Hodge Spectral Concentration in Simplicial Complex Neural Networks

Simplicial neural networks extend learning on graphs to higher-order topological structures, but feature propagation in these networks remains less well understood. We develop a spectral framework and characterize these dynamics through the Hodge decomposition. Rather than summarizing feature evolution with a single energy, as used to characterize over-smoothing and -sharpening in graph neural networks, we track the feature-normalized lower and upper Hodge Rayleigh quotients separately. This decouples spectral concentration from changes in feature magnitude and retains information about the gradient and curl subspaces, allowing us to distinguish three forms of spectral concentration: harmonic-dominant, gradient-dominant, and curl-dominant dynamics. For linear simplicial convolutions with independent, symmetric lower- and upper-channel mixing weights, we show that the dynamics are a gradient flow of a learnable energy and derive three competing spectral growth rates associated with the harmonic and highest-frequency gradient and curl eigenspaces. We then show that these rates determine the asymptotic regime for almost every input. Since they are computed from the learned weights and Hodge frequencies, these rates provide a post-training diagnostic of which eigenspace the normalized features concentrate in. We instantiate our framework with normalized propagation choices, including an adjacency-like form that recovers graph convolution at rank zero. Finally, experiments on synthetic edge classification and ocean-drifter trajectory prediction tasks validate the spectral predictions and illustrate how supervision and the choice of propagation operator select among the qualitatively distinct spectral concentration regimes.