Nonequilibrium phase transitions in a racism-spreading model with interaction-driven dynamics

TL;DR

This work analyzes a three-state, interaction-driven framework for racism spread on networks, combining mean-field analysis on fully-connected graphs with agent-based simulations on Barabási–Albert and Watts–Strogatz topologies. The model yields three stationary regimes: two racism-free absorbing states and one endemic active phase, with continuous, DP-like transitions governed by thresholds $\beta_c^{(1)}=\mu$ and $\beta_c^{(2)}=(\delta+\mu)\frac{\eta}{\mu}-\delta$. Network structure modulates the precise thresholds and prevalence but leaves the qualitative phase structure robust across topologies. The results highlight how hubs and clustering influence intervention efficacy, and suggest topology-aware strategies (e.g., targeting hubs or enhancing local counter-speech) to suppress racist content in online environments. The analytical appendix provides explicit stationary solutions and stability conditions, reinforcing the connection between microscopic interaction rules and macroscopic phase behavior in nonequilibrium social contagion models.

Abstract

Racism remains a persistent societal issue, increasingly amplified by the structure and dynamics of online social networks. In this work, we propose a three-state compartmental model to study the spreading and suppression of racist content, drawing from epidemic-like dynamics and interaction-driven transitions. We analyze the model on fully-connected (homogeneous mixing) networks using a set of coupled differential equations, and on Barabási-Albert (BA) scale-free and Watts-Strogatz (WS) small-world networks through agent-based simulations. The system exhibits three distinct stationary regimes: two racism-free absorbing states and one active phase with persistent racist content. We identify and characterize the phase transitions between these regimes, discuss the role of network topology, and highlight the emergence of absorbing states. Our findings illustrate how statistical physics tools can help uncover the macroscopic consequences of microscopic social interactions in digital environments.

Figures (7)

• Figure 1: Illustration of the model's compartments: Susceptible (S), Infected (I) and Deniers (D).
• Figure 2: Time evolution of the agents' fractions in the three compartments $s, i$ and $d$ for the mean-field formulation of the model, obtained from the numerical integration of Eqs. (\ref{['eq5']}) - (\ref{['eq7']}). The fixed parameters are $\alpha=0.20, \delta=0.15$ and $\eta=0.18$, and we varied $\beta$ and $\mu$: (a) $\mu=0.10, \beta=0.20$, (b) $\mu=0.10, \beta=0.40$, (c) $\mu=0.30, \beta=0.20$ and (d) $\mu=0.30, \beta=0.40$.
• Figure 3: Stationary fractions $s, i$ and $d$ as functions of the contagion probability $\beta$ for the mean-field formulation of the model, obtained from the numerical integration of Eqs. (\ref{['eq5']}) - (\ref{['eq7']}). The fixed parameters are $\alpha=0.20, \delta=0.15$ and $\eta=0.18$, and we varied $\mu$: (a) $\mu=0.02$, (b) $\mu=0.10$, (c) $\mu=0.30$ and (d) $\mu=0.50$. The vertical (orange) dotted lines in panels (b) - (d) indicate analytically predicted critical values $\beta_{c}^{(2)}$ (panel (b)) and $\beta_{c}^{(1)}$ (panels (c) and (d)) associated with phase transitions (see the main text).
• Figure 4: Stationary fractions $s, i$ and $d$ as functions of the birth/death probability $\mu$ for the mean-field formulation of the model, obtained from the numerical integration of Eqs. (\ref{['eq5']}) - (\ref{['eq7']}). The fixed parameters are $\alpha=0.20, \delta=0.15$ and $\eta=0.18$, for two distinct values of $\beta$: (a) $\beta=0.10$ and (b) $\beta=0.30$. The vertical (orange) dotted lines indicate analytically predicted critical values associated with phase transitions (see the main text).
• Figure 5: Phase diagram of the model on the plane $\beta$ versus $\mu$. The fixed parameters are $\alpha=0.20, \delta=0.15$ and $\eta=0.18$. The diagram show the three macroscopic collective states, represented by phases I, II and III. The lines denote the critical points $\beta_c^{(1)}$ (dotted line) and $\beta_c^{(2)}$ (full line) given by Eqs. (\ref{['betac1']}) and (\ref{['betac2']}), respectively, as well the special value of $\mu=\eta$ where both critical points are equal (dashed line). This last line separates the two racism-free stationary solutions.