Collapsed Effective Operators for Higher-order Structures

Research Question and Approach

Traditional spectral graph theory relies on the graph Laplacian to analyze pairwise interactions. However, many complex systems—such as protein complexes, neural circuits, and collaborative networks—exhibit polyadic (higher-order) interactions that simple graphs cannot capture. While higher-order structures like simplicial complexes and hypergraphs exist, their spectral operators (e.g., the Hodge Laplacian) are rank-specific, meaning they operate on edges, faces, or higher-dimensional cells separately. This creates a "fusion problem": practitioners must use ad hoc heuristics to aggregate these multi-rank signals back to the vertex level for node-level tasks like classification or clustering.

The authors propose a principled solution: Collapsed Effective Operators. By treating the entire topological structure as a graded system, they construct a "Graded Laplacian" that couples all ranks. They then use the Schur complement to "integrate out" the higher-order degrees of freedom, effectively collapsing the multi-rank information into a single, vertex-level operator. This operator acts as a topologically-aware filter that encodes how higher-order cells mediate vertex dynamics.

The authors demonstrate that their collapsed operator is positive semi-definite and provides a clear spectral interpretation. By reducing the effective conductance between nodes that share common higher-order cells, the operator captures non-local dependencies that standard graph Laplacians miss.

Empirically, the method shows significant utility across several domains:

• Spectral Clustering: It improves the identification of clusters by incorporating higher-order connectivity.
• Signal Smoothing: It acts as a selective filter that preserves topological features while removing noise.
• Protein Analysis: It successfully recovers secondary protein structures (like helices and sheets) that are otherwise invisible to purely geometric, graph-based methods.