Opinion Dynamics in Learning Systems

Abstract

We propose and analyze a unified framework that interleaves peer-to-peer opinion dynamics with performative effects of learning systems. While network theory studies how opinions evolve via social connections, and performative prediction examines how learning systems interplay with individuals' opinions, neither captures the emergent dynamics when these forces co-evolve. We model this interplay as a recursive feedback loop: a platform's predictions influence individual opinions, which then evolve through social interactions before forming the training data for the next platform model update. We demonstrate that this co-evolution induces a novel equilibrium that qualitatively differs from standard network equilibria. Specifically, we show that standard predictive objectives act as a ``homogenizing force" driving networks toward consensus even under conditions where classical opinion-dynamics models lead to disagreement. Further, we demonstrate how learning under partial observations creates spillover effects among individuals, even if individuals are not susceptible to peer-influence. Finally, we study a platform that systematically deviates from standard predictive objectives, and demonstrate how classical opinion-dynamics models underestimate the equilibrium response to node-level interventions. We complement our theoretical findings with semi-synthetic simulations on social network data. Combined, our results illuminate performativity as an important, so far neglected, qualifying factor in social networks.

Figures (5)

• Figure 1: Peer-platform co-influence in opinion dynamics: (A) Individuals interact with a digital platform and update their opinion based on the recommended content they are exposed to. (B) Individuals' opinions evolve through peer interaction. (C) The platform observes expressed opinions and updates their recommendations.
• Figure 2: In $(a)$, we show how performativity homogenizes opinions across retraining steps. The $x$-axis denotes the retraining step $t$, and the $y$-axis corresponds to individuals' expressed opinions in each times step, where the error bars indicate the variance. In (b), we show how platform susceptibility decreases variance of opinions at equilibrium. The $x$-axis denotes the varying homogeneous platform susceptibility while the $y$-axis denotes the variance of opinions in equilibrium.
• Figure 3: In $(a)$, we show how a stubborn individual $q\in U$ with $\alpha_q=0$ and $x^*_q=0$ is influenced by their peers across retraining steps for two different scenarios. In $(b)$, we show how indirect platform influence increases over retraining steps. We compare different the opinion of the stubborn individual $l$ with $\beta_l=0$. The purple dashed line denotes the opinion of the stubborn individual if we set $\beta_k=0$, $k\notin \{l\}\cup S$.
• Figure 4: We show how peer susceptibility decreases variance at equilibrium. The $x$-axis denotes the varying homogeneous peer susceptibility while the $y$-axis denotes the variance of opinions in equilibrium.
• Figure 5: We show the individual $l\in U$ with $\alpha_l=0$ converges to different equilibria under different innate opinions of individuals in set $O$. The $x$-axis denotes the retraining steps, and the $y$-axis denotes the opinions of individual $l$. Left: The platform uses the mean estimation as in Theorem \ref{['theorem:consensus_mean_estimation']}. Right: The platform uses MLP for platform predictions.

Theorems & Definitions (26)

• Proposition 1: Existence of a unique performatively stable equilibrium
• Proposition 2: Convergence under repeated perfect prediction
• Proposition 3: Homogenizing force of performativity
• Theorem 1: Consensus through performativity
• Theorem 2: Peer influence via platform
• Proposition 4: Consensus in the limit
• Theorem 3: Indirect platform influence
• Lemma 1
• proof
• Lemma 2