Geometric Features of Higher-Order Networks via the Spectral Triplet

TL;DR

The paper develops a non-commutative, Connes-inspired geometric framework for higher-order networks by embedding simplicial complexes into a spectral triplet $(\mathcal{A},\mathcal{H},D)$ and using the Dirac operator $D$ to derive dimension, distance, and curvature via spectral data and the heat kernel. It defines a multi-order spectral-dimension vector $d_s$ and employs the spectral action to extract curvature terms, alongside a Connes-type distance $\mathrm{dist}_{C,\mathcal{N}}$ for simplicial complexes. The approach is demonstrated on Bach’s musical datasets, where dimension estimates, spectral vs. Forman curvature comparisons, and Connes distances between notes reveal how harmonic structure maps to a curved, higher-dimensional space. The results suggest that geometric measures provide complementary, scalable descriptors for higher-order networks and can guide understanding of complex system structure and dynamics across domains.

Abstract

Our work is concerned with simplicial complexes that describe higher-order interactions in real complex systems. This description allows to go beyond the pairwise node-to-node representation that simple networks provide and to capture a hierarchy of interactions of different orders. The prime contribution of this work is the introduction of geometric measures for these simplicial complexes. We do so by noting the non-commutativity of the algebra associated with their matrix representations and consequently we bring to bear the spectral triplet formalism of Connes on these structures and then notions of associated dimensions, curvature, and distance can be computed to serve as characterizing features in addition to known topological metrics.

Figures (28)

• Figure 1: To compute the Forman edge-level curvature, we start with the Hodge Laplacian $L$ over edges, and construct the Bochner Laplacian $L'$ obtained by zeroing out the diagonal of $L$ and replacing it with the row-wise sum of the absolute value of the off-diagonal entries. The Forman curvature matrix is then obtained by subtracting $L'$ from $L$: $F = L - L'$. To compute vertex-level curvature, or scalar curvature, the curvature contribution of all edges incident to a vertex is summed.
• Figure 2: Visualization of the temporal evolution and cumulative construction of the simplicial complex
• Figure 3: Evolution of the heat kernel expansion scalar curvatures over vertices for Fugue from Sonata No. 3
• Figure 4: Evolution of Forman-Ricci node curvatures versus time for Fugue from Sonata No. 3
• Figure 5: Evolution of the average scalar curvature using our proposed method for the Fugue from Sonata No. 3.