# Polarization in Political Rallies: A Markovian Agent-Based Model for Opinion Dynamics
###### Abstract
Moreover, agent-based models can incorporate mechanisms such as bounded confidence, social reinforcement, and external media influence, which are difficult to represent within traditional social dynamics frameworks [5; 6]. As such, agent-based models offer a complementary and often indispensable approach to studying opinion dynamics, particularly when the goal is to understand how local interaction rules scale up to system-wide behavioral patterns. Traditional models, such as the Friedkin--Johnsen framework, have been extensively used to analyze opinion dynamics. However, these models often face limitations in capturing complex interactions and achieving the scalability required for real-world scenarios. This paper proposes a novel approach to studying opinion polarization using the Markovian agents modelling paradigm. This method offers enhanced scalability and flexibility, enabling the dynamic analysis of polarized opinions within heterogeneous groups. By studying a political rally scenario--where a speaker attempts to influence a diverse audience--our analysis explores how spatial distribution, individual predispositions, and interaction patterns shape the formation and evolution of polarized opinions. The findings aim to provide deeper insights into opinion dynamics and contribute to more effective strategies for managing polarization in social systems. The proposed Markovian agent-based model is a computational framework exploiting continuous-time Markov chains (CTMCs). Here, we want to use it to analyze how individuals in structured groups influence each other's opinions. Agents transit between discrete states (e.g., Support or Oppose) based on interaction rules, spatial proximity, and predefined influence rates. The framework exhibits the following key features:  
* Markovian Agents: Each agent (e.g., a politician or rally attendee) is modeled as a CTMC. Transitions between opinion states occur at specific rates (e.g., a politician persuades listeners at rate \(\lambda_{s}\)) (i.e., a speaker at a rally emits messages to convert opponents; listeners switch opinions stochastically, influenced by the speaker's rate (\(\lambda_{s}\)) and their neighbors' opinions--Figure 1a). * Spatial Interactions: Agents interact based on their physical or network proximity (e.g., grid neighbors) (i.e., listeners in a crowd influence only adjacent neighbors; central listeners receive more messages and shift opinions faster than those at the edge--Figure 1b). * Message-Passing: Agents send \"messages\" (e.g., arguments) during state transitions or while holding an opinion. Reception depends on perception probabilities (i.e., a listener in Oppose sends messages to nearby agents to convert them, but neighboring Support agents might reject these with probability \(\alpha\)--Figure 1c). * Analytical Solution: Unlike simulation-heavy models, the proposed approach uses coupled differential equations to compute opinion distributions over time (i.e., with 400 listeners and one speaker, the proposed approach solves 801 equations to predict how many support the speaker at any given moment, avoiding costly Monte Carlo simulations). In addition to offering a solid theoretical framework for analyzing opinion dynamics, the proposed approach also presents several key strengths that highlight its versatility and effectiveness across a range of applications:  
* Transient Analysis: It enables the tracking of opinion evolution in real time, allowing questions such as \"How many supporters will the speaker gain in the next two hours?\" to be addressed. * Spatial Dynamics: The model captures spatial phenomena such as opinion clustering (e.g., resistant groups forming in corners of a crowd) and can represent hierarchical roles within the population, distinguishing between leaders and followers. * Scalability: It efficiently handles large populations (up to 10,000 agents) by relying on analytical equations rather than computationally expensive simulations.