# Polarization in Political Rallies: A Markovian Agent-Based Model for Opinion Dynamics
## 4 Reference Scenario and Proposed Model  
The scenario analyzed in this paper is inspired by an electoral rally, where a political candidate addresses an audience, promotes the ideas of their platform, and seeks to persuade individuals to vote for them. The audience is made up of people with different initial political beliefs: the same political party as the speaker and the opposite one. When the rally begins, listeners belonging to the speaker's political faith will have his/her same opinion and a high propensity to accept the message; listeners of contrary faith, on the other hand, will have a lower propensity to change their opinion by listening to the speaker's arguments. Since the listeners share the same environment, they influence one another in an attempt to sway those holding different opinions. Obviously, the candidate could have higher oratory ability and a greater strength of conviction than the people in the audience. Spatial distribution also plays a role: while the candidate influences everyone, the attendees can only interact with neighborhoods. For representing the scenario mentioned above, we used three classes of agents: one for modeling the speaker and two for representing the two types of listeners as shown in Figure 3. Emission and perception of messages are detailed in the following. Class \(Sp\) is used to model the speaker. The only agent in class \(Sp\) is modeled by a single state CTMC representing the speaker's opinion, let's say \(Opinion_{0}\), that cannot change. The classes \(L_{0}\) and \(L_{1}\) are used for modeling individuals; they may have one opinion between two: either \(Opinion_{0}\), the same as the speaker, or \(Opinion_{1}\). Accordingly, agents in both classes \(L_{0}\) and \begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Model** & **Main Assumptions** & **Special Features** & **Limitations** & **Typical Applications** \\ \hline
**DeGroot** & Continuous opinions; fixed network; deterministic averaging & Simple consensus dynamics; analytical tractability in steady state & No transient analysis & Trust modeling; information diffusion; collaborative decision-making \\
**[PERSON]** & Stubbornness; fixed influence weights; continuous time & Models resistance to influence; convergence dynamics & Requires symmetric networks for closed-form solutions & Group influence analysis; sociological modeling \\
**Hegselmann-Krause** & Bounded confidence; synchronous updates; discrete time & Opinion clustering; tolerance modeling & Relies on simulations; limited analytical results for realistic cases & Political fragmentation; echo chamber formation; public discourse analysis \\
**Deffuant-Weisbuch** & Pairwise bounded confidence; random interactions & Captures gradual belief change; probabilistic awareness & Primarily simulation-based; lacks spatial awareness & Cultural drift; polarization dynamics; social contagion \\ MAM (This work) & Markovian agents; message-driven transitions; spatially distributed & & More complex model message-level modeling; fully analyzable mapping & Political rallies; emergency crowd dynamics; spatio-temporal influence propagation \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of representative opinion dynamics models. \(L_{1}\) are characterized by two states, _op0_ and _op1_, modeling the two possible opinions an individual may have. Classes \(L_{0}\) and \(L_{1}\) differ for different initial probability vectors \(\mathbf{\pi}_{0}^{L_{0}}\) and \(\mathbf{\pi}_{0}^{L_{1}}\), so that individuals with different opinions are easily implemented in the model. \(Sp\) class agent generates messages \(m_{sp}\) through a self-loop with rate \(\lambda_{S}\); \(\lambda_{S}\) is the rate characterizing the ability of the speaker to change the other's mind. Attendees also try to move others toward their own opinions, interacting with them. However, we assumed that a single individual cannot spontaneously change his own opinion.