Analysis of a continuous opinion and discrete action dynamics model coupled with an external observation dynamics

TL;DR

This work studies how a continuous-opinion, discrete-action (CODA) model (on a fixed network) interacts with an external observation dynamics that represents urban pollution. It introduces a coupled discrete-time framework with a linear pollution state $p(k+1)=\gamma p(k) + \sum_i e_i(k)$ and a quantized environmental signal $q_p(k)$, with a coupling parameter $\beta$ controlling the influence of environment versus neighbors on opinions. The authors derive explicit opinion equilibria $\theta_i^* = (1-\beta)\frac{2 n_i^{+*}-n_i}{n_i} + \beta q_p^*$ (i.e., $f_i^*$) and the corresponding external equilibrium $p^*$ under stationarity, and analyze when the external signal is stationary leading to CODA-like behavior; they further classify regimes where actions are preserved via weak/strongly robust polarized clusters and study the Fully Synchronized (FS) regime, showing possible chaos or limit cycles for $\beta>0.5$. Numerical experiments on square lattices and complete graphs corroborate the theoretical predictions, illustrating polarization, action preservation, and the transition between steady states, chaos, and limit cycles as $\beta$ varies. The results highlight how external information signals can qualitatively reshape opinion dynamics and action stability in pollution-driven social systems, with potential implications for information design and policy intervention in urban settings.

Abstract

We consider a set of consumers in a city or town (who thus generate pollution) whose opinion is governed by a continuous opinion and discrete action (CODA) dynamics model. This dynamics is coupled with an observation signal dynamics, which defines the information the consumers have access to regarding the common pollution. We show that the external observation signal has a significant impact on the asymptotic behavior of the CODA model. When the coupling is strong, it induces either a chaotic behavior or convergence towards a limit cycle. When the coupling is weak, a more classical behavior characterized by local agreements in polarized clusters is observed. In both cases, conditions under which clusters of consumers don't change their actions are provided.Numerical examples are provided to illustrate the derived analytical results.

Figures (4)

• Figure 1: Bifurcation diagram of the opinion for $0.5<\beta<1$. $N=20$, $\theta_{}(0)= 0.4$, $p(0)=100$, $\bar{p} = 15$, $e_\text{min} = 0$, $e_\text{max} =1$ and $\gamma = 0.5$,
• Figure 2: Visualization of Opinion Dynamics on a 50 × 50 Square Lattice for $\beta = 0.45$: Initial opinions are randomly distributed as i.i.d. uniform variables between -1 and 1, with the resultant opinions after 100 iterations represented by each colored square cell. Agents engage in communication with their adjacent cells (above, below, left, and right). The cells marked with crosses indicate the presence of strongly robust polarized clusters; black crosses correspond to action -1, and white crosses denote action 1.
• Figure 3: Depiction of Dynamical Evolution in the 50 × 50 Square Lattice from Figure \ref{['fig:beta_inf_05']}: The upper panel showcases the trajectory of each agent's opinion over iterations, while the lower panel illustrates the corresponding state evolution. The red dashed line marks the state threshold.
• Figure 4: Trajectory dynamics. Left: two possible equilibria. Center: no equilibrium or limit cycle. Right: limit cycle.

Theorems & Definitions (23)

• Lemma 1
• proof
• Proposition 1
• proof
• Corollary 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof