# Polarization in Political Rallies: A Markovian Agent-Based Model for Opinion Dynamics
## 4 Reference Scenario and Proposed Model
\begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt}} \hline
**Parameter** & **Definition** & **Notes** \\ \hline \(\pi_{0}^{Sp}\) & Initial distribution of the speaker’s state probability & The speaker has a single state associated with opinion \(op0\), therefore the initial state probability distribution will be \([1]\) and will remain unchanged throughout the observation period. \\ \hline \(\pi_{0}^{Li}\) & Listeners in class \(L_{i}\) initial state distribution & Listeners have two states, each associated with opinion \(op0\) and \(op1\), respectively. The initial state probability distribution is \([1\;0]\) for listeners in class \(L_{0}\) and \([0\;1]\) for listeners in class \(L_{1}\). This distribution will be influenced by the opinions of the speaker and neighboring listeners throughout the observation period. \\ \hline \(\mathbf{Q}^{Sp}\) & Intrinsic behavior of the speaker & Infinitesimal generator matrix of the CTMC is equal to \(\mathbf{0}\). The speaker cannot spontaneously change their opinion throughout the entire observation period. This is a trivial consequence of the fact that the speaker was modeled with a single state. \\ \hline \(\mathbf{Q}^{L_{i}}\) & Intrinsic behavior of listeners in class \(L_{i}\) & Infinitesimal generator matrix of the CTMC is equal to \(\mathbf{0}\). We assumed the listeners in both classes \(L_{0}\) and \(L_{1}\) cannot spontaneously change their opinion throughout the entire observation period. \\ \hline \(\mathbf{\Lambda}^{Sp}\) & The rate at which the speaker attempts to influence the listeners’ opinions & It is a vector of size 1, representing the rate at which the speaker’s Markov chain reenters its only state, \(op0\), and sends messages to all listeners. Higher rates imply stronger influence \\ \hline \(\mathbf{\Lambda}^{L_{i}}\) & The rate at which the listeners in class \(L_{i}\) attempt to influence their neighbors’ opinions & It is a \(2\times 2\) matrix, representing the rate at which the listener’s Markov chain reenters both states, \(op0\) and \(op1\), and sends messages to neighboring listeners. The diagonal element at position \((0,0)\) represents the rate to induce \(op0\), while the diagonal element at position \((1,1)\) represents the rate to induce \(op1\). Off-diagonal elements must be 0. Higher rates imply stronger influence \\ \hline \(\mathbf{\Lambda}^{L_{i}}(m_{sp})\) & The probability with which a listener belonging to class neighbor with opinion \(op0\) and changes their opinion from \(op1\) to \(op0\) \\ \hline \(\mathbf{\Lambda}^{L_{i}}(m_{nc0})\) & The probability with which a listener belonging to class \(L_{i}\) accepts the influence of a neighbor with opinion \(op0\) and changes their opinion from \(op1\) to \(op0\). \\ \hline \(\mathbf{\Lambda}^{L_{i}}(m_{nc1})\) & The probability with which a listener belonging to class \(L_{i}\) accepts the influence of a neighbor with opinion \(op1\) and changes their opinion from \(op1\) to \(op1\). \\ \hline \end{tabular}
\end{table} 
Table 2: Model parameters. When these agents are deployed over a space \(\mathcal{V}\), the perception function \(\mathbf{u}_{m}()\) has to be defined between each couple of them for all the exchanged messages \(m\). The values of \(u_{m}(\mathbf{v},C,s,\mathbf{v}^{\prime},C^{\prime},s^{\prime})\) represent the probability that an agent of class \(C\), in position \(\mathbf{v}\), and in state \(s\), perceives a message \(m\) generated by an agent of class \(C^{\prime}\) in position \(\mathbf{v}^{\prime}\) in state \(s^{\prime}\). We assume the spread of the agents belonging to both the classes \(L_{0}\) and \(L_{1}\) over a \((n\times n)\) squared grid with one agent per grid cell; on the other hand, only one agent in the class \(Sp\) is deployed. A position will be denoted by a couple \(\mathbf{v}=(x,y)\), with \(x,y\in\{0,\cdots,n-1\}\). Whereas the messages emitted by the agent in class \(Sp\) are perceived by all the others, we assume agents in the grid interact only with their immediate neighborhoods.