Neural Hypernetworks: A Topological Framework

• Neural hypernetworks are high-order structures that generalize graphs to encode complex multi-way interactions using hypergraphs, posets, and simplicial complexes.
• They incorporate intrinsic geometric invariants, such as Forman Ricci curvature and the Euler characteristic, to quantify network structure.
• The framework facilitates persistent homology through canonical filtrations, enabling systematic analysis of hierarchical and multi-scale connectivity.

Neural hypernetworks are high-order mathematical structures that generalize traditional networks (graphs) to encode complex, multi-way relationships among entities. In recent research, hypernetwork theory has been systematically formalized using poset and simplicial complex machinery, which enables rigorous geometric and topological analysis. This abstraction provides intrinsic notions of curvature, notably Forman Ricci curvature, and allows for canonical association with simplicial complexes, thereby facilitating systematic applications to computational topology, such as persistent homology.

1. Hypernetworks as Hypergraphs, Posets, and Simplicial Complexes

Formally, a hypernetwork is modeled as a hypergraph $H = (V, E)$, where $V$ consists of hypervertices—typically sets of nodes—and %%%%2%%%% is a set of hyperedges, which connect groups of hypervertices. The principal mathematical insight is that every hypernetwork admits a canonical representation as a partially ordered set (poset) under set inclusion.

Within this framework, each element of the poset is assigned a rank that encodes hierarchical complexity, and totally ordered chains in the poset map directly to simplices in a canonical associated simplicial complex $\mathcal{A}(P)$. Specifically, an $m$-simplex corresponds to a chain of $m+1$ nested elements in the poset. This correspondence preserves the combinatorial and higher-order connectivity structure of the original hypernetwork and unifies the treatment of both directed and undirected situations.

2. Topological Invariants: Euler Characteristic

The Euler characteristic, a fundamental topological invariant, is canonically defined for the poset and its associated simplicial complex. For a ranked poset $P$, the Euler characteristic is computed as: $\chi_g(P) = \sum_{j=0}^{r} (-1)^j F_j$ where $F_j$ is the number of elements of rank $j$ and $r$ is the largest rank present. This definition is invariant under the specific model—whether poset, simplicial, or polyhedral complex—and reflects essential topological structure, including the number of components, cycles, and higher-dimensional holes.

3. Forman Ricci Curvature: Geometric Quantification

The geometric analysis of hypernetworks is advanced via Forman Ricci curvature, which is formulated for discrete cell complexes and extends naturally to hypernetworks represented as simplicial complexes. In the unweighted, combinatorial case for 2-dimensional complexes, the curvature of an edge $e$ is: $$\mathrm{Ric}_F(e) = \#\{\text{triangles } t^2 \supset e\} - \#\{\text{edges parallel to } e\} + 2$$ Parallel edges are those sharing a common parent (higher face) or child (lower face). For vertices and triangles, curvature-like terms $R_0(v)$ and $R_2(t)$ are defined. The Euler characteristic is then connected to local curvatures by a discrete Gauss-Bonnet-type theorem: $$\sum_{v \in F_0} R_0(v) - 2 \sum_{e \in F_1} \mathrm{Ric}_F(e) + \sum_{t \in F_2} R_2(t) = \chi(X)$$ where $F_0, F_1, F_2$ denote the sets of vertices, edges, triangles, respectively, and $X$ represents the simplicial complex.

4. Persistent Homology and Geometric Filtrations

This theory provides a robust foundation for geometric persistent homology. The poset-to-simplicial complex correspondence canonically constructs the filtered complexes required for persistent homology, enabling automated and systematic capture of higher-order connectivity. Filtrations can be controlled not only by purely combinatorial features (e.g., edge addition) but also by geometric invariants such as curvature, allowing the selection of subcomplexes based on refined geometric criteria.

This approach ensures that the persistent homology computation is sensitive to true multi-scale, hierarchical structure in hypernetworks, avoiding the limitations of ad hoc or local constructions. For low-rank hypernetworks, such as certain chemical reaction networks, both curvature and Euler characteristic are efficiently computable.

5. Extension to Higher Dimensions and Directed Structures

The mathematical machinery generalizes straightforwardly to higher-dimensional complexes beyond triangles, and to directed hypernetworks. The same principles—poset representation, associated simplicial complex, curvature definitions, and topological invariants—extend to any dimension, providing a powerful framework for modeling complex, multi-way interactions in various domains.

For directed hypernetworks, appropriate modifications of Forman Ricci curvature and the combinatorial structure have been established, as demonstrated in follow-up works. This renders the geometric and topological tools applicable in contexts requiring directionality or orientation, such as reaction networks or information flow systems.

6. Analytical Summary Table

Structure	Mathematical Representation	Main Invariant/Formulas
Hypernetwork	Hypergraph $H = (V, E)$	Poset via set inclusion
Poset	Ranked poset from hypernetwork	Euler characteristic $\chi_g(P)$
Simplicial complex	$\mathcal{A}(P)$ from poset	Chains map to simplices
Forman Ricci curvature	For edges and faces in $\mathcal{A}(P)$	$\mathrm{Ric}_F(e)$, $R_0(v), R_2(t)$
Gauss-Bonnet-type theorem	Discrete combinatorial sum	$$\sum_v R_0(v) - 2\sum_e \mathrm{Ric}_F(e) + \sum_t R_2(t) = \chi(X)$$
Persistent homology complexes	Canonical filtration via poset→simplicial	Geometric filtering by curvature/Euler characteristic

7. Significance and Directions

The poset-simplicial complex framework~(Saucan, 2021) enables rigorous translation of hypernetwork structure into topological and geometric domains. This directly supports systematic analysis of higher-order relationships, quantification of structural invariants, and efficient computation of persistent homology, while providing a pathway to discrete analogues of classical geometric theorems. The mathematical construction admits generalization to arbitrary dimension and directionality, and its practical simplicity facilitates application in computational topology and network science.

The approach substantially enriches hypernetwork analysis by moving beyond combinatorial tools: it canonically encodes hierarchy and multi-way relations and leverages discrete curvature for network characterization and filtering. The resulting geometric perspective strengthens both theoretical understanding and computational tractability, advancing persistent homology and curvature-based methods for complex system analysis.