suggesting a transition. This figure shows simulations results for a one-
dimensional lattice with $N=2^{10}$ sites, Moore neighborhood, and vision one.
However, the results obtained for all visions, neighborhoods, and sites
considered collapse on the same curve.

##  III Simulations and phase diagrams

This paper aims to characterize the Epstein’s model for a civil disorder in a
one-dimensional lattice with periodic boundary conditions as a first
approximation to understand the dynamics of social protests. Thus, we perform
simulations on a one-dimensional lattice with $N=2^{8}$ and $N=2^{10}$ sites,
considering a system without and with police officers. The first one has
agents with two possible states, active and passive. The second one has police
officers then the agents can be active, passive, or jailed.

Furthermore, to study the effects of interactions in the system dynamics, we
consider agents with visions one and seven interacting in Moore and random
neighborhoods. Note that vision determines the neighborhood’s size, so when we
consider a one-dimensional lattice, the vision represents the number of pairs
of agents to consider to evaluate an agent’s state switch. For example, when
the agent’s vision is one, the Moore neighborhood of the agents consists of
its nearest neighbors. Thus each agent considers two sites, one to their left
and one to their right. When the agent’s vision is seven, the Moore
neighborhood of the agents counts seven sites on the left and seven sites on
the right, with fourteen agents in total.

On the other hand, a random neighborhood is when an agent can randomly select
other agents to form their neighborhood. Then, when the agent’s vision is one,
it chooses two agents randomly. When its vision is seven, the agent can
randomly select fourteen agents from the lattice.

We have not used Epstein’s motion rule from the original model in this work.
So the agents occupying the whole of the one-dimensional lattice and the
system can enter a stationary regime, as shown in Fig. 2. As we can see, the
dynamics converge quickly around the same value for the agents’ concentration
in any state and remains constant on average after some time. For this reason,
in this paper, we can study the asymptotic properties of the dynamics and
characterize the model on the steady-state. In this way, we use $20$
realizations with $5500$ time steps for all the study cases. In each
realization, all agents have different initial positions and state variables.
Then, we discard the first $500$ time steps to obtain a steady state. Finally,
we calculated the average quantities over $20$ realizations to characterize
the model.

###  III.1 System without police officers

When we study the system without police officers, the state switch equation
(1) changes to

$H(1-L)>T.$ (4)

Hence, the agents’ state only depends on its local parameters and is
independent of their neighborhood. Therefore, changes in the system’s dynamics
depend on the threshold and the initial simulation’s conditions. To study the
whole system, we run simulations for threshold and legitimacy values between
$0.00$ and $0.99$ with a step of $0.01$ for both variables.

Figure 4: The global average energy variations for different threshold fixed
values in a system without police officers. The energy shows two minimum
values. The first is a local minimum, where the active agents predominate in
the system. The other one is a global minimum or the ground state, and all
agents in the system are passive. On the other hand, the system reaches a
maximum value when the active and passive agent concentrations are similar.
This figure shows simulations results for a one-dimensional lattice with
$N=2^{10}$ sites, Moore neighborhood, and vision one. However, the results
obtained for all visions, neighborhoods, and sites considered collapse on the
same curve.

To obtain a first idea of the model dynamics, we study the concentration and
energy variations for different threshold fixed values. We observe the
variation of the concentration of agents when the legitimacy increases in Fig.
3. With low values to legitimacy, the active agents are predominant. Then as
legitimacy increases, the passive agents are predominant. When the threshold
is $T=0.10$, the system’s dominant state changes, as shown in Fig. 3(a). Note
that when the legitimacy is $L<0.80$, the active agents predominate, when
$L=0.80$, the concentration for two states are similar, and when $L>0.80$, the
passive agents are dominant. For $L\geq 0.90$, all agents of the system are in
the passive state. We can see a translation of the point of concentration
similarity and the point when all agents of the system are passive states when
the threshold increases in 3(a), 3(b), 3(c), and 3(d) figures. The
translations of these points indicate transitions in the system. It is
important to note that all showed results collapsed on the same curves for all
visions, neighborhoods, and sites considered in our simulations. This behavior
is because the state switch equation (4) is independent of the neighborhood
and indicates the system’s dynamic depending on the threshold and the initial
simulation’s conditions.

We present the global average energy versus legitimacy with different
threshold values in Fig. 4. When the threshold is $T=0.10$, the energy starts
with a lower value of around $\left<E\right>\approx-0.8$. As the legitimacy
increases, the energy reaches a maximum value around
$\left<E\right>\approx-0.5$ when the legitimacy is $L=0.80$. Next, the energy
converges to the minimum value when the legitimacy is $L\geq 0.90$. The
initial energy value is a local minimum and indicates when the active agents
predominate. Then, when the agents’ concentrations are similar, the energy
reaches a maximum. Finally, the energy minimizes when the system has only
passive agents and reaches the absolute minimum or the ground state.

On the other hand, when the system possesses higher thresholds values, the
energy reaches a maximum and then converges to the minimum quickly. Besides,
we can see a translation of these energy points as the threshold increases
suggesting a transition.

To build the phase diagram, we search the coordinates $(T,L)$ where the
agents’ concentrations are similar, and all agents are passive. As shown in
Fig. 5, these points define the phase boundaries. Following the phases
described in opinion models, the results show order-disorder transitions.
Phase AP and PA are ordered phases with a majority agent state. Active agents
are dominant in phase AP and passives in phase PA. As a result of crossing the
dashed line between these two phases, we observe a disordered phase with
similarly active and passive agents concentrations. Note that our numerical
results are consistent with the assumption for an average agent. We can find
the critical legitimacy to obtain equal concentrations of active and passive.
As a result, we obtained this dashed line of critical legitimacy $L_{c}=1-2T$.
The consensus phase is a particular ordered phase when all agents are passive.
The solid line shows the transition from majority order to consensus order.

Figure 5:  Phase diagram for a system without police officers. Phases AP and
PA are ordered phases with a majority agent state. Active agents are
predominant in the AP phase and passives in phase PA. The dashed line between
these two phases shows when the system has similar concentrations, therefore
disordered. The consensus phase is a particular ordered phase when all agents
are passives. The solid line shows the transition from the order with a
majority to consensus. Every point in this diagram corresponds to an $L$ and
$T$ value when the concentrations of active and passive agents are similar or
when the system reaches a consensus. The points for all visions,
neighborhoods, and sites considered collapse on the same curve. Figure 6:
Stationary probability density function of the agents’ concentration for a
system without police officers. We see two transitions when the system
increases legitimacy and threshold fixed. A continuous transition from AP
phase to PA phase across a disordered phase in $L=0.80$, as seen in figures
(a),(b), and (c). Then, a continuous transition from PA phase to consensus, as
seen in figures (d), (e), and (f). This figure shows simulations results for a
one-dimensional lattice with $N=2^{10}$ sites, Moore neighborhood, and vision
one. However, these behaviors are the same for all visions, neighborhoods, and
sites considered in our simulations. Figure 7: Agents concentration variations
in a system with police officers for different values of police officers
concentrations, legitimacy, vision one, and threshold fixed. When the
interactions occur in a Moore neighborhood, the predominance of active agents
decreases when the police officers’ concentrations increase because of
increased jailed agents. As a result, we can see a translation of the
concentration similarity points (segmented lines) in figures (a), (b), (c),
and (d). These translations show changes of the predominant state and the
existence of a point in which the three states of the system are similar,
suggesting a phase change. In a random neighborhood, we can see the same
dynamics of translation of the concentration similarity point in figures (e),
(f), (g), and (h). However, police officers can capture more active agents
because of the random selection of their neighborhoods. Thus, the predominance
of jailed agents results until the system reaches a high legitimacy. The
vertical solid line depends on the fixed threshold and indicates when the
system reaches a consensus. This figure shows simulations results for a one-
dimensional lattice with $N=2^{10}$ sites, but with $N=2^{8}$ sites, we
observe the same result.

To observe the system transition, we study the stationary probability density
function of the agents’ concentration. We show distributions for a system with
Moore neighborhood with vision one and $T=0.10$ in Fig. 6 because we observe
the same behavior independent of the neighborhood, vision and sites considered
in our simulations. We can see the system transition from the order with
active agents majority in $L=0.75$ to a disordered phase in $L=0.80$, then a
change to order with passive agent majority in $L=0.85$, in figures 6(a),
6(b), and 6(c). As for legitimacy increases, we can observe a transition to a
consensus in $L=0.90$ in figures 6(d), 6(e), and 6(f).

Figure 8: The global average energy variations in a system with police
officers for different values of police officers concentrations, legitimacy,
vision one, and threshold fixed. In a Moore neighborhood, the energy reaches a
minimum value for low police officers concentrations and increases as
legitimacy increases, as shown in figure (a). When the system reaches a
consensus, the energy shows a global minimum. As it increases the police
officers’ concentration, it is possible to observe that the energy maintains a
constant value before reaching a consensus of passive agents. The maximum
energy value indicates when the majority agent states have similar
concentrations. In a random neighborhood, the energy has a constant value and
increases as legitimacy increases, as shown in figure (b). In both cases, the
global minimum of energy increase as a police officer’s concentration
increases. This figure shows simulations results for a one-dimensional lattice
with $N=2^{10}$ sites, but with $N=2^{8}$ sites, we observe the same result.

###  III.2 System with police officers

To study the model with police officers, we used legitimacy values and the
concentration of police officers from $0.00$ to $0.99$ with a step of $0.01$
for both variables. We vary the concentrations of police officers as an
initial condition because its value determines the system’s dynamics. The
police officers’ role is to dissuade a social protest, preventing citizen
agents from becoming active agents and arrest active agents in the system
producing jailed agents. Besides, the police officers’ action depends on the
vision, so we study the system separately with two different visions. On the
other hand, the police officers’ inclusion makes the change of the agents
state depends on the state parameters and the neighborhood conditions, as we
can see in Eq. (1). We used a fixed threshold value in $T=0.10$ for these
simulations because its role is to determine a limit value to the state switch
equation. Furthermore, this value coincides with Epstein’s reported value in
the five study cases from the original model for a generalized rebellion.
Thus, this value allows us to characterize this context with a more
significant parameters set.

Figure 9: Phase diagrams for a system with police officers and vision one.
Every point in this diagram corresponds to police officers’ concentration and
legitimacy value when the different states are at similar concentrations. For
example, the $C_{ap}$ points ( ) are the points at which the concentrations of
active and passive agents are similar. These points determine different
ordered phases with a majority, and each label indicates the order of the
predominant state. For example, in the PAJ phase, passive agents are
predominate, followed by active and jailed agents, and so on. The triple point
in these curves’ intersection indicates when the three states are in similar
concentrations, so we observe a disorder in the system. The system reaches a
consensus when legitimacy is greater than or equal to $0.90$. The black region
indicates when there are only police officers in the system. With Moore
neighborhood show six differents order phases with a majority, one point where
the system reaches a disorder, and a consensus phase. With random
neighborhoods, order phases with majority change because police officers can
capture more active agents and predominantly jailed agents. However, the
system reaches a consensus phase in the same conditions because the fixed
threshold determines this change. This figure shows simulations results for a
one-dimensional lattice with $N=2^{10}$ sites, but with $N=2^{8}$ sites, we
observe the same result. Figure 10: Stationary probability density function of
the concentration of agents for a system with police officers and vision one.
We see two transitions when the system increases legitimacy and fixes police
officers’ concentration. In the Moore neighborhood, we can see a disordered
phase, then order with passive agents majority, and finally, a consensus phase
in figures (a), (b), and (c). The same dynamics occur in the random
neighborhood, with an order with active agents majority, a disordered phase,
and then a consensus phase in figures (d), (e), and (f). This figure shows
simulations results for a one-dimensional lattice with $N=2^{10}$ sites, but
with $N=2^{8}$ sites, we observe the same result.

#### Results with vision one

We show the agents’ concentration variations for different values of police
officers’ concentrations and vision one in Fig. 7. When the interactions occur
in a Moore neighborhood, we observe a variation of the concentration of agents
when the legitimacy increases in figures 7(a), 7(b), 7(c), and 7(d). With low
values to legitimacy, the active agents are predominant. Then as legitimacy
increases, the passive agents are dominant. The jailed agents’ concentration
depends on the police officers’ concentration. Then their variations only
occur as police officers’ concentration increases and produce a change of
active agents concentration. With interactions in the random neighborhood, the
jailed agent concentrations have predominant values for low legitimacy, as we
can see in figures 7(e), 7(f), 7(g), and 7(h). Then, passive agents are
predominant as legitimacy increases. The active agent concentrations depend on
the police officers’ concentrations and decrease as the number of police
officers in the system increases.

Now, we can observe the global average energy versus legitimacy for a Moore
and random neighborhood in figures 8(a) and 8(b), respectively. On the one
side, when the interactions occur in the Moore neighborhood, and police
officers’ concentration equals $C_{pol}=0.10$, the energy starts around
$\left<E\right>\approx-0.6$. Next, it has an increasing behavior to around
$\left<E\right>\approx-0.4$, and a legitimacy value is close to $L=0.80$.
Then, the energy decreases quickly to the lower value
$\left<E\right>\approx-0.8$. We observe the local energy minimum when active
agents are predominant. Then, the energy maximum shows when the agents’ states
have similar concentrations. In particular, the active and passive agents
concentration is approximately $0.4$, and jailed agents and police officers
concentrations are close to $0.10$. Finally, we see the global energy minimum
when all agents are passive. Note that this global minimum is not absolute
because the police officers’ concentration equals $C_{pol}=0.10$. As the
police officers’ concentration increases, the energy maintains a constant
value before reaching the minimum energy value. Note that this global energy
minimum increases as a police officer’s concentration. On the other side, for
a random neighborhood, the energy started around $\left<E\right>\approx-0.4$,
with a police officer concentration equal to $0.10$, $0.30$, and $0.50$. Next,
the energy had an increasing behavior until it reached a maximum around
$\left<E\right>\approx-0.3$ and eventually converged rapidly to different
energy minima. We observe similar behavior in both neighborhoods for police
officer concentration is $C_{pol}=0.70$. Furthermore, the global minimum of
energy increases as a police officers’ concentration increases for all cases.

As we noticed in the results for a system without police officers, in the
systems with police officers, there are also points where the state
concentrations are similar. Their positions move as the police officers’
concentrations increase. These translations suggest a change in the state
predominant in the system. Furthermore, there is a point when the three states
of the systems are similar, configuring the order-disorder transitions like
opinions models [36, 12].

To verify this idea, we search for phase boundaries defined by the points in
which the agents’ concentrations are similar and when all agents reach a
passive state. Every point corresponds to police officers’ concentration and
legitimacy value and depends on a pair of similar agent states. Thus, the
$C_{ap}$ coordinate ( ) is when the concentrations of active and passive
agents are similar. The $C_{ja}$ coordinate ( ) indicates similarity in the
jailed and passive state, and the $C_{ja}$ point ( ) when jailed and actives
agents are similar concentration. Each point formed a curve defining different
regions on the phase diagram shown in Fig. 9. For both the Moore neighborhood
in Fig. 9(a) and the random neighborhood in Fig. 9(b), we observe phases
classified according to the transitions described for the system without
police officers. There are six ordered phases with a majority state. Each one
has a label indicating the order of the predominant state. For example, the
PAJ phase has dominant passives agents, followed by active and the jailed
agents, and so on for the other phases. The system reaches a consensus in the
passive phase when all agents are passives and legitimacy equal $0.90$. This
value is determined by the threshold value selected. The black region
indicates when there are only police officers in the system.

The system reaches a disordered phase labeled a triple point when the active,
passive, and jailed states are in similar concentrations. The position of this
point and the regions of the phases depends on the neighborhood. Most phases
have an observable area for the Moore neighborhood. The part where jailed
agents predominate is minor because police officers can only capture active
agents among their nearest neighbors. In contrast, police officers are more
likely to catch an active agent in a random neighborhood. Thus, we can observe
a translation of the triple point and an increase in the regions’ size with
predominant jailed agents and a decrease in the areas where active agents are
dominant. It is important to note that now it is more difficult to see the
order-disorder transition. Unlike the system without police officers, where we
observe the transition for all parameters, in the system with police officers,
the disordered phase is only a point in this diagram.

We can observe the system transition with the stationary probability density
function of the agents’ concentration in Fig. 10. As for legitimacy increases,
the system changes from the disordered phase to an ordered phase with a
passive state majority. It then reaches a consensus phase in a Moore
neighborhood in figures 10(a), 10(b), and 10(c). For a random neighborhood,
the transition from the ordered phase with the active state to a disordered
phase, then the consensus phase, as we see in figures 10(d), 10(e), and 10(f).
The final passive agents’ concentration depends on the police officers’s
concentration fixed to observe the transition. So, in the Moore neighborhood,
the final concentration is $0.6$ and, in a random neighborhood, the passive
agents’ concentration is $0.9$.

Figure 11: Agents concentration variations in a system with police officers
for different values of police officers concentrations, legitimacy, vision
seven, and threshold fixed. The first and second-row figures show interactions
in a Moore and random neighborhood respectably. We can see the predominance of
jailed agents with low police officers concentrations in Moore and random
neighborhoods in figures (a) and (e). Note that the concentration of jailed
agents is more significant in the random neighborhood because police officers
can capture more agents. However, as the police officers’ concentrations
increase, we can see the predominance of passive agents for both kinds of
neighborhoods. Besides, we observe the same dynamics of translation of the
concentration similarity (segmented lines) as in a system with vision one but
only for low police officers concentrations. The vertical solid line depends
on the fixed threshold and indicates when the system reaches a consensus. This
figure shows simulations results for a one-dimensional lattice with $N=2^{10}$
sites, but with $N=2^{8}$ sites, we observe the same result. Figure 12: The
global average energy variations in a system with police officers for
different values of police officers concentrations, legitimacy, vision seven,
and threshold fixed. As for legitimacy increases, the energy reaches a global
minimum in Moore and random neighborhoods. However, the energy only shows a
maximum with police officers’ concentration values lower or equal to $0.10$.
Then, as police officers’ concentration increases, the energy constantly
decreases until the system reaches a consensus. Besides, in both cases, the
global minimum of energy increase as a police officer’s concentration
increases, as we can see in figures (a) and (b). This figure shows simulations
results for a one-dimensional lattice with $N=2^{10}$ sites, but with
$N=2^{8}$ sites, we observe the same result.

#### Results with vision seven

We show the agents’ concentration variations for different police officers’
concentrations and vision seven in Fig. 11. When the interactions occur in a
Moore neighborhood, we can see a variation of the concentration of agents when
the legitimacy increases in figures 11(a), 11(b), 11(c), and 11(d). With low
legitimacy, active agents exist, but jailed agents are predominant. Then as
legitimacy increases, the passive agents are dominant. As police officers’
concentration increases, the active agents’ concentration minimizes. Passive
agents increase constantly, and jailed agents decrease.

With interactions in the random neighborhood, the jailed agent concentration
has predominant values for low legitimacy, as we can see in figures 11(e),
11(f), 11(g), and 11(h). Furthermore, the active agent’s concentration starts
initially with a low value, and passive agents are predominant as legitimacy
increases. As police officers’ concentration increases, the concentration of
active agents disappears, passive agents increase constantly, and jailed
agents decrease. This behavior indicates a change in the importance of the
police officers’ role to dissuade a protest. When the agents have vision
seven, the relevant police officers’ role is to prevent citizen agents from
becoming active and prevent the emergence of a protest.

Now, we can observe the global average energy versus legitimacy for a Moore
and random neighborhood in figures 12(a) and 12(b), respectively. On the one
side, when the interactions occur in the Moore neighborhood, and police
officers’ concentration equals $C_{pol}=0.10$, the energy starts close to
$\left<E\right>\approx-0.4$. Next, it maintains a constant behavior until the
energy decreases quickly to the lower value of around
$\left<E\right>\approx-0.8$. On the other side, in the random neighborhood,
and police officers’ concentration equals $C_{pol}=0.10$, the energy has a
lower value, around $\left<E\right>\approx-0.3$. Then increases to reach a
maximum value close to $\left<E\right>\approx-0.4$ and decreases quickly to
the lower value around $\left<E\right>\approx-0.8$. Note that, in both cases,
we observe the initial energy value when jailed agents are predominant. Then,
the energy maximum shows when the jailed and passive agents concentration is
approximately $0.4$, and police officers concentration is $0.10$. Finally, we
see the global energy minimum when all agents are passive. However, in both
neighborhoods, as police officers’ concentrations increase, the energy has a
decreasing behavior until reaching the lower energy value. We note that the
global energy minimum increases as a police officer’s concentration for all
cases.

Figure 13: Phase diagrams for a system with police officers and vision seven.
Every point in this diagram corresponds to police officers’ concentration and
legitimacy value when the different states are at similar concentrations. We
can see the same phases and transitions order-disorder from the system with
vision one in the Moore neighborhood. Nevertheless, the size of each phase
change. With random neighborhood, the disordered phase and the JAP phase
disappear due to increased police officers’ action with vision seven, as shown
in figure (b). In both cases, the system reaches a consensus when the
legitimacy is greater than or equal to $0.90$ because the fixed threshold
determines this change. The black region indicates only police officers in the
system. This figure shows simulations results for a one-dimensional lattice
with $N=2^{10}$ sites, but with $N=2^{8}$ sites, we observe the same result.
Figure 14: Stationary probability density function of the concentration of
agents for a system with police officers and vision seven. We can see two
transitions in the Moore neighborhood when the system increased legitimacy and
fixed police officers’ concentration. A disordered phase, then orders with
passive agents majority, and finally, a consensus phase in figures (a), (b),
and (c). However, in the random neighborhood, only observe order-with majority
phases and then a consensus phase in figures (d), (e), and (f). This figure
shows simulations results for a one-dimensional lattice with $N=2^{10}$ sites,
but with $N=2^{8}$ sites, we observe the same result.

For low police officers concentrations, points where the state concentrations
are similar exist, suggesting changes in the predominant state in the system.
To search for a point where the system shows an order-disorder transition, we
built a phase diagram. Every point corresponds to police officers’
concentration and legitimacy value and depends on a pair of similar agent
states. Then, we can observe the phases formed for the system with police
officers’s vision seven in Fig. 13. We see the same six phases observed in the
system for the Moore neighborhood with vision one in Fig. 13(a). The region’s
size for every phase changes notably because of the increase in agents’ vision
and the police officers’ activity. As a result, we note that the regions with
the predominance of passive agents are more significant than the others.
Besides, we can observe a triple point where the disordered phase occurs, the
area where the system only has police officers, and a consensus phase with
only passive agents. For the random neighborhood, the effect of vision seven
is more significant, as shown in Fig. 13(b). Although the size of the regions
dominated by passive agents is similar to those of the Moore neighborhood, we
can notice that the JAP phase and the point at which all concentrations are
similar disappears. As a result, we observe that there is no order-disorder
transition. However, the system change between different order phases with a
majority state depending on the police officers’ concentration and legitimacy
values. It reaches a consensus when all agents are in the passive state at the
value of legitimacy is $0.90$ because this depends on the threshold value
fixed at the beginning of simulations.

We show the stationary probability density function of the agents’
concentration in Fig. 14 to observe the system transition. For the Moore
neighborhood, we selected the police officers’ concentrations $0.06$ and
varied the legitimacy. We can see, as legitimacy increases, the system change
from the disordered phase to an ordered phase with a passive state majority in
figures 14(a) and 14(b). Then reaches a consensus phase in Fig. 14(c). We
selected a lower value of police officers’ concentrations for the random
neighborhood to observe the possibility of finding similar concentrations for
the three states as legitimacy increases. Nevertheless, only find order with
majority phases, as shown in figures 14(d), 14(e), and the consensus phase in
Fig. 14(f).

##  IV Discussion and Concluding Remarks

This paper studied the one-dimensional civil disorder model with the whole
lattice occupied to characterize their evolution on the steady-state. To do
this, we performed extensive numerical simulations of the model with and
without police officers, considering visions one and seven in Moore and random
neighborhoods to study the effects of interactions on the system’s dynamics.
We used the agent state concentration and introduced a Potts-like energy
function as global quantities to characterize the model.

In the system without police, the dynamics only depend on values assigned as
initial conditions. One of them is hardship, a local parameter uniformly
distributed between values zero and one for each agent. In the model, this
parameter allows for a heterogeneous society of agents. The other two,
legitimacy and threshold, are global parameters that we use as control
parameters for our simulations. The threshold is a quantity defined as non-
negative and determines a limit for an agent’s state change.

Given that the product of hardship and legitimacy symbolizes the grievance in
the state change equation, we interpret this threshold as a tolerance for
grievance. The higher the threshold value, the more disposed agents are to
tolerate grievance before rebelling against authority. This parameter takes
different values between zero and one, which we interpret as an essential
property of a community that depends on its culture or way of life. Thus, we
find communities with a low threshold and quickly protest for a grievance and
communities with a very high threshold that lives with a minimal grievance and
does not rebel against authority. On the other hand, legitimacy is the
community’s perception of the regime or the system’s authority. Thus, a low
legitimacy produces more significant grievance, and high legitimacy favors the
passivity of the system.

In our results for the variations in the concentration of agents as a function
of legitimacy, we observe that active agents are predominant for low values of
legitimacy. As legitimacy increases, we find a point where the concentrations
of active and passive agents are similar, and then passive agents become
predominant. According to the legitimacy variation, the predominance changes
depend on the threshold values. We note that passive agents are always
dominant for threshold values greater than $0.50$ and coexist with active
agents. However, for high legitimacy values, we notice that all agents in the
system become passive. These changes in the predominance of a state among the
agents are indicators of phase changes. We build the phase diagram based on
the concentration points of similar agents or when all agents are passive, and
we observe order-disorder transitions. We identify the AP and PA phases as
order phases with a majority state. Active agents are predominant in the first
and passive agents in the second. We find a disordered phase in which the
agents’ concentrations are similar when crossing between these phases. When
all the agents are passive, the system reaches the consensus phase. We study
the transitions with the stationary probability density function of the
concentration of the agents, and we observe a typical scenario of continuous
transitions [36].

The energy shows one maximum and two minimum values for low threshold values.
One of them is a local minimum, where the active agents predominate in the
system. The other is a global minimum or ground state, where all agents in the
system are passive. The system reaches the maximum value when the
concentrations of active and passive agents are similar. In contrast, when the
system has higher threshold values, the energy is maximum and quickly
converges to a minimum. We can see the translation of these energy points as
the threshold increases, allowing us to identify the phase changes. Thus, the
local minimum indicates the ordered phase with an active agents majority, the
maximum the disordered phase, and the global minimum or ground state is the
consensus phase.

The local energy minimum observed for threshold values greater than zero and
less than $0.50$ is a metastable point. The active agents’ predominance
generates energy and stops the system from reaching the ground state. In other
words, the system is not entirely stable due to grievance. However, as
legitimacy increases, the system can reach a consensus. On the other hand,
when the threshold values are greater than $0.50$, the local minimum
disappears and becomes an energy maximum. The maximum is an unstable point for
all threshold values because the system can fall to the local or global
minimum depending on how legitimacy varies. Since the concentrations of active
and passive agents are similar, the system increases energy due to grievance
generating a scenario of instability comparable to a polarized society. In
this context, we understand polarization as a situation of equal opinion
searching for a consensus. It is important to note that the maximum energy
that our results show is the maximum possible given the initial conditions of
random positioning of the agents. However, it is possible to find an absolute
maximum for the energy by positioning the agents deliberately to form the
checkerboard appearance.

Finally, the global minimum is a stable point of the system since all the
agents are in the same passive state in the consensus phase. When threshold
values exceed $0.50$, the system can reach the global minimum for low
legitimacy values. Nevertheless, for threshold values less than $0.50$, we
find that reaching consensus requires higher legitimacy values. Indeed, the
greater the tolerance for grievance, the less legitimacy is required to reach
consensus. This result reflects the existence of societies with high
thresholds, in which reaching consensus requires low legitimacy values, unlike
other societies in which reaching or maintaining consensus requires high
legitimacy.

From these results, we can reveal that the principle underlying the dynamics
of the model is a Principle of Minimum Grievance, equivalent to that observed
in the model of worker protest in a factory. This principle allows us to
interpret that the system seeks minimal grievance or a consensus. Naturally,
reaching and maintaining consensus requires high values of legitimacy or high
values of tolerance, the latter being the one that determines the value of
legitimacy necessary in a heterogeneous society. Thus, the emergence of
protests is due to global conditions of legitimacy or threshold that generate
a grievance, increasing the system’s energy. This grievance generates energy
variations that can lead the system to instability or metastability.

In the system with police, the agents can be active, passive, or jailed, and
the dynamic depends on the vision and the neighborhood. For this reason, we
study the system separately, considering two different views. We fix the
threshold value and consider the legitimacy and concentration of police
officers in the system as a control parameter because the police officer’s
role is to deter a protest. Besides, the agents’ state switch depends on the
number of active agents and police officers in their neighborhood.

The concentration of agents allows us to identify the predominance of states
in the system. We find that active agents predominate in the low legitimacy
regime only when the system has vision one and Moore neighborhood. For the
rest of the cases, the jailed agents always predominate. As in the system
without police, as legitimacy increases, the predominant state changes, and
passive agents become the majority. In addition, we again note points where
concentrations are similar. For legitimacy values greater than or equal to
$0.90$, all agents in the system are passive. It is essential to mention that
the increased concentration of police officers in the system facilitates the
predominance of passive agents. Therefore, in general, the concentration of
passive agents increases as legitimacy increases. In particular, this increase
is linear in the system with vision seven because the agents do not switch to
the active state as police presence increases in a neighborhood. In addition,
when the system reached the legitimacy value in which jailed and passive
concentrations are similar, the number of jailed agents began to drop rapidly.

The phase diagrams allow us to identify the domains’ limits based on the
changes in the predominance of a state in the system. The diagrams show order-
disorder transitions for the system with vision equal to one. We identify six
ordered phases with a majority state, a disordered phase at a specific point
in the diagram, and a consensus phase where all agents in the system are
passive. The disorder point’s position and the phases’ sizes depend on the
type of neighborhood. The random neighborhood makes it easier to capture
active agents. So, the disorder point requires less concentration of police
than in the Moore neighborhood, and the most extensive phases are those with a
predominance of passive agents. When the system has vision seven, the
differences depend on the neighborhood. In a Moore neighborhood, we observe
order-disorder transitions with the same phases observed in the system with
vision one. However, in a random neighborhood, we only observe 5 phases of
order with a majority, and we do not observe the order-disorder transition. In
this neighborhood, police officers significantly increase activity. So, they
can capture many active agents in the low legitimacy regime, preventing the
emergence of active agents with high legitimacy values. We show a summary of
the different scenarios of the system with police officers in steady-state in
Fig. 15. Finally, the study of the stationary probability density functions of
the concentration of the officers shows us that the phase changes are
continuous at the same as in the system without police.

Figure 15: Schematic visualization of different phases found at the one-
dimensional civil disorder model with police officers on the steady-state. We
observe six ordered phases with a majority state, a disordered phase, and a
consensus with vision one. With vision seven and Moore neighborhood, we
observe the same phases. However, there is neither the disordered phase nor
the ordered phase, with most jailed agents followed by active and passive
agents (JAP phase) in a random neighborhood.

From the results we obtained for energy, we can note essential differences in
behavior according to the concentration of police officers in the system. When
the police concentration has values less than or equal to $0.10$, we can see
that the energy shows a local minimum, a maximum, and a global minimum. The
local minimum shows an ordered state with a majority. We observe the most
active agents in the case of vision one and a Moore neighborhood. In the other
cases, the jailed agents always predominate. As for legitimacy increases, the
energy reaches a maximum when at least two agent states are in similar
concentrations. For example, for the system with vision one and a Moore
neighborhood, this maximum shows when the concentration of active and passive
agents are similar. In the other cases, we observe that the jailed and the
passive are in a similar concentration. Therefore, the maximum energy only
shows changes between phases of order with a majority and cannot be identified
with the state of disorder of the system. Finally, we observe the global
energy minimum when all agents in the system are passive, indicating that the
system is in the consensus phase.

When the concentration of police officers has values greater than $0.10$ but
less than $0.70$, we observe different behavior of energy depending on the
vision and the type of neighborhood. For vision one and a Moore neighborhood,
the initial energy is maximum. Then, as legitimacy increases, energy remains
constant until the legitimacy value indicates the phase with passive agents
majority. Then, the energy decays quickly to the minimum in the consensus
phase. The initial energy for vision one and a random neighborhood is a local
minimum. We observe a minimal increase until it reaches a maximum and quickly
decays to a global minimum. The maximum shows the transition from jailed
agents’ majority phase to the passive agents’ majority phase and then reaches
the consensus phase. For vision seven and both neighborhoods, we see that the
energy starts at the maximum possible value, indicating a phase where the
prisoners predominate. Then, energy steadily decays towards the global minimum
in the consensus phase.

Finally, for regimes where the police concentration is greater than or equal
to $0.70$, we see for all cases that the initial energy has a closer value to
the minimum. As legitimacy increases, the energy constantly decreases until it
reaches the minimum associated with consensus. In general, we can note that
the minimum energy for all cases is not an absolute minimum. Despite reaching
the consensus phase and all the agents being passive, there are police
officers in the system. Therefore, its value depends on the police officers’
concentration and their random position on the lattice. This result leads us
to conjecture that the energy value will be closer to the absolute minimum for
an arbitrary initial configuration with two clusters of police officers and
passive agents.

Regarding the observed stability points of energy, we see similarities for the
system without police officers. The local energy minimum observed for low
police concentration is a metastable point for the same reason as the system
without police. For all visions and neighborhoods, this point shows the
existence of grievance, either because there are active agents or prisoners.
Therefore, the system can change to the global minimum seeking consensus as
legitimacy increases. The point of maximum energy is generally unstable for
low values of police concentration because it can fall to the local or global
minimum with variations in legitimacy. For high police concentrations, the
maximum remains constant at low legitimacy, and as legitimacy increases, it
reaches the minimum rapidly. Finally, the global energy minimum is stable for
all cases because the system reaches a consensus. However, the increase of
police officers’ concentration makes its value change. Based on the results
obtained, we can still interpret the system based on the principle of minimum
grievance because the system tends towards a global minimum as legitimacy
increases. However, the energy is insufficient to identify the system’s
prevailing state and the effects of police concentration on the dynamics.

Since the system now considers three possible states for agents and police
officers, we must complement the analysis with the concentration of agents.
With these two macroscopic quantities together, we can identify the most
relevant function of the police based on the vision and the type of
neighborhood. When considering vision one, the capture of active agents is the
most relevant police function to dissuade a protest. In the Moore neighborhood
with low police concentration, the system requires high legitimacy values for
active officers are not predominant. As the concentration of police officers
increases, we observe a significant decrease in the activity of active agents,
so the system needs a lower value of legitimacy to change the active majority.
In the random neighborhood, police officers increase the capture of active
agents, allowing them not to be a dominant state in the system for low values
of legitimacy. In addition, as the concentration of police officers increases,
the activity of active agents rapidly decreases, increasing the concentration
of jailed agents. Nevertheless, high values of legitimacy are still needed for
passive agents to be predominant, and subsequently, the system reaches
consensus.

When the system has vision seven, we observe that there are still active
agents with a low concentration of police officers, but they are not
predominant. Therefore, the system requires increasing legitimacy to reach the
predominance of passive agents. As the concentration of police officers
increases, the activity of active agents disappears. As for legitimacy
increases, the passive agents increase, and the jailed agents decrease
linearly. Hence, in these cases, the relevant role of police officers is to
prevent the appearance of a protest by preventing the officers from becoming
active.

Although the police activity considerably reduces the active agents’ activity
in both cases, the permanent presence of jailed agents indicates a grievance
in the population and an increase in the system’s energy. Hence, we can
conclude that the system will not be utterly stable if there is an internal
grievance. This affirmation indicates that dissuading a protest by capturing
active agents is ineffective in reducing grievance. Nevertheless, our results
show that as the concentration of police officers in the system increases, the
system needs a lower value of legitimacy to reach consensus or minimum energy.
Observing the minimum obtained, we note that it is not an absolute minimum and
depends directly on the concentration of police officers in the system.
Therefore, the greater the concentration of police officers, the value of the
minimum energy increases. In other words, dissuading protests to facilitate or
maintain consensus has an energy cost for the system proportional to the
number of police officers or the amount of force used.

We do not observe any significant variation of the system’s behavior without
police officers in terms of the variations in lattice size, the vision of the
agents, or the type of neighborhood. So, we confirm that the dynamics only
depend on the initial random conditions, such as the positioning of agents in
the lattice and the definition of local parameters. Therefore, we conjecture
that the system’s dynamics will be the same regardless of the topology or
dimension of the grid. In addition, the use of the movement rule will not
generate state changes in the agents because the defined rules do not depend
on the neighborhood in the system without police. In this context, the global
quantities introduced and the phase diagram present new elements to analyze
the model’s dynamics and make new interpretations to understand the dynamics
of social protests.

In the results obtained for the system with police officers, we did not notice
significant variations of the dynamics with the variation of the size of the
system. However, the dynamics dependent directly on the vision and the
neighborhood product of the definition of the agents’ rules. The police
officers’ activity increased significantly with the elections of a random
neighborhood and increased vision. The global quantities introduced allowed us
to identify the most relevant role and effect of the police officers in the
system. It is essential to mention that the size of the system determines the
maximum possible vision of the agents. Hence, for visions close to the maximum
possible, we conjecture that the results will be similar to those shown for
vision seven.

On the other hand, considering the movement of the agents and other topologies
or lattice dimensions, we expect considerable changes in the system’s dynamics
with police officers. When Epstein presented this model in two dimensions, he
reported very particular dynamics for specific parameters, such as the
punctuated equilibrium phenomenon. Therefore, our results can serve future
research as a first approximation to characterize the results reported by
Epstein or even find new states or phenomena not yet reported in the current
literature.

It is essential to note that we base our interpretations on the numerical
results obtained from this simplified model. However, in the real world, the
dynamics of social protests are more complex and involve many other factors.
Nevertheless, studying this model from the perspective of sociophysics can
yield new elements that allow us to address the complexities of the dynamics
of social protest.

###### Acknowledgements.

I.O. thanks ANID (Chile) for financial support through Doctoral Scholarship
[BECAS-ANID/Doctorado Nacional/2019-21191380]. F.U. thanks FONDECYT (ANID-
Chile) for financial support through Postdoctoral N∘ $3180227$. We acknowledge
anonymous referees for their recommendations and constructive comments.

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