Affective Polarization on Small-World and Scale-Free Networks

Abstract

Affective polarization, the emotional divide characterized by in-group love (trust towards fellow partisans) and out-group hate (mistrust towards those with opposite political views), has become prevalent in the current society. Despite its prevalence, the role of social network structure in the dynamics of affective polarization is yet to be well-understood. We provide a mean-field approximation of opinion dynamics under affective polarization on Watts-Strogatz and power-law (scale-free) networks. Our results show that consensus is fragile in social networks with power-law degree distributions, and the smaller average path length of the network (resembling a small-world network) makes achieving the consensus further difficult. Simulations and numerical experiments on real-world networks indicate that the mean-field model is aligned with the actual dynamics. Our findings shed light on how real-world network properties shape the dynamics of affective polarization and why consensus remains elusive in the real-world.

Figures (26)

• Figure 1: Four example trajectories illustrating each case of Theorem \ref{['TheoremSW']} for $\theta^\mathcal{B}(0) = \theta^\mathcal{R}(0) = 0.6$. The long-term outcomes are: (case 1) No Polarization, (case 2/case 3) Partisan Polarization, and (case 4) Non-Partisan Polarization.
• Figure 2: Two example trajectories illustrating how a smaller rewiring constant $p$ facilitates reaching consensus. In all trajectories $\theta^\mathcal{B}(0) = \theta^\mathcal{R}(0) = 0.6, r = 0.8, \alpha = 0.4, \beta = 0.2$.
• Figure 3: Discrete simulation of the model (column-2) and mean-field approximation of the model (column-1). Fraction of people in the blue (row-2) and red (row-1) group of each bin that adopted choice-1 in the final state is plotted. For each case the following conditions were fixed: $d_{min} = 1, d_{max} = 100, \alpha = 0.6, \beta = 0.6, \eta_\mathcal{R} = 2.3, \eta_\mathcal{B} = 2.1, \gamma_\mathcal{R} = \gamma_\mathcal{B} = 2.4, \theta_{f,e}^\mathcal{B}(0) = 0.2, \theta_{f,e}^\mathcal{R}(0) = 0.8$. This figure shows that the mean-field approximation replicates the dynamics of the discrete model under the conditions above.
• Figure 4: Discrete simulation of the model (column-2) and mean-field approximation of the model (column-1). Fraction of people in the blue (row-2) and red (row-1) group of each bin that adopted choice-1 in the final state is plotted. For each case the following conditions were fixed: $d_{min} = 1, d_{max} = 100, \alpha = 0.6, \beta = 0.6, \eta_ \mathcal{R}= \eta_\mathcal{B} = 2.2, \gamma_\mathcal{R} = \gamma_\mathcal{B}= 2.4, \theta_{f,e}^\mathcal{B}(0) = \theta_{f,e}^\mathcal{R}(0) = 0.6$. This figure shows that the mean-field approximation is similar to the dynamics of the discrete model under the conditions above.
• Figure 6: Dynamics on Configuration Model for the following conditions: $d_{min} = 1, d_{max} = 50, \alpha = \beta = 0.6, \eta_\mathcal{R} = \eta_\mathcal{B} = 2.1, \gamma_\mathcal{R} = \gamma_\mathcal{B} = 2.5, \theta_{f,e}^\mathcal{B}(0) = \theta_{f,e}^\mathcal{R}(0) = 0.7$.

Theorems & Definitions (2)

• Theorem 1
• Proposition 1.1