Facebook: Non-Commuting Operators Explain Survey Response Order Effects

Facebook’s understanding of how opinions form is being challenged by a new model employing concepts from quantum physics; researchers report that existing frameworks fail to adequately explain phenomena like cognitive ambivalence and order effects. Weiqi Chu of the Department of Mathematics and Statistics, University of Massachusetts Amherst, and colleagues propose representing each person’s cognitive state not as a simple value, but with a density matrix encoding both expressed opinion and cognitive ambivalence. This approach treats survey questions as non-commuting operators, providing a principled explanation for order effects. The team observes that quantum coherence decays exponentially at a rate independent of the network, suggesting a universal dynamic in opinion formation beyond individual connections. They tested the framework on the Facebook-100 dataset and identified quantities without classical counterparts, such as quantum coherence and pairwise opinion covariances. Under a product state approximation, the quantum model reduces to the classical Friedkin, Johnsen opinion model. Numerical simulations revealed that while initial pairwise correlations between opinions exhibited network-dependent behavior, these dynamics consistently converged to a shared steady state regardless of the network.

Density Matrix Encoding of Opinion and Ambivalence

Human opinions may be better understood through the lens of quantum mechanics than classical probability. Researchers including Weiqi Chu of the Department of Mathematics and Statistics, University of Massachusetts Amherst, have proposed a novel model representing individual cognitive states not as simple values, but as density matrices, mathematical objects typically used to describe quantum systems. This approach directly addresses limitations in existing opinion dynamics models, which struggle to account for phenomena like cognitive ambivalence and order effects. The team’s work, published recently, moves beyond representing opinions as fixed points on a scale, instead encoding both expressed beliefs and internal conflicts within the density matrix. This allows for a more nuanced representation of how individuals genuinely hold seemingly contradictory views simultaneously. The researchers state that “survey questions become non-commuting self-adjoint operators, which provides a principled explanation for order effects.”

The model identifies quantities without classical counterparts, including quantum coherence and pairwise opinion covariances. Under a product state approximation, the quantum model reduces to the classical Friedkin, Johnsen opinion model. Numerical tests on both simulated and real-world networks, including data from the Facebook-100 dataset, demonstrate that quantum coherence decays exponentially at a rate independent of the network. This suggests a universal dynamic governing opinion formation, irrespective of social connections. The researchers found that pairwise correlations follow network-dependent transient dynamics but converge to the same steady state regardless of the network. The model provides a principled explanation for order effects, as the matrix encodes a stated opinion while simultaneously capturing cognitive ambivalence, acknowledging the inherent uncertainty in human judgment. This model gives access to quantities without classical counterparts, such as coherence magnitude and phase.

Current models of opinion dynamics typically treat individual beliefs as fixed points on a scale, or as vectors representing a stance on a particular issue. These classical approaches, while computationally efficient, struggle to account for complexities inherent in human cognition, specifically phenomena like cognitive ambivalence and order effects. Researchers have long observed that individuals can simultaneously hold conflicting views, and that the sequence in which survey questions are presented demonstrably alters responses, behaviors these simpler models cannot explain. A key limitation stems from the assumption that opinions are definite at any given time, adhering to the principles of classical probability. This framework fails to capture the nuanced uncertainty and internal conflict frequently present in human belief systems, allowing for a more sophisticated understanding of how opinions genuinely form and evolve. The team’s approach frames survey questions as providing a principled explanation for these observed inconsistencies, identifying quantities without classical counterparts, such as quantum coherence and pairwise opinion covariances.

This departs significantly from conventional models relying on scalar or vector representations of opinion. Weiqi Chu of the Department of Mathematics and Statistics, University of Massachusetts Amherst, and co-authors’ innovation lies in framing survey questions as a concept borrowed from quantum mechanics. Unlike classical models, this framework accounts for phenomena like the disjunction fallacy, where people irrationally assess probabilities. The model observes quantum coherence decays exponentially at a rate independent of the network, and identifies quantities without classical counterparts, such as quantum coherence and pairwise opinion covariances. Under a product state approximation, the quantum model reduces to the classical Friedkin, Johnsen opinion model. The model provides a principled explanation for order effects; this density matrix doesn’t just encode a stated opinion, it simultaneously captures cognitive ambivalence. This finding introduces a characteristic of the model, quantum coherence, without a classical counterpart. Numerical simulations revealed that while initial pairwise correlations between opinions exhibited network-dependent transient dynamics, these dynamics consistently converged to the same steady state regardless of the network, and the team tested the framework on the Facebook-100 dataset.

The ability to model shifting opinions with greater accuracy has significant implications for fields ranging from political forecasting to public health campaigns, and a new approach leverages tools from quantum physics to achieve this. Researchers, including Weiqi Chu of the Department of Mathematics and Statistics, University of Massachusetts Amherst, are moving beyond traditional opinion models which treat individual beliefs as fixed values, instead employing the Lindblad master equation, a framework originally developed to describe the behavior of open quantum systems, to account for cognitive ambivalence and the perplexing phenomenon of order effects in surveys. Unlike previous quantum models of opinion dynamics which focused on reversible processes, this framework explicitly addresses the irreversible nature of social influence and opinion crystallization. Under a product state approximation, the quantum model reduces to the classical Friedkin, Johnsen opinion model. This finding introduces quantities without classical counterparts, such as quantum coherence and pairwise opinion covariances, highlighting the potential for a unified theory of opinion dynamics by borrowing insights from the seemingly disparate realm of quantum mechanics.

Conventional models of opinion formation struggle to account for the nuances of human belief, particularly cognitive ambivalence and the perplexing effects where the sequence of questioning alters responses. These effects suggest opinions aren’t simply fixed values but fluid states. Researchers are now applying tools from quantum mechanics to address these limitations, moving beyond scalar or vector representations of opinion to a more sophisticated framework. Weiqi Chu of the Department of Mathematics and Statistics, University of Massachusetts Amherst, and colleagues’ approach centers on representing each individual’s cognitive state using a concept borrowed from quantum physics. The density matrix doesn’t just encode a stated opinion; it simultaneously captures internal conflicting views, acknowledging the inherent uncertainty in human judgment. This mathematical framing treats questions not as neutral probes, but as measurements that inherently influence the observed state.

A surprising link between quantum mechanics and social influence has emerged, revealing that the formation of opinions may be governed by principles traditionally reserved for the subatomic world. Researchers, including Weiqi Chu of the Department of Mathematics and Statistics, University of Massachusetts Amherst, have shown that under a product state approximation, the quantum model reduces to the classical Friedkin, Johnsen opinion model, suggesting a deeper underlying structure to how beliefs evolve. The implications extend beyond sociology, potentially offering new insights into the fundamental principles governing information processing and decision-making in complex systems, as well as a new way to model the evolution of beliefs.

Researchers are now moving beyond these limitations, exploring how cognitive ambivalence, simultaneous holding of conflicting views, and the order in which questions are presented impact expressed opinions. This density matrix approach allows for a more sophisticated analysis of relationships between individuals. Further investigation demonstrates that the model accurately reflects established classical opinion dynamics when simplified, specifically reducing to the Friedkin, Johnsen model under a product state approximation. Weiqi Chu of the Department of Mathematics and Statistics, University of Massachusetts Amherst, and colleagues tested their framework on both artificially constructed networks and the Facebook-100 dataset, confirming the model’s applicability across diverse social landscapes. This model gives access to quantities without classical counterparts, such as coherence magnitude and phase, offering new avenues for quantifying the subtleties of belief formation and influence.

This work, detailed in a recent publication, introduces a novel framework where each person’s cognitive state isn’t a single opinion, but a density matrix encoding both their expressed opinion and cognitive ambivalence, a significant departure from scalar or vector representations used previously. This model gives access to quantities without classical counterparts, such as coherence magnitude and phase, hinting at deeper underlying mechanisms driving opinion dynamics and potentially offering new avenues for understanding collective belief systems.

Numerical simulations revealed that while initial pairwise correlations between opinions exhibited network-dependent transient dynamics, these dynamics consistently converged to the same steady state regardless of the network, and that quantum coherence decays exponentially at a rate independent of the network. This suggests universal principles govern opinion formation, irrespective of specific social connections. A key finding centered on the decay of a uniquely quantum mechanical property absent in classical models, providing a quantifiable measure of how quickly individuals resolve conflicting viewpoints, offering a potential metric for understanding belief polarization. The researchers also explored how the model accounts for phenomena like order effects, where the sequence of survey questions influences responses. By framing survey questions as non-commuting self-adjoint operators, the model provides a principled explanation for order effects, moving beyond descriptive observations toward a predictive framework. These numerical tests confirm the model’s capacity to represent complex cognitive states and provide a foundation for future investigations into the quantum underpinnings of social dynamics.