Phase diagram in a one-dimensional civil disorder model

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.

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 among 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.

Figure 2

Agents' concentration variation on time. Panel (a) corresponds to a system without police officers and (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.

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.

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 passive in phase PA. The dashed line between these two phases shows when the system has similar concentrations and is 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 (a), (b), and (c). Then, a continuous transition from PA phase to consensus, as seen in (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 (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 (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.

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 (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 (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.

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 $C_{\mathrm{ap}}$ 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 (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 (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.

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 (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 (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.

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 in (a). 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 (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 (a), (b), and (c). However, in the random neighborhood, only observe order-with majority phases and then a consensus phase in (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.

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.