Universality of noise-induced transitions in nonlinear voter models

TL;DR

By mapping a broad family of nonlinear voter models with symmetric absorbing states to a canonical GV framework and then extending it with noise that removes absorbing states, the paper reveals two primary phase-transition lines: a continuous Ising-type line and a discontinuous MGV line meeting at a tricritical point. Finite-size scaling across complete graphs, random networks, and 2D lattices shows that continuous transitions belong to the Ising universality class for the respective dimensionalities, while the tricritical point obeys mean-field tricritical exponents. The authors unify noisy and nonnoisy variants of nonlinear voter dynamics, including partisan and q-voter generalizations, aging, and Sznajd-like interactions, under a single analytical scheme. This framework clarifies how nonlinearity and noise shape collective opinion dynamics and provides a practical tool for identifying universality classes in stochastic opinion-models.

Abstract

We analyze the universality classes of phase transitions in a variety of nonlinear voter models. By mapping several models with symmetric absorbing states onto a canonical model introduced in previous studies, we confirm that they exhibit a Generalized Voter (GV) transition. We then propose a canonical mean-field model that extends the original formulation by incorporating a noise term that eliminates the absorbing states. This generalization gives rise to a phase diagram featuring two distinct types of phase transitions: a continuous Ising transition and a discontinuous transition we call Modified Generalized Voter (MGV). These two transition lines converge at a tricritical point. We map diverse noisy nonlinear voter models onto this extended canonical form. Using finite-size scaling techniques above and below the upper critical dimension, we show that the continuous transition of these models belongs to the Ising universality class in their respective dimensionality. We also find universal behavior at the tricitical point. Our results provide a unifying framework for classifying phase transitions in stochastic models of opinion dynamics with both nonlinearity and noise.

Figures (15)

• Figure 1: Left panel: Phase diagram in the space $(B,A)$ for the different regimes of the potential $V(m)$ sketched at each region. Transition lines: Ising, DP: Directed Percolation, GV: Generalized Voter. Dashed lines correspond to $A=0$ and $A=B$ delimiting the region for which the potential presents three minima. Right panel: Location of the minimum of the potential $V(m)$ versus the parameter $A$ for $B=-0.5$ (top) and $B=0.5$ (bottom) corresponding, respectively, to the gold and silver vertical arrows of the left panel. For $B\leq0$, solid (resp. dotted) line corresponds to the absolute (resp. relative) minimum of $V(m)$.
• Figure 2: Paths in the parameter space $(B,A)$ for the nonlinear voter model (n$\ell$ VM, $\varepsilon=0$) and the nonlinear partisan voter model (n$\ell$ PVM, $\varepsilon>0$). Paths are obtained with Eqs. (\ref{['eq:NLPVM']}), which reduce to Eqs. (\ref{['eq:NLVM']}) for $\varepsilon=0$, varying $\alpha$ for several values of the preference $\varepsilon$, as indicated in the legend. The vertical grey dashed line is generated fixing $\alpha=1$ and increasing $\varepsilon$, namely $A=-\varepsilon,\,B=0$. Inset: Zoom of the path for $\varepsilon=0$. Intersections of the path with the vertical axis $B = 0$, occurring at $A=1/2, B=0$.
• Figure 3: Phase diagram in the parameter space $(\alpha,\varepsilon)$ for the nonlinear partisan voter model. Transition line correspond to the GV transition given by Eq. \ref{['eq:NLPVM_epsilonc']}.
• Figure 4: Paths in the parameter space $(B,A)$ for the $q$-voter model, generated from Eqs. (\ref{['eq:q-voter_A']},\ref{['eq:q-voter_B']}) increasing the switching probability $\epsilon$, as indicated by the arrows, for several values of $q$. For $q=2,3$ the paths are vertical at $B=0$, namely $A=\frac{1}{2}-\epsilon$ for $q=2$ and $A=\frac{1}{2}-\frac{3}{2}\epsilon$ for $q=3$, and cross the VM point $A=B=0$.
• Figure 5: Paths in the parameter diagram $(B,A)$ for the voter model with aging, obtained by increasing $p_\infty$ as indicated by the arrows, for $\tau^*=1$ and several values of $p_0$. For $p_\infty>p_0$, all curves nearly overlap.