Modeling Public Opinion Dynamics: The Spiral of silence in clustered homophilic networks

TL;DR

The paper addresses how public opinion voice emerges in networks shaped by choice homophily and triadic closure, formalizing Noelle-Neumann’s spiral of silence within a realistic, co-evolving network framework. It couples a network-formation model with mean-field and Q-learning opinion-expression dynamics, and validates insights through Monte Carlo simulations. Key findings show that moderate clustering supports minority expression by strengthening local cohesion, while excessive clustering intensifies asymmetries and majority dominance, with hysteresis arising from saddle bifurcations and learning dynamics. The results have practical implications for online platforms and offer a foundation for empirical validation and extension to richer behavioral strategies.

Abstract

Public discourse emerges from the interplay between individuals' willingness to voice their opinions and the structural features of the social networks in which they are embedded. In this work we investigate how choice homophily and triadic closure shape the emergence of the spiral of silence, the phenomenon whereby minority views are progressively silenced due to fear of isolation. We advance the state of the art in three ways. First, we integrate a realistic network formation model, where homophily and triadic closure co-evolve, with a mean-field model of opinion expression. Second, we perform a bifurcation analysis of the associated Q-learning dynamics, revealing conditions for hysteresis and path dependence in collective expression. Third, we validate our theoretical predictions through Monte Carlo simulations, which highlight the role of finite-size effects and structural noise. Our results show that moderate triadic closure can foster minority expression by reinforcing local cohesion, whereas excessive closure amplifies asymmetries and entrenches majority dominance. These findings provide new insights into how algorithmic reinforcement of clustering in online platforms can either sustain diversity of opinion or accelerate its suppression.

Figures (17)

• Figure 1: Pure-strategy Nash equilibrium regimes. The equilibria are abbreviated by e for opinion expression and by s for silence. The first entry in the '(,)' label correspond to the behaviour of $G_1$ and the second for $G_2$. When the structural parameter $T_{ii}$ is smaller that $\xi$ (for reference, $\xi = 0.2$ in the figure), the group is silent. As $T_{ii}$ grows, the group is more willing to express if the other group is silent. When both $T_{11}$ and $T_{22}$ are bigger than $\frac{\xi + 1}{2}$, a new phase is possible where both groups expresses.
• Figure 2: Phase diagram for the expression of one group. Colored regions represent parameter combinations where the phase of the group is guaranteed, regardless of the behavior of the other group. The hatched region indicates that a specific phase is guaranteed, but coexistence is also possible depending on the state of the other group.
• Figure 3: Hysteresis cycle in expression dynamics. As the intra-connectivity grows, new state arise. However, for lower temperatures, the system might remain trapped in previous states, leading to hysteresis.
• Figure 4: Network formation mechanism. (Left) At each time step an edge is created and another is removed. In this example, an edge between an agent in $G_1$ and another in $G_2$ is created while an edge between the agent in $G_1$ and another agent in $G_3$ is removed. This is a process $1-2-3$. (Top right) The node in $G_1$ connects with the node in $G_2$ through a common neighbour in the process known as triadic closure. (Bottom right) The node in $G_1$ now connects with the node in $G_2$ due to other possible mechanism.
• Figure 5: Homophilic amplification network. Predicted fraction of neighbours in the same group $G_{1}$ as a function of choice homophily. Groups have equal size ($n_1= n_2=0.5$) and equal choice homophily ($s_{11}=s_{22} = s$, $s_{12}=s_{21}=1-s$). Inset: example of the typical structure that appears in homophilic amplification fixed-points. Both groups are well connected internally and a only a few inter-groups links are created.