Persistence probability based dynamics and phase diagrams in biased q-voter models

Technical Summary: Persistence Probability in Biased q-Voter Models

Problem Statement
This study investigates the persistence probability within opinion dynamics, specifically focusing on two nonlinear $q$-voter models with binary opinions ($S = \pm 1$) on a fully connected network. Persistence probability, $P(t)$, is defined as the fraction of agents who have not changed their initial opinion up to time $t$. While persistence has been extensively studied in physical systems (e.g., Ising models, diffusion) and various opinion dynamics models (e.g., standard Voter model), there has been no prior analysis of persistence in $q$-voter models. The authors aim to determine how the persistence probability evolves over time, how it depends on model parameters (specifically bias mechanisms and panel size $q$), and how its long-time behavior correlates with the stationary states (fixed points) of the underlying dynamics.

Methodology
The authors analyze two specific extensions of the $q$-voter model:

1. The DMSS Model: A generalized model where, if a $q$-panel is not unanimous, the target agent flips their opinion with a probability dependent on their current state ($\epsilon_\uparrow$ for $-1 \to +1$ and $\epsilon_\downarrow$ for $+1 \to -1$).
2. The MS Model: A model where non-unanimous panels influence the target agent based on a weighted average of opinions, with positive opinions carrying weight $p$ and negative opinions carrying weight $1-p$.

The study employs two primary approaches:

• Mean Field Theory (MFT): Analytical rate equations are derived for the density of positive opinions $c(t)$ and the persistence probabilities $P_+(t)$ and $P_-(t)$. Exact closed-form solutions are obtained for specific cases: $q=2$, $q=3$, and the limit $q \to \infty$. For $q \ge 4$, the integrals required for exact solutions generally involve elliptic functions, so the authors restrict exact analytical treatment to the solvable cases.
• Monte Carlo Simulations: Numerical simulations are performed to compute persistence probabilities for various parameter ranges, including $q=2, 3, 5, 50$, and to validate the analytical results.

The authors define separate persistence probabilities for agents starting with positive ($P_+$) and negative ($P_-$) opinions and examine their long-time behavior ($t \to \infty$).

Key Contributions and Results

• Analytical Solutions: The paper provides closed-form analytical expressions for $P_+(t)$ and $P_-(t)$ for both models in the limits of $q=2$, $q=3$, and $q \to \infty$. These solutions are expressed in terms of the time-dependent density $c(t)$ or directly as functions of time for specific parameter regimes.
• Long-Time Behavior: The study finds that the long-time behavior of persistence probability is generally either a saturation to a finite value or an exponential decay.
  • Saturation: Occurs when the system reaches a consensus state (all agents adopting the same opinion) that is stable. In these cases, the persistence probability for the consensus opinion saturates to a non-zero value.
  • Exponential Decay: Occurs when the system evolves toward a non-consensus (mixed) steady state or when the initial opinion is not the one leading to consensus. The decay is well-approximated by the form $P(t) \sim \alpha + \beta e^{-\gamma t}$.
• Phase Diagrams: The authors construct phase diagrams in the parameter space (e.g., $\epsilon_\uparrow$ vs. $\epsilon_\downarrow$ for DMSS, and $p$ for MS). These diagrams map regions where persistence saturates (indicating a stable consensus state for that opinion) versus regions where it decays to zero.
  • For the DMSS model, the phase boundaries depend on the initial density $c_0$. The persistence phase boundary aligns with the unstable fixed point separating the basins of attraction for positive and negative consensus.
  • For the MS model, the asymptotic behavior is determined solely by the parameter $p$ (specifically whether $p > 1/2$ or $p < 1/2$), regardless of the initial density $c_0$ (provided $c_0 \neq 0, 1$).
• Correlation with Fixed Points: A central finding is the strong correlation between the persistence probability's long-time behavior and the stability of the system's fixed points. If a consensus state is a stable fixed point, the persistence probability for that opinion saturates; otherwise, it decays. This relationship is described as non-trivial, as persistence is a dynamic property not typically determined solely by equilibrium properties.
• Model Comparison:
  • In the DMSS model, the decay rate of the minority persistence varies continuously with the bias parameters.
  • In the MS model, the decay rate of the minority persistence is found to be nearly independent of the bias parameter $p$ and the initial density $c_0$, exhibiting a universal exponential decay rate in the studied range.
  • The MS model shows negligible dependence on $q$ for persistence behavior, consistent with its steady-state behavior, whereas the DMSS model exhibits significant $q$-dependence.

Significance and Claims
The paper claims to provide the first study of persistence probability in $q$-voter models. It highlights that unlike in some other opinion dynamics models (like the standard Voter model in 1D or Ising models) where power-law behaviors are observed, the biased $q$-voter models on fully connected networks exhibit exponential decay or saturation.

The authors emphasize that while persistence probability is generally not determined by equilibrium properties, these specific models present an "interesting exception" where the saturation of persistence is directly linked to the existence of stable consensus fixed points. The study demonstrates that in realistic scenarios where no full consensus is reached (non-consensus steady states), opinions formed purely through interaction are bound to change within finite times, leading to an exponential decay of persistence. The work bridges the gap between the static properties of fixed points and the dynamic evolution of agent persistence, offering a nuanced view of how bias and network structure (via $q$) influence the longevity of initial opinions.