Persistent Tri-State Message Passing

In stochastic message passing, an edge's sampled role changes the node states used to compute adaptive weights at later layers. Weight averaging therefore depends on whether edge roles persist across layers or are resampled at each layer. We study this dependence in Persistent Tri-State Message Passing (P3MP), which combines persistent additive, subtractive, and inactive roles with weights computed from each sample's endpoint states. For two layers, we separate target-state feedback from weight-source covariance and derive the condition under which local and shared weights reverse their preactivation ordering. Shared weights retain a 1/K fraction of the feedback for K samples. The interaction is zero for a source-only scorer. Permuting complete weight vectors between samples quantifies the same role-state dependence while preserving their empirical distribution. Exact enumeration and feature-only checkpoint measurements agree with the analytical predictions. Across independently trained feature-only models, local weights yield higher accuracy and lower negative log-likelihood under persistence, while shared weights yield higher accuracy and lower loss under layerwise resampling. The same accuracy ordering holds on deduplicated Chameleon and Squirrel.