Bounded-Confidence Models of Multi-Dimensional Opinions with Topic-Weighted Discordance

People's opinions on a wide range of topics often evolve over time through their interactions with others. Models of opinion dynamics primarily focus on one-dimensional opinions, which represent opinions on one topic. However, opinions on various topics are rarely isolated; instead, they can be interdependent and correlated. In a bounded-confidence model (BCM) of opinion dynamics, agents are receptive to each other only if their opinions are sufficiently similar. We extend classical agent-based BCMs -- namely, the Hegselmann--Krause BCM, which has synchronous interactions, and the Deffuant--Weisbuch BCM, which has asynchronous interactions -- to a multidimensional setting, in which the opinions are multidimensional vectors representing opinions of different topics and opinions on different topics are interdependent. To measure opinion differences between agents, we introduce topic-weighted discordance functions that account for opinion differences in all topics. We define regions of receptiveness for our models, and we use them to characterize the steady-state opinion clusters and provide an analytical approach to compute these regions. In addition, we numerically simulate our models on various networks with initial opinions drawn from a variety of distributions. When initial opinions are correlated across different topics, our topic-weighted BCMs yield significantly different results in both transient and steady states compared to baseline models, where the dynamics of each opinion topic are independent.