Where Latents Belong and How They Flow: Joint Identification of Structural Roles

Latent factors in dynamical systems can play different roles across observations and time. For example, an idea in a group meeting may be expressed by only some participants, inherit information from earlier discussions, and shape what is said next. We formalize this notion as the structural role of a latent factor, describing where it is expressed, whether it receives influence from the past, and whether it transmits influence to the future. The main challenge is correspondence, since the three structures may be identified under different permutations, while a structural role is defined only when they are aligned to the same latent coordinate. We show that, under suitable structural and variability conditions, all three roles are jointly identifiable under a common permutation at each time point. Guided by this theory, we develop RoleTrace, a framework that traces structural roles in a shared latent coordinate system. Synthetic experiments validate structural recovery, while experiments on multiparty meetings examine the identified roles in real sequential interactions.