Wisdom of Crowds Effects under Antagonistic Interactions and Correlated Opinions

TL;DR

This work analyzes when and how the wisdom of crowds emerges in linear opinion dynamics on signed networks, extending DeGroot, Friedkin–Johnsen, and concatenated FJ models to antagonistic interactions. By leveraging PF properties for signed matrices and geometric wisdom regions, it derives conditions for mean accuracy and variance concentration, showing that signed networks generally enlarge the region of possible wisdom improvement compared to unsigned cases. It further characterizes how dependence among initial opinions reshapes optimal social-power allocations, sometimes necessitating negative weights, and provides continuous-time analogs with analogous wisdom criteria. The results offer a principled framework for understanding opinion aggregation under polarization, with implications for designing interventions in polarized or adversarial social systems.

Abstract

This paper investigates the wisdom of crowds of linear opinion dynamics models evolving on signed networks. Conditions are given under which models such as the DeGroot, Friedkin-Johnsen (FJ) and concatenated FJ models improve or undermine collective wisdom. The extension to dependent initial opinions is also presented, highlighting how the correlation structure influences the feasibility and geometry of the wisdom-improving regions.

Figures (4)

• Figure 1: Concentration regions $\Gamma_1$ and $\Gamma_2$ corresponding to an hyperellipsoid determined by the variances $\sigma_i^2 = \{ 6, \, 1, \, 1\}$. The hyperplane $\Psi_1$ is shown in pink, the $3$-simplex is shown in red, the region $\Gamma_1$ is in green while $\Gamma_2$ includes both the regions in blue and in green.
• Figure 2: Variance propagation for Example \ref{['example_fjconcatenatedfj']}
• Figure 3: Variance propagation (population-based average case) for Example \ref{['example_signedbipartitedegroot']}
• Figure 4: The green line represent the hyperplane $\Psi_1= \{ \mathds{1}^\top \mathbf{y} = 1 \}$ in $n=2$. The red line represents instead the hyperplane $\Psi_2= \{ \mathbf{v}^\top \mathbf{y} = 1 \}$ with $\mathbf{v}=[-1,\,1]^\top$. The convex region $\Gamma_6$ is the intersection of the green line and the ellipse $\Phi_3=\{n^2\mathbf{y}^\top\Sigma\mathbf{y}\leq \mathds{1}^\top\Sigma\mathds{1}\}$, the convex region $\Gamma_9$ is the intersection of the red line and the ellipse $\Phi_3$ with variances $\sigma_1^2=1$, $\sigma_2^2=4$ and covariances (a) $\rho_{12}=0$ (b) $\rho_{12}=0.55$ (c) $\rho_{12}=-0.4$. The associated optimal social powers (denoted $\mathbf{y}^\ast_{\rm uni}$ and $\mathbf{y}^\ast_{\rm bi}$ for the unipartite and bipartite (bipartition-based average) case), are also shown.

Theorems & Definitions (41)

• Definition 1
• Lemma 1
• Proposition 1
• Lemma 2
• Theorem 1
• Remark 1
• Example 1
• Example 2
• Example 3
• Lemma 3