When heterogeneity drives hysteresis: Anticonformity in the multistate $q$-voter model on networks

TL;DR

The paper addresses whether quenched anticonformity preserves discontinuous phase transitions in the multistate $q$-voter model beyond the complete graph. It combines pair approximation and Monte Carlo simulations on random, random regular, and Barabási–Albert networks to compare quenched and annealed implementations. The main finding is that quenched anticonformity induces discontinuous transitions with hysteresis for $S\ge 3$ and $q\ge 2$, and the hysteresis width grows with network density, whereas annealed dynamics tend to continuous transitions with limited hysteresis in sparse graphs; PA predictions align with simulations mainly in the quenched dense regime. These results imply that fixed heterogeneity in social responses can qualitatively alter collective opinion dynamics and tipping-point behavior on realistic networks.

Abstract

Discontinuous phase transitions are closely linked to tipping points, critical mass effects, and hysteresis, phenomena that have been confirmed empirically and recognized as highly important in social systems. The multistate $q$-voter model, an agent-based approach to simulate discrete decision-making and opinion dynamics, is particularly relevant in this context. Previous studies of the $q$-voter model with anticonformity on complete graphs uncovered a counterintuitive result. Changing the model formulation from the annealed (homogeneous agents with varying behavior) to quenched (heterogeneous agents with fixed behavior) produces discontinuous phase transitions. This is contrary to the common expectation that quenched heterogeneity smooths transitions. To test whether this effect is merely a mean-field artifact, we extend the analysis to random graphs. Using pair approximation and Monte Carlo simulations, we show that the phenomenon persists beyond the complete graph, specifically on random graphs and Barabási-Albert scale-free networks. The novelty of our work is twofold: (i) we demonstrate for the first time that replacing the annealed with the quenched approach can change the type of phase transitions from continuous to discontinuous not only on complete graphs but also on sparser networks, and (ii) we provide pair-approximation results for the multistate $q$-voter model with competing conformity and anticonformity mechanisms, covering both quenched and annealed cases, which had previously been studied only in binary models.

Figures (6)

• Figure 1: Symbolic scheme of the updating procedure in the annealed (left panel) and quenched (right panel) three-state $q$-voter model with anticonformity, where $\blacktriangle$, $\bullet$, and $\blacksquare$ denote different states, black-filled voters mark anticonformists, white-filled voters mark conformists. In the annealed version the voters are gray to highlight that their behavior is randomly decided during an update. The neighbors chosen for the $q$-panel are placed inside gray ellipses. In this example, $q=2$.
• Figure 2: Schematic representation of relevant variables tracked in the 3-state $q$-voter model with annealed disorder.
• Figure 3: Schematic representation of relevant variables tracked in the 3-state $q$-voter model with conformists $\circ$ and anticonformists $\bullet$ in quenched disorder.
• Figure 4: The influence of graph density on phase transitions within PA and simulations. Stationary values of state concentrations (of the highest and lowest occupied state) obtained from PA (solid lines) and Monte Carlo simulations on random graphs (markers) for the three-state ($S=3$) $q$-voter model with $q=2$ in the annealed (left panels: a, c, e) and quenched (right panels: b, d, f) approaches. Empty symbols correspond to results obtained from an initially disordered state ($1/3$ of voters in each state at $t=0$), whereas filled symbols correspond to the initial condition where all voters are in the same state. Panels (a)-(b) show results for $\langle k \rangle = 16$, (c)-(d) for $\langle k \rangle = 50$, and (e)-(f) for $\langle k \rangle = 150$. The shaded area highlights the hysteresis region obtained from PA.
• Figure 5: The influence of graph type with a fixed density of $\langle k\rangle = 16$ on phase transitions within PA and simulations. Stationary values of state concentrations (of the highest and lowest occupied state) obtained from PA (solid lines) and Monte Carlo simulations (markers) for the three-state $q$-voter model with $q=3$ and $\langle k\rangle = 16$ on Barabási–Albert (top panels: a, b), random regular (middle panels: c, d) and random (bottom panels: e, f) graphs in the annealed (left: a, c, e) and quenched (right: b, d, f) disorder. Empty shapes mark results obtained from initial disorder, i.e. $1/3$ of voters in each state at $t=0$, while filled shapes correspond to the initial condition in which all voters are in the same state. Shaded area highlights the hysteresis region obtained from PA.