Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs

TL;DR

The paper argues that graphs and even standard higher-order models like hypergraphs fail to simultaneously capture set-type and hierarchical interior-to-boundary relations in complex data. It introduces combinatorial complexes (CCs) as a flexible framework that combines a rank-based hierarchy with set-type relations, enabling a Dirac-like operator $ ext{D}_ ext{CC}$ and joint modeling across multiple orders. CCs generalize both hypergraphs and cell complexes, clarifying when each perspective is advantageous and offering a practical learning advantage demonstrated by a signal-processing experiment. This approach has implications for spectral topological signal processing by providing richer, multi-layer representations that better reflect the geometry and topology of data.

Abstract

Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively represent the complex relations found in high-dimensional data. Such higher-order domains are typically modeled either as hypergraphs, or as simplicial, cubical or other cell complexes. In this context, cell complexes are often seen as a subclass of hypergraphs with additional algebraic structure that can be exploited, e.g., to develop a spectral theory. In this article, we promote an alternative perspective. We argue that hypergraphs and cell complexes emphasize \emph{different} types of relations, which may have different utility depending on the application context. Whereas hypergraphs are effective in modeling set-type, multi-body relations between entities, cell complexes provide an effective means to model hierarchical, interior-to-boundary type relations. We discuss the relative advantages of these two choices and elaborate on the previously introduced concept of a combinatorial complex that enables co-existing set-type and hierarchical relations. Finally, we provide a brief numerical experiment to demonstrate that this modelling flexibility can be advantageous in learning tasks.

Figures (2)

• Figure 1: Abstractions of relational data via graphs, hypergraphs and cell complexes.Left: A graph may be presented via a bipartite structure comprising a set of vertices and a set of edges. Each edge is associated with exactly two nodes. Middle: A hypergraph can also be represented by a bipartite structure with a set of vertices and a set of (hyper)edges. In contrast to the case of graphs, the hyperedges do not have a cardinality constraint, i.e., we relax the constraints on the connectivity of each (hyper)edge in the bipartite representation. Right: a cell complex can in general not be presented via a bipartite structure, but represents a multi-partite structure, in which all but the out-most layers are connected both "downwards" and "upwards" via boundary-to-interior type relations (this is simply the Hasse diagram of the complex). Note in particular that, e.g. a 2-simplex (triangle (1,2,3) above) does not correspond to a direct relation between three nodes, rather it is a relation between three edges (1-cells). In general, there is only an indirect coupling between cells of order $k$ and $k'$ if $|k-k'|>1$. Similarly to the case of graphs, there are certain cardinality constraints with respect to the connectivity between the layers (e.g., a 1-simplex always just links two nodes).
• Figure 2: Illustration of a combinatorial complex.Left: Schematic drawing of a combinatorial complex. Black dots indicate rank-0 cells (vertices), cyan shapes indicate rank-1 cells, and light green indicates rank-2 cells. Note that vertex 2 is directly attached to the left rank-2 cell, without being part of a rank-1 cell that is subordinated to this rank-2 cell. Such a set-type hierarchical configuration is not possible for both cell complexes and hypergraphs. Right: a Hasse-like diagram of the multi-partite structure viewpoint of the combinatorial complex. Note that there are maps between cells of adjacency rank ($B_{01}$, $B_{12}$) similar to standard cell complexes. Furthermore, there is also a mapping $B_{02}$ from the rank-$2$ cells directly to rank-$0$ cells (in this case signified by the dashed violet link). The transposes of these maps naturally correspond to the (co-boundary) maps in the reverse direction.

Theorems & Definitions (6)

• Definition 1: Hypergraph
• Definition 2: Abstract simplicial complex
• Definition 3: Regular cell complex
• Definition 4: Set-type relation
• Definition 5: Rank function
• Definition 6: Combinatorial complex