Topology, noise, and parallel updates in circular opinion dynamics

Abstract

We study a circular opinion dynamics model with local midpoint interactions, extended to allow parallel updates of multiple sites. On a ring, the dynamics admits twisted states associated with integer winding numbers. We investigate how bi-modal noise, which drives opinions toward two antipodal directions, affects these configurations. Numerically, we find that noise both destabilizes winding states and induces a flip--flop regime, characterized by macroscopic switching between preferred orientations. We introduce order parameters that distinguish topological trapping from symmetry breaking, providing a simple macroscopic description of the dynamics.

Figures (7)

• Figure 1: One update of the ACCA dynamics under Algorithm 2. (a) A randomly chosen edge $(i,j)$ is updated by the circular midpoint rule, producing an intermediate state $\theta_{t}^{*}$. (b) Then a site $k$ is selected uniformly and its opinion is moved a fraction $\varepsilon$ toward one of the two antipodal targets $a\in\{0,\pi\}$, each chosen with probability $1/2$. In the schematic, the displacement is shown with $\varepsilon=0.35$ for visibility.
• Figure 2: ACCA dynamics with $N=100$ and $\varepsilon=0$ under periodic boundary conditions. (a) Configuration snapshots at four different times. (b) In the four panels are respectively plotted: the heatmap of the time evolution, the order parameters $R(t)$, $Y(t)$ and $\tau_1$ as function of time
• Figure 3: Heatmap of the time evolution, the order parameters $R(t)$, $Y(t)$ and $\tau_1$ as function of time for the ACCA dynamics with $N=100$ and $\varepsilon\neq 0$ under periodic boundary conditions. (a) $\varepsilon=0.002$; (b) $\varepsilon=0.02$.
• Figure 4: ACCA dynamics with $N=100$ and $\varepsilon=0$ under open boundary conditions. (a) Configuration snapshots at four different times. (b) In the four panels are respectively plotted: the heatmap of the time evolution, the order parameters $R(t)$, $Y(t)$ and $\tau_1$ as function of time
• Figure 5: ACCA dynamics with $N=100$ and $\varepsilon=0.002$ under open boundary conditions. (a) Configuration snapshots at four different times. (b) In the four panels are respectively plotted: the heatmap of the time evolution, the order parameters $R(t)$, $Y(t)$ and $\tau_1$ as function of time

Theorems & Definitions (3)

• definition 1: Order parameters $R$ and $\psi$
• definition 2: Weighted toroidal Kendall coefficient
• theorem 1