Controlling representation evolution in deep graph neural networks.

Deep Graph Neural Networks (GNNs) often degrade when many propagation layers are stacked. Although this phenomenon is commonly associated with over-smoothing, node-wise similarity alone does not fully describe deep representation failure. During propagation, node representations form a layer-wise trajectory that may become unstable, lose traceability to input information, or collapse into a low-dimensional subspace. Motivated by this view, we formulate deep GNN degradation as a problem of representation trajectory control. The framework integrates three complementary controls along a shared representation trajectory. The Stable Reaction-Diffusion encoder (SRD) provides a trajectory-stabilizing forward evolution mechanism that explicitly controls propagation drift and finite-depth sensitivity, rather than simply stacking additional message-passing layers. A deterministic depth-normalized traceability constraint (RDC) encourages intermediate input-space targets to remain traceable to the original features through graph-aware residual consistency. A Rank-Rate Constraint (RRC) penalizes excessive inter-layer decay of logarithmic numerical rank to slow premature subspace degeneration. We position the method as an integration and diagnostic framework rather than a new propagation operator. Theoretical analysis and experiments demonstrate the effectiveness, robustness, and stability of the proposed trajectory-control framework.