Aggregation Dispersion: An Information-Geometric Diagnostic of Oversmoothing in Graph Neural Networks

Message-passing neural networks (MPNNs) learn node representations by iteratively aggregating information from neighbors, but stacking too many layers causes all representations to converge, a phenomenon known as oversmoothing. We propose a geometric perspective on this phenomenon. At each depth $k$, the propagation operator assigns to every node $v$ a walk distribution $p_v^{(k)}$, a probability distribution over all nodes describing how $v$ distributes its attention across the graph. These walk distributions form a point cloud on the probability simplex, and oversmoothing is the contraction of this cloud to a single point. To measure this contraction, we exploit the intrinsic geometry of the simplex: the Fisher--Rao metric, the unique Riemannian metric invariant under sufficient statistics, whose chordal distance under the square-root embedding $p \mapsto \sqrt{p}$ is the Hellinger distance. The resulting diagnostic, the \emph{aggregation dispersion} $\mathcal{D}_k$, is the variance of the embedded point cloud on the unit sphere. We prove that: (i) $\mathcal{D}_k = 0$ if and only if the $k$-th step propagation matrix has rank one, providing a necessary and sufficient condition for representational collapse; (ii) $\mathcal{D}_k$ is monotonically non-increasing with depth, so each layer irreversibly spends a finite diversity budget; (iii) the spectral gap of the propagation matrix controls the rate of this decay; and (iv) for contractive MPNNs, the aggregation dispersion upper-bounds the representation dispersion up to architecture-dependent constants, connecting the geometry of the simplex to the geometry of the feature space. In experiments across 14 model configurations and 11 datasets, $\mathcal{D}_k$ achieves the highest average Pearson correlation with accuracy degradation among all training-free diagnostics ($|r| = 0.721$), outperforming the post-training effective rank ($|r| = 0.688$).