Standard models of opinion dynamics operate on a single dimension — each agent has a number representing their position on an issue, and interactions shift these numbers toward consensus or polarization. Real opinions are multidimensional: a person has views on economics, social policy, foreign affairs, and technology, and these views interact.
A multi-layer framework models multidimensional opinions with confirmation bias (arXiv:2603.21081). Each opinion dimension is a layer. Agents on one layer influence each other based on proximity in that dimension and in other dimensions. Confirmation bias means agents weight information from similar others more heavily — but “similar” is computed across all layers, not just the one being updated. Your economic opinion is influenced by people who agree with you on social policy.
The cross-layer coupling creates a mechanism for artificial correlation. Two opinion dimensions that are logically independent — say, views on climate change and views on gun control — become statistically correlated through the confirmation bias coupling. People who agree on one dimension preferentially influence each other on the other dimension. The correlation is not caused by any logical relationship between the topics. It's caused by the network structure of who listens to whom.
The resulting polarization is not just extremism on each dimension independently. It's the emergence of ideological packages — correlated bundles of opinions that cluster together despite no logical necessity for the clustering. The model reproduces the empirical pattern where positions on unrelated issues become predictable from knowing someone's position on a single issue.
The structural insight: ideological coherence is a network effect, not a logical one. Opinions on unrelated topics become correlated because the confirmation bias mechanism treats agreement on any dimension as evidence of trustworthiness on all dimensions. The clustering of opinions into “left” and “right” packages is a geometric consequence of how people filter information, not a reflection of any underlying structure in the issues themselves. The geometry of influence creates the geometry of belief.