SuperHyperGraph Foundations for Artificial Intelligence, Machine Learning, and Neural Networks

1. Introduction

1.1. Graph, Hypergraph, and Superhypergraph Theory

Graph theory offers a rigorous framework for analyzing discrete relational systems: a set of vertices is connected by edges, and the resulting structure naturally captures pairwise interactions [1,2]. Owing to the clarity and flexibility of this abstraction, graphs have become standard models in many disciplines, including data mining, algorithm design, artificial intelligence, neural computation, quantum information, and chemistry (see, e.g., [3,4]). However, requiring every edge to link exactly two vertices may be inadequate when the underlying phenomenon involves genuine multiway relations or layered (hierarchical) organization. This motivates the parallel development of hypergraph and superhypergraph theories, which enlarge the classical setting in complementary directions.

A hypergraph replaces ordinary edges by hyperedges, where each hyperedge may connect an arbitrary nonempty subset of the vertex set, thus encoding higher-order relations directly [5,6,7]. As a strict generalization of graphs, hypergraph theory has grown into a mature area with substantial structural and algorithmic results and with a wide range of applications (see, e.g., [8,9,10]). Beyond this, a superhypergraph introduces explicit nesting by iterating the powerset operation; this produces multi-level collections of vertex-sets and edge-sets and thereby allows one to represent hierarchical groupings and multi-layer connectivity within a single combinatorial object [11,12,13,14,15]. Accordingly, superhypergraphs extend hypergraphs and are particularly suitable for modeling complex systems with several interacting layers [16,17,18].

Table 1 highlights the principal differences among graphs, hypergraphs, and superhypergraphs. Throughout, n denotes a finite positive integer. We also emphasize that extensive algorithmic literature exists for these models, ranging from classical graph algorithms to modern extensions and adaptations (see, e.g., [19,20]).

Note that graphs, hypergraphs, and superhypergraphs admit natural directed counterparts—Directed Graphs [21,22], Directed HyperGraphs [23,24], and Directed SuperHyperGraphs [25]—and these directed models can be further extended to bidirected versions, namely Bidirected Graphs [26,27,28], Bidirected HyperGraphs [22], and Bidirected SuperHyperGraphs [22]. Such directed and bidirected variants enrich the theory by explicitly incorporating orientation information into the underlying combinatorial structures.

1.2. HyperGraph and SuperHyperGraph for Artificial Intelligence

It is needless to say that research in machine learning and artificial intelligence has become extremely active in recent years and now plays a very significant role in modern society. Graph, HyperGraph, and SuperHyperGraph are also used in the field of Artificial Intelligence. Examples of related concepts are presented in Table 2.

1.3. Our Contributions

From the above discussion, research based on SuperHyperGraphs is important and is expected to play a major role in fields such as AI. Accordingly, this paper extends several fundamental frameworks-including GCN, GraphRAG, causal graphs, graph embedding, graph-based natural kanguage processing and graph generation-by incorporating the SuperHyperGraph viewpoint. Some of these parallel notions are summarized in Table 2.

2. Preliminaries

This section introduces the notation and basic terminology used in the sequel. Unless explicitly stated otherwise, all graphs and hypergraphs in this paper are assumed to be finite.

2.1. SuperHyperGraphs

A hypergraph generalizes an ordinary graph by allowing an edge to connect any nonempty subset of the vertex set. Hence, hypergraphs provide a direct language for modeling multiway interactions [5,6,86,87]. By iterating the powerset operation one step further, one obtains nested (higher-order) vertex objects and, consequently, a finite SuperHyperGraph whose vertices and edges may themselves be set-valued at multiple levels [16,18,40,88]. Such hierarchical representations are useful, for instance, in molecular design, complex-network analysis, and related applications [57,89,90]. Throughout, the index n in ${\mathrm{PS}}_n(·)$ and in an n-SuperHyperGraph is always taken to be a nonnegative integer.

Definition 1 

(Base set). A base set S is the underlying universe of discourse:

$$S=\{x\mid x\mathrm{is}\mathrm{an}\mathrm{admissible}\mathrm{object}\mathrm{in}\mathrm{the}\mathrm{context}\mathrm{under}\mathrm{consideration}\}.$$

All sets appearing in $\mathrm{PS}\left(S\right)$ and in the iterated powersets ${\mathrm{PS}}_n\left(S\right)$ are ultimately built from elements of S.

Definition 2 

(Powerset). (see [91,92,93]) For a set S, the powerset of S is

$$\mathrm{PS}\left(S\right)=\{A\mid A\subseteq S\}.$$

In particular, $\varnothing \in \mathrm{PS}\left(S\right)$ and $S\in \mathrm{PS}\left(S\right)$.

Definition 3 

(Hypergraph). [29,94] A hypergraph is a pair $H=(V,E)$ such that:

• V is a finite set (thevertices), and
• E is a finite family of nonempty subsets of V (thehyperedges).

Thus, a hyperedge may involve more than two vertices, capturing genuinely multiway relations.

Definition 4 

(n-th powerset). [95,96] Let X be a set. Define ${\mathrm{PS}}_1\left(X\right):=\mathrm{PS}\left(X\right)$ and, for every $n\ge 1$,

$${\mathrm{PS}}_{n+1}\left(X\right)=\mathrm{PS}\left({\mathrm{PS}}_n\left(X\right)\right).$$

When it is convenient to exclude the empty set, we write

$${\mathrm{PS}}_n^{*}\left(X\right)={\mathrm{PS}}_n\left(X\right)\setminus \{\varnothing \}.$$

Definition 5 

(n-SuperHyperGraph). (see [15,97]) Let $V_0$ be a finite, nonempty base set. Define

$${\mathrm{PS}}^0\left(V_0\right):=V_0,{\mathrm{PS}}^{k+1}\left(V_0\right):=\mathrm{PS}\left({\mathrm{PS}}^k\left(V_0\right)\right)(k\in \mathbb{N}).$$

For $n\ge 0$, ann-SuperHyperGraph on $V_0$ is a pair

$${\mathrm{SHG}}^{\left(n\right)}=(V,E)$$

satisfying

$$V\subseteq {\mathrm{PS}}^n\left(V_0\right)\mathrm{and}E\subseteq \mathrm{PS}\left(V\right)\setminus \{\varnothing \}.$$

Elements of V are calledn-supervertices, and elements of E are calledn-superedges (i.e., each n-superedge is a nonempty subset of V).

3. Review: SuperHyperGraph Neural Network

A HyperGraph Neural Network learns node representations by aggregating features through hyperedges, capturing higher-order interactions among multiple vertices for prediction tasks [5]. A SuperHyperGraph Neural Network learns base-vertex representations by aggregating through nested hyperedges, capturing hierarchical hyperedge-of-hyperedge interactions for complex reasoning tasks [33].

Definition 6  (HyperGraph Neural Network (HGNN)).

[5] Let $G=(V,E,W)$ be a finite undirected hypergraph, where

$$V=\{v_1,\dots ,v_n\},E=\{e_1,\dots ,e_m\},W=\mathrm{diag}(w_1,\dots ,w_m),$$

with $w_j>0$ the (hyperedge) weight of $e_j$. Let $X\in {\mathbb{R}}^{n\times d}$ be the input feature matrix whose i-th row $x_i\in {\mathbb{R}}^d$ is the feature vector of vertex $v_i$.

Define the incidence matrix $H\in {\{0,1\}}^{n\times m}$ by

$$H_{ij}=\left\{\begin{array}{cc}1,\hfill & v_i\in e_j,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Define the diagonal degree matrices $D_V\in {\mathbb{R}}^{n\times n}$ and $D_E\in {\mathbb{R}}^{m\times m}$ by

$${\left(D_V\right)}_{ii}=\sum _{j=1}^m{H}_{ij}{w}_j,{\left(D_E\right)}_{jj}=\sum _{i=1}^n{H}_{ij}.$$

Let $\Theta \in {\mathbb{R}}^{d\times c}$ be a learnable weight matrix and let $\sigma (·)$ be a nonlinearity (e.g., ReLU). A single HGNN convolutional layer is the map

$${\mathrm{HGNN}}_{\Theta }:{\mathbb{R}}^{n\times d}\to {\mathbb{R}}^{n\times c},Y={\mathrm{HGNN}}_{\Theta }\left(X\right):=\sigma \left(D_V^{-{\displaystyle \frac{1}{2}}}HW{D}_E^{-1}{H}^{\top }{D}_V^{-{\displaystyle \frac{1}{2}}}X\Theta \right).$$

By stacking L layers, one obtains embeddings $X^{\left(0\right)}=X$ and

$$X^{(\ell +1)}=\sigma \left(D_V^{-{\displaystyle \frac{1}{2}}}HW{D}_E^{-1}{H}^{\top }{D}_V^{-{\displaystyle \frac{1}{2}}}{X}^{(\ell )}{\Theta }^{(\ell )}\right),\ell =0,1,\dots ,L-1.$$

Definition 7 

(n-SuperHyperGraph Neural Network (n-SHGNN)). Let $H^{\left(n\right)}=\left(V^{\left(n\right)},E^{\left(n\right)}\right)$ be an n-SuperHyperGraph over a finite base vertex set $V_0$ (so the ultimate objects of interest are the base vertices in $V_0$). Define the expanded hypergraph$H^{\prime }=(V_0,E^{\prime })$ by declaring that a subset $e^{\prime }\subseteq V_0$ belongs to $E^{\prime }$ if and only if there exists a superhyperedge $e\in E^{\left(n\right)}$ such that

$$e^{\prime }=\bigcup _{v\in e}v.$$

(Here each $v\in e$ is treated as a subset of $V_0$ via the natural “flattening/union” operation.)

Let $X\in {\mathbb{R}}^{|V_0|\times d}$ be the input feature matrix whose i-th row $x_i\in {\mathbb{R}}^d$ is the feature vector of base vertex $v_i\in V_0$. Let $E^{\prime }=\{e_1^{\prime },\dots ,e_{m^{\prime }}^{\prime }\}$ and define the incidence matrix $H^{\prime }\in {\{0,1\}}^{|V_0|\times m^{\prime }}$ by

$$H_{ij}^{\prime }=\left\{\begin{array}{cc}1,\hfill & v_i\in e_j^{\prime },\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let $w\left(e_j^{\prime }\right)>0$ be a learnable weight for $e_j^{\prime }$, and set

$$W=\mathrm{diag}\left(w\left(e_1^{\prime }\right),\dots ,w\left(e_{m^{\prime }}^{\prime }\right)\right)\in {\mathbb{R}}^{m^{\prime }\times m^{\prime }}.$$

Define diagonal degree matrices $D_V\in {\mathbb{R}}^{|V_0|\times |V_0|}$ and $D_E\in {\mathbb{R}}^{m^{\prime }\times m^{\prime }}$ by

$${\left(D_V\right)}_{ii}=\sum _{j=1}^{m^{\prime }}{H}_{ij}^{\prime }w\left(e_j^{\prime }\right),{\left(D_E\right)}_{jj}=\sum _{i=1}^{|V_0|}{H}_{ij}^{\prime }.$$

Let $\Theta \in {\mathbb{R}}^{d\times c}$ be learnable and let $\sigma (·)$ be a nonlinearity. A single n-SHGNN convolutional layer is the map

$$\mathrm{n}-{\mathrm{SHGNN}}_{\Theta }:{\mathbb{R}}^{|V_0|\times d}\to {\mathbb{R}}^{|V_0|\times c},Y=\mathrm{n}-{\mathrm{SHGNN}}_{\Theta }\left(X\right):=\sigma \left(D_V^{-{\displaystyle \frac{1}{2}}}{H}^{\prime }W{D}_E^{-1}{H^{\prime }}^{\top }{D}_V^{-{\displaystyle \frac{1}{2}}}X\Theta \right),$$

where $Y\in {\mathbb{R}}^{|V_0|\times c}$ is the updated base-vertex feature matrix.

4. Review and Results: HyperGraphRAG and SuperHyperGraphRAG

We examine HyperGraphRAG and SuperHyperGraphRAG. Retrieval-augmented generation combines external knowledge retrieval with neural text generation, grounding model outputs in relevant, up-to-date context for accurate responses[98,99,100,101]. GraphRAG is retrieval-augmented generation using a graph linking context chunks or entities; retrieval seeds similar nodes, expands along edges to gather connected evidence, serializes the subgraph, then an LLM generates answers [58,59,102,103,104].

Definition 8 

(GraphRAG: graph-based retrieval-augmented generation). Let $\mathcal{D}$ be a finite corpus and let $\mathcal{C}=\{c_1,\dots ,c_N\}$ be a finite set of atomic contexts (e.g., chunks, passages, tables, figure captions) extracted from $\mathcal{D}$. Let $\mathcal{Q}$ be a set of queries.

A GraphRAG system is a tuple

$$\mathrm{GRAG}=\left(\mathcal{C},\mathrm{\Phi },G,{Score}_V,Expand,Select,Serialize,\mathcal{M}\right),$$

where:

(i) 

$\mathrm{\Phi }:\mathcal{C}\cup \mathcal{Q}\to {\mathbb{R}}^d$ is an embedding map.

(ii) 

$G=(V,E)$ is a finite graph whose vertex set V indexes retrievable units (e.g., $V=\mathcal{C}$, or $V=\mathcal{C}\cup \mathcal{E}$ with entities $\mathcal{E}$). Each edge $\{u,v\}\in E$ encodes a binary relation between u and v (e.g., co-mention, hyperlink, citation, temporal adjacency, or a knowledge-graph triple).

(iii) 

${Score}_V:V\times \mathcal{Q}\to \mathbb{R}$ is a vertex scoring function; a typical choice is

$${Score}_V(v;q)={\displaystyle \frac{\langle \mathrm{\Phi }\left(v\right),\mathrm{\Phi }\left(q\right)\rangle }{\parallel \mathrm{\Phi }\left(v\right)\parallel ·\parallel \mathrm{\Phi }\left(q\right)\parallel }}(v\in V,q\in \mathcal{Q}),$$

with the convention that $\mathrm{\Phi }\left(v\right)$ means the embedding of the context (or entity) represented by v.

(iv) 

$Expand:\mathcal{P}\left(V\right)\times \mathcal{Q}\to \mathcal{P}\left(V\right)$ is an expansion operator that enlarges a seed set of vertices using the graph structure (e.g., k-hop neighborhood, weighted random walk, or personalized PageRank).

(v) 

$Select:\mathcal{P}\left(V\right)\times \mathcal{Q}\to \mathcal{P}\left(V\right)$ is a selection operator producing the final vertex set to be used as evidence.

(vi) 

$Serialize:(G,S)\mapsto \mathrm{a}\mathrm{finite}\mathrm{text}\mathrm{string}$ converts a vertex-induced subgraph $G\left[S\right]$ (and optionally edge labels) into an LLM-readable context (e.g., ordered passages, entity summaries, relation triples).

(vii) 

$\mathcal{M}$ is a conditional generator (LLM) that maps $(q,\mathrm{context})$ to an answer.

Given $q\in \mathcal{Q}$, GraphRAG computes:

(a) 

Seeding:choose $S_0\left(q\right)\subseteq V$ (e.g., top-k vertices by ${Score}_V(·;q)$).

(b) 

Graph expansion: set $S_1\left(q\right)=Expand(S_0\left(q\right);q)$.

(c) 

Final selection: set $S\left(q\right)=Select(S_1\left(q\right);q)$.

(d) 

Generation: output

$$\mathrm{GRAG}\left(q\right)=\mathcal{M}\left(q,Serialize(G,S(q\left)\right)\right).$$

Remark 1 

(How GraphRAG differs from classical RAG). Classical RAG typically retrieves a set of contexts by similarity alone. GraphRAG additionally uses E to expand and organize evidence, so that retrieved contexts are not only similar to q, but also structurally connected via relations encoded in G.

HyperGraphRAG is retrieval-augmented generation using a hypergraph, where vertices are context chunks and hyperedges encode multiway relations; retrieval expands through relevant hyperedges, selects evidence, then an LLM generates grounded answers [60].

Definition 9 

(HyperGraphRAG: hypergraph-based retrieval-augmented generation). [60] Let $\mathcal{D},\mathcal{C},\mathcal{Q},\mathrm{\Phi }$ be as in Definition 8. A HyperGraphRAG system is a tuple

$$\mathrm{HGRAG}=\left(\mathcal{C},\mathrm{\Phi },H,{Score}_V,{Score}_E,{Expand}_H,{Select}_H,{Serialize}_H,\mathcal{M}\right),$$

where:

(i) 

$H=(V,\mathcal{E})$ is a hypergraph (Definition 3) whose vertices V index retrievable units, and whose hyperedges $\mathcal{E}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}$ encode multiway relations. A hyperedge $e=\{v_1,\dots ,v_m\}\in \mathcal{E}$ represents that $v_1,\dots ,v_m$ participate in a shared interaction (e.g., same event, same topic, same table row, same co-author group, or the same multi-entity mention).

(ii) 

${Score}_V:V\times \mathcal{Q}\to \mathbb{R}$ is a vertex scoring function (e.g., cosine similarity as in Definition 8).

(iii) 

${Score}_E:\mathcal{E}\times \mathcal{Q}\to \mathbb{R}$ is a hyperedge scoring function; one concrete choice is

$${Score}_E(e;q)=Agg\left(\{{Score}_V(v;q)\mid v\in e\}\right),$$

where Agg is a fixed aggregator such as max, mean, or a softmax-weighted mean.

(iv) 

${Expand}_H:\mathcal{P}\left(V\right)\times \mathcal{Q}\to \mathcal{P}\left(V\right)$ is a hypergraph expansion operator. A canonical expansion is:

$${Expand}_H(S;q)=S\cup \bigcup \left\{e\in \mathcal{E}|e\cap S\ne \varnothing \mathrm{and}{Score}_E(e;q)\ge \tau \left(q\right)\right\},$$

for some threshold $\tau \left(q\right)\in \mathbb{R}$ (possibly query-dependent).

(v) 

${Select}_H:\mathcal{P}\left(V\right)\times \mathcal{Q}\to \mathcal{P}\left(V\right)$ selects the final evidence vertices.

(vi) 

${Serialize}_H$ serializes the induced subhypergraph $H\left[S\right]$ (optionally including hyperedge descriptions) into a finite text context.

(vii) 

$\mathcal{M}$ is an LLM used for generation, as before.

Given $q\in \mathcal{Q}$, HyperGraphRAG computes:

(a) 

Seeding:choose $S_0\left(q\right)\subseteq V$ (e.g., top-k by ${Score}_V(·;q)$).

(b) 

Hyperedge-aware expansion: set $S_1\left(q\right)={Expand}_H(S_0\left(q\right);q)$.

(c) 

Final selection: set $S\left(q\right)={Select}_H(S_1\left(q\right);q)$.

(d) 

Generation: output

$$\mathrm{HGRAG}\left(q\right)=\mathcal{M}\left(q,{Serialize}_H(H,S\left(q\right))\right).$$

Proposition 1 

(GraphRAG is a special case of HyperGraphRAG). Assume that every hyperedge of $H=(V,\mathcal{E})$ has size 2, i.e.,

$$\forall e\in \mathcal{E},\left|e\right|=2.$$

Define the graph $G=(V,E)$ by

$$E=\left\{\{u,v\}\subseteq V\mid \{u,v\}\in \mathcal{E}\right\}.$$

Then any HyperGraphRAG instance on H induces a GraphRAG instance on G that yields the same expanded vertex sets, provided Expand uses the same threshold rule via ${Score}_E(\{u,v\};q)$ and neighborhood expansion.

Proof. 

If $\left|e\right|=2$ for all $e\in \mathcal{E}$, then each hyperedge is of the form $e=\{u,v\}$ with $u,v\in V$, so $\mathcal{E}$ is in bijection with the edge set E of a simple graph $G=(V,E)$ defined above. For any seed set $S\subseteq V$,

$$\bigcup \{e\in \mathcal{E}\mid e\cap S\ne \varnothing \}=\bigcup \left\{\right\{u,v\}\in E\mid \{u,v\}\cap S\ne \varnothing \},$$

which is exactly the one-step graph-neighborhood expansion (up to the same score/threshold filter applied to edges). Therefore the expanded vertex sets coincide, and hence the selected substructures and serialized contexts can be chosen identical. □

SuperHyperGraphRAG is retrieval-augmented generation over an n-superhypergraph whose supervertices are nested context-sets, expanded via superedges; it generalizes GraphRAG and HyperGraphRAG by enabling hierarchical multiway retrieval and structured evidence serialization. As a reference, the comparison of GraphRAG, HyperGraphRAG, and SuperHyperGraphRAG is presented in Table 3.

Definition 10 

(Iterated powerset and flattening). Let $V_0$ be a finite nonempty base set of atomic contexts (chunks, passages, entity cards, etc.). Define ${\mathrm{PS}}^0\left(V_0\right):=V_0$ and ${\mathrm{PS}}^{n+1}\left(V_0\right):=\mathrm{PS}\left({\mathrm{PS}}^n\left(V_0\right)\right)$ for $n\ge 0$.

For each $n\ge 0$, define a map

$${Flat}_n:{\mathrm{PS}}^n\left(V_0\right)\setminus \{\varnothing \}⟶\mathrm{PS}\left(V_0\right)\setminus \{\varnothing \}$$

recursively by

$${Flat}_0\left(x\right):=\left\{x\right\}(x\in V_0),{Flat}_{n+1}\left(X\right):=\bigcup _{Y\in X}{Flat}_n\left(Y\right)(X\in {\mathrm{PS}}^{n+1}\left(V_0\right)\setminus \{\varnothing \}).$$

Thus, each n-level object $v\in {\mathrm{PS}}^n\left(V_0\right)$ is a nested set of depth n, and ${Flat}_n\left(v\right)\subseteq V_0$ is the set of atomic contexts contained in v after unnesting.

Remark 2 

(Where the SuperHyperGraph idea enters). In SuperHyperGraphRAG, retrievable units are allowed to besupervertices$v\in {\mathrm{PS}}^n\left(V_0\right)$, so a single retrieved unit can represent a multi-level group of atomic contexts. Edges then connect these supervertices, enabling retrieval over nested (iterated-powerset) structures rather than only flat units.

Definition 11 

(SuperHyperGraphRAG). Fix a finite set $V_0$ of atomic contexts and a set $\mathcal{Q}$ of queries. Fix an integer $n\ge 0$. Let ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$ be an n–SuperHyperGraph on $V_0$ in the sense of Definition 5, and assume $V\subseteq {\mathrm{PS}}^n\left(V_0\right)\setminus \{\varnothing \}$.

A SuperHyperGraphRAG system of depth nis a tuple

$${\mathrm{SHGRAG}}^{\left(n\right)}=\left(V_0,\mathcal{Q},{\mathrm{\Phi }}_0,{\mathrm{SHG}}^{\left(n\right)},{Score}_V,{Score}_E,{Expand}_n,{Select}_n,{Serialize}_n,\mathcal{M}\right),$$

where:

(i) 

${\mathrm{\Phi }}_0:V_0\cup \mathcal{Q}\to {\mathbb{R}}^d$ is an embedding map on atomic contexts and queries.

(ii) 

$\mathcal{M}$ is a conditional generator (LLM) that maps a pair $(q,\mathrm{context})$ to an output string.

(iii) 

The induced embedding of a supervertex $v\in V\subseteq {\mathrm{PS}}^n\left(V_0\right)$ is defined by the explicit average

$$\mathrm{\Phi }\left(v\right):={\displaystyle \frac{1}{|{Flat}_n\left(v\right)|}}\sum _{x\in {Flat}_n\left(v\right)}{\mathrm{\Phi }}_0\left(x\right)\in {\mathbb{R}}^d,$$

where ${Flat}_n$ is from Definition 10.

(iv) 

The vertex score is the cosine similarity

$${Score}_V(v;q):={\displaystyle \frac{\langle \mathrm{\Phi }\left(v\right),{\mathrm{\Phi }}_0\left(q\right)\rangle }{\parallel \mathrm{\Phi }\left(v\right)\parallel ·\parallel {\mathrm{\Phi }}_0\left(q\right)\parallel }}(v\in V,q\in \mathcal{Q}),$$

assuming $\mathrm{\Phi }\left(v\right)\ne 0$ and ${\mathrm{\Phi }}_0\left(q\right)\ne 0$.

(v) 

The superedge score uses a fixed aggregator Agg (already declared in the preamble):

$${Score}_E(e;q):=Agg\left(\{{Score}_V(v;q)\mid v\in e\}\right)(e\in E,q\in \mathcal{Q}).$$

(vi) 

The expansion operator is

$${Expand}_n(S;q):=S\cup \bigcup \left\{e\in E|e\cap S\ne \varnothing \mathrm{and}{Score}_E(e;q)\ge \tau \left(q\right)\right\},$$

for a threshold function $\tau :\mathcal{Q}\to \mathbb{R}$.

(vii) 

The selection operator ${Select}_n:\mathrm{PS}\left(V\right)\times \mathcal{Q}\to \mathrm{PS}\left(V\right)$ returns the final evidence set.

(viii) 

The serialization operator ${Serialize}_n$ converts the induced sub-superhypergraph ${\mathrm{SHG}}^{\left(n\right)}\left[S\right]$ (with optional labels and summaries) into a finite text string.

Given $q\in \mathcal{Q}$, the SuperHyperGraphRAG answer is computed by:

(a) 

Seed $S_0\left(q\right)\subseteq V$ (e.g., top-k supervertices by ${Score}_V(·;q)$).

(b) 

Expand for $T\in \mathbb{N}$ steps:

$$S_{t+1}\left(q\right):={Expand}_n(S_t\left(q\right);q)(t=0,1,\dots ,T-1).$$

(c) 

Select $S\left(q\right):={Select}_n(S_T\left(q\right);q)$.

(d) 

Generate

$${\mathrm{SHGRAG}}^{\left(n\right)}\left(q\right):=\mathcal{M}\left(q,{Serialize}_n({\mathrm{SHG}}^{\left(n\right)},S\left(q\right))\right).$$

Theorem 1 

(HyperGraphRAG is a special case of SuperHyperGraphRAG). Let $H=(V,\mathcal{E})$ be a hypergraph with $V=V_0$ (atomic contexts as vertices). Fix any HyperGraphRAG instance on H whose expansion rule is

$${Expand}_H(S;q)=S\cup \bigcup \left\{e\in \mathcal{E}|e\cap S\ne \varnothing \mathrm{and}{Score}_E(e;q)\ge \tau \left(q\right)\right\}.$$

Define an 0–SuperHyperGraph ${\mathrm{SHG}}^{\left(0\right)}=(V^{\prime },E^{\prime })$ by

$$V^{\prime }:=V_0,E^{\prime }:=\mathcal{E}.$$

Then the depth-0 SuperHyperGraphRAG system ${\mathrm{SHGRAG}}^{\left(0\right)}$ built on ${\mathrm{SHG}}^{\left(0\right)}$, with the same

$${\mathrm{\Phi }}_0,Agg,\tau ,Select,Serialize,\mathcal{M}$$

, produces exactly the same expanded sets and the same final generated output as the given HyperGraphRAG system, for every query $q\in \mathcal{Q}$ and every step $t\in \{0,\dots ,T\}$.

Proof. 

Since $n=0$, we have ${\mathrm{PS}}^0\left(V_0\right)=V_0$, hence $V^{\prime }=V_0\subseteq {\mathrm{PS}}^0\left(V_0\right)$. For any $v\in V^{\prime }$, Definition 10 gives ${Flat}_0\left(v\right)=\left\{v\right\}$. Therefore the induced supervertex embedding in Definition 11 satisfies

$$\mathrm{\Phi }\left(v\right)={\displaystyle \frac{1}{|{Flat}_0\left(v\right)|}}\sum _{x\in {Flat}_0\left(v\right)}{\mathrm{\Phi }}_0\left(x\right)={\displaystyle \frac{1}{\left|\right\{v\left\}\right|}}\sum _{x\in \left\{v\right\}}{\mathrm{\Phi }}_0\left(x\right)=1·{\mathrm{\Phi }}_0\left(v\right)={\mathrm{\Phi }}_0\left(v\right).$$

Hence for all $v\in V^{\prime }$ and $q\in \mathcal{Q}$,

$${Score}_V(v;q)={\displaystyle \frac{\langle \mathrm{\Phi }\left(v\right),{\mathrm{\Phi }}_0\left(q\right)\rangle }{\parallel \mathrm{\Phi }\left(v\right)\parallel ·\parallel {\mathrm{\Phi }}_0\left(q\right)\parallel }}={\displaystyle \frac{\langle {\mathrm{\Phi }}_0\left(v\right),{\mathrm{\Phi }}_0\left(q\right)\rangle }{\parallel {\mathrm{\Phi }}_0\left(v\right)\parallel ·\parallel {\mathrm{\Phi }}_0\left(q\right)\parallel }},$$

which is exactly the vertex scoring used by the HyperGraphRAG instance (under the shared choice of cosine scoring).

Now $E^{\prime }=\mathcal{E}$, and for every hyperedge $e\in \mathcal{E}$ we also have $e\in E^{\prime }$ and the same aggregator Agg, so the edge scores coincide:

$${Score}_E^{\mathrm{SHG}}(e;q)=Agg\left(\{{Score}_V(v;q)\mid v\in e\}\right)={Score}_E^H(e;q).$$

Consequently, for any $S\subseteq V^{\prime }=V$ and any $q\in \mathcal{Q}$,

$$\begin{array}{cc}\hfill {Expand}_0(S;q)& =S\cup \bigcup \left\{e\in E^{\prime }|e\cap S\ne \varnothing \mathrm{and}{Score}_E^{\mathrm{SHG}}(e;q)\ge \tau \left(q\right)\right\}\hfill \\ \hfill & =S\cup \bigcup \left\{e\in \mathcal{E}|e\cap S\ne \varnothing \mathrm{and}{Score}_E^H(e;q)\ge \tau \left(q\right)\right\}\hfill \\ \hfill & ={Expand}_H(S;q).\hfill \end{array}$$

By induction on t, starting from the same seed $S_0\left(q\right)$, we obtain $S_t^{\mathrm{SHG}}\left(q\right)=S_t^H\left(q\right)$ for all $t\le T$. Since Select, Serialize, and $\mathcal{M}$ are assumed identical, the final output strings are identical as well. □

Theorem 2 

(GraphRAG is a special case of SuperHyperGraphRAG). Let $G=(V,E_G)$ be a (finite, undirected) graph on atomic contexts $V=V_0$. Assume a GraphRAG instance whose expansion is realized by iterating one-step neighbor expansion T times, i.e.,

$$N_G\left(S\right):=S\cup \{u\in V\mid \exists v\in S:\{u,v\}\in E_G\},S_{t+1}\left(q\right):=N_G\left(S_t\left(q\right)\right).$$

Define a depth-0 superhypergraph ${\mathrm{SHG}}^{\left(0\right)}=(V^{\prime },E^{\prime })$ by

$$V^{\prime }:=V_0,E^{\prime }:=\left\{\{u,v\}\subseteq V_0\mid \{u,v\}\in E_G\right\}.$$

Assume further that the SuperHyperGraphRAG expansion ${Expand}_0$ uses $\tau \left(q\right)=-\infty$ (no threshold filtering), so that every incident edge expands the set. Then, for every query $q\in \mathcal{Q}$ and every $t\in \{0,\dots ,T\}$, the SuperHyperGraphRAG iterates satisfy

$$S_t^{\mathrm{SHG}}\left(q\right)=S_t^G\left(q\right),$$

and hence the induced evidence substructures (and outputs, if $Select,Serialize,\mathcal{M}$ coincide) match the GraphRAG instance.

Proof. 

Because $E^{\prime }$ consists only of 2-element subsets, ${\mathrm{SHG}}^{\left(0\right)}$ is a 2-uniform hypergraph on $V_0$, and by construction it encodes exactly the same adjacencies as G.

Fix $S\subseteq V_0$. Since $\tau \left(q\right)=-\infty$, the expansion rule becomes

$${Expand}_0(S;q)=S\cup \bigcup \{e\in E^{\prime }\mid e\cap S\ne \varnothing \}.$$

We prove the set equality

$${Expand}_0(S;q)=N_G\left(S\right).$$

First, let $x\in {Expand}_0(S;q)$. Then either $x\in S$, which implies $x\in N_G\left(S\right)$, or else

$$x\in \bigcup \{e\in E^{\prime }\mid e\cap S\ne \varnothing \},$$

so there exists $e\in E^{\prime }$ such that $x\in e$ and $e\cap S\ne \varnothing$. Because $\left|e\right|=2$, we may write $e=\{u,v\}$ with $\{u,v\}\in E_G$ and $x\in \{u,v\}$. Also $e\cap S\ne \varnothing$ means either $u\in S$ or $v\in S$. Hence x is either in S or adjacent in G to a vertex in S, so $x\in N_G\left(S\right)$. Thus ${Expand}_0(S;q)\subseteq N_G\left(S\right)$.

Conversely, let $x\in N_G\left(S\right)$. If $x\in S$, then $x\in {Expand}_0(S;q)$ holds immediately. Otherwise, by definition of $N_G\left(S\right)$, there exists $v\in S$ with $\{x,v\}\in E_G$. Then $e:=\{x,v\}\in E^{\prime }$, and $e\cap S\ne \varnothing$ since $v\in e\cap S$. Therefore $x\in e\subseteq \bigcup \{e\in E^{\prime }\mid e\cap S\ne \varnothing \}\subseteq {Expand}_0(S;q)$. So $N_G\left(S\right)\subseteq {Expand}_0(S;q)$.

Combining both inclusions gives ${Expand}_0(S;q)=N_G\left(S\right)$. Starting from the same seed $S_0\left(q\right)$, induction on t yields

$$S_{t+1}^{\mathrm{SHG}}\left(q\right)={Expand}_0(S_t^{\mathrm{SHG}}\left(q\right);q)=N_G\left(S_t^G\left(q\right)\right)=S_{t+1}^G\left(q\right),$$

hence $S_t^{\mathrm{SHG}}\left(q\right)=S_t^G\left(q\right)$ for all $t\le T$. If $Select,Serialize,\mathcal{M}$ are shared between the two systems, the final outputs coincide as well. □

5. Review and Results: SuperHyperGraph Generation

Graph generation produces new graphs satisfying constraints by learning node–edge patterns, enabling molecule design, networks synthesis, and simulation [63,64,65,105,106]. Hypergraph generation produces new hypergraphs by learning multi-vertex interaction patterns, enabling group-relation synthesis in chemistry, text, and systems [107,108,109,110]. Superhypergraph generation produces nested higher-order hypergraphs by learning hyperedge-of-hyperedge patterns, enabling hierarchical interaction synthesis and complex structure design. The comparison of Graph Generation, HyperGraph Generation, and SuperHyperGraph Generation is presented in Table 4.

Definition 12 (Graph Generation (probabilistic / model-based)).

Fix a finite vertex set V and a measurable condition space $\mathcal{C}$ (e.g., labels, text, constraints, statistics). A graph generator is a triple

$${\mathsf{Gen}}_G=(\mathcal{Z},{\pi }_{\theta },g_{\theta }),$$

where

• $\mathcal{Z}$ is a latent (random-seed) space,
• ${\pi }_{\theta }$ is a probability distribution on $\mathcal{Z}$,
• $g_{\theta }:\mathcal{Z}\times \mathcal{C}\to \mathcal{G}\left(V\right)$ is a measurable map.

Given $c\in \mathcal{C}$,graph generation conditioned on cis the random output

$$G=g_{\theta }(Z,c)\mathrm{where}Z\sim {\pi }_{\theta }.$$

Equivalently, ${\mathsf{Gen}}_G$ induces a conditional distribution $P_{\theta }(·|c)$ on $\mathcal{G}\left(V\right)$ by

$$P_{\theta }(G\in A\mid c)={\pi }_{\theta }\left(\{z\in \mathcal{Z}:g_{\theta }(z,c)\in A\}\right)\mathrm{for}\mathrm{all}\mathrm{measurable}A\subseteq \mathcal{G}\left(V\right).$$

Definition 13  (Hypergraph Generation (probabilistic / model-based)).

Fix a finite vertex set V and a measurable condition space $\mathcal{C}$. Ahypergraph generatoris a triple

$${\mathsf{Gen}}_H=(\mathcal{Z},{\pi }_{\theta },h_{\theta }),$$

where $\mathcal{Z}$ is a latent space, ${\pi }_{\theta }$ is a probability distribution on $\mathcal{Z}$, and $h_{\theta }:\mathcal{Z}\times \mathcal{C}\to \mathcal{H}\left(V\right)$ is a measurable map. Given $c\in \mathcal{C}$,hypergraph generation conditioned on cis the random output

$$H=h_{\theta }(Z,c)\mathrm{where}Z\sim {\pi }_{\theta },$$

which induces a conditional distribution on $\mathcal{H}\left(V\right)$ in the same way as above.

Definition 14  (Constraint-based (deterministic) Generation).

Let V be a finite vertex set and let Φ be a predicate (constraint) on $\mathcal{G}\left(V\right)$. Aconstraint-based graph generation procedureis an algorithm $\mathsf{Alg}$ that, given Φ, outputs a graph $G\in \mathcal{G}\left(V\right)$ such that

$$\mathrm{\Phi }\left(G\right)=\mathrm{true}.$$

The enumerative variant outputs the entire solution set

$$Sol\left(\mathrm{\Phi }\right)=\{G\in \mathcal{G}(V):\mathrm{\Phi }(G)=\mathrm{true}\}.$$

Replacing $\mathcal{G}\left(V\right)$ by $\mathcal{H}\left(V\right)$ gives the corresponding notions for hypergraphs.

Definition 15 

(n-SuperHypergraph generation (probabilistic / model-based)). Fix $n\ge 0$, a finite, nonempty base set $V_0$, and a measurable condition space $\mathcal{C}$. Let

$$\mathcal{S}{\mathcal{H}}^{\left(n\right)}\left(V_0\right):=\left\{{\mathrm{SHG}}^{\left(n\right)}=(V,E)|V\subseteq {\mathcal{P}}_{*}^n\left(V_0\right),E\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}\right\}$$

denote the set of all n-SuperHypergraphs on $V_0$.

Ann-SuperHypergraph generatoris a triple

$${\mathsf{Gen}}_n=(\mathcal{Z},{\pi }_{\theta },g_{\theta }^{\left(n\right)}),$$

where

(i) 

$\mathcal{Z}$ is a latent (random-seed) space,

(ii) 

${\pi }_{\theta }$ is a probability distribution on $\mathcal{Z}$,

(iii) 

$g_{\theta }^{\left(n\right)}:\mathcal{Z}\times \mathcal{C}\to \mathcal{S}{\mathcal{H}}^{\left(n\right)}\left(V_0\right)$ is a measurable map.

Given $c\in \mathcal{C}$,generation conditioned on cis the random output

$${\mathrm{SHG}}^{\left(n\right)}=g_{\theta }^{\left(n\right)}(Z,c)\mathrm{where}Z\sim {\pi }_{\theta }.$$

Equivalently, ${\mathsf{Gen}}_n$ induces a conditional distribution $P_{\theta }^{\left(n\right)}(·|c)$ on $\mathcal{S}{\mathcal{H}}^{\left(n\right)}\left(V_0\right)$ by

$$P_{\theta }^{\left(n\right)}(A\mid c)={\pi }_{\theta }\left(\{z\in \mathcal{Z}:g_{\theta }^{\left(n\right)}(z,c)\in A\}\right)\mathrm{for}\mathrm{all}\mathrm{measurable}A\subseteq \mathcal{S}{\mathcal{H}}^{\left(n\right)}\left(V_0\right).$$

Definition 16  (Constraint-based (deterministic) n-SuperHypergraph generation).

Fix $n\ge 0$ and $V_0$. Let Ψ be a predicate (constraint) on $\mathcal{S}{\mathcal{H}}^{\left(n\right)}\left(V_0\right)$. Aconstraint-based n-SuperHypergraph generation procedureis an algorithm $\mathsf{Alg}$ that outputs ${\mathrm{SHG}}^{\left(n\right)}\in \mathcal{S}{\mathcal{H}}^{\left(n\right)}\left(V_0\right)$ such that

$$\Psi \left({\mathrm{SHG}}^{\left(n\right)}\right)=\mathrm{true}.$$

The enumerative variant outputs the entire solution set

$$Sol\left(\Psi \right)=\{{\mathrm{SHG}}^{\left(n\right)}\in \mathcal{S}{\mathcal{H}}^{\left(n\right)}\left(V_0\right):\Psi \left({\mathrm{SHG}}^{\left(n\right)}\right)=\mathrm{true}\}.$$

Theorem 3 

(Hypergraph generation is a special case of n-SuperHypergraph generation). Let $V_0$ be a finite vertex set, and let $\mathcal{H}\left(V_0\right)$ be the set of hypergraphs on $V_0$:

$$\mathcal{H}\left(V_0\right)=\{(V_0,\mathcal{E})\mid \mathcal{E}\subseteq \mathcal{P}\left(V_0\right)\setminus \{\varnothing \}\}.$$

Then $\mathcal{H}\left(V_0\right)$ is naturally identified with the subset of $\mathcal{S}{\mathcal{H}}^{\left(0\right)}\left(V_0\right)$ consisting of those 0-SuperHyperGraphs ${\mathrm{SHG}}^{\left(0\right)}=(V,E)$ with $V=V_0$. Consequently, every hypergraph generator ${\mathsf{Gen}}_H=(\mathcal{Z},{\pi }_{\theta },h_{\theta })$ on $V_0$ is an instance of a 0-SuperHypergraph generator ${\mathsf{Gen}}_0=(\mathcal{Z},{\pi }_{\theta },g_{\theta }^{\left(0\right)})$.

Proof. 

By the Definition, ${\mathcal{P}}^0\left(V_0\right)=V_0$ and ${\mathcal{P}}_{*}^0\left(V_0\right)=V_0$ (since $V_0\ne \varnothing$). Thus, by the Definition, a 0-SuperHypergraph on $V_0$ is a pair $(V,E)$ with

$$V\subseteq V_0,E\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}.$$

If we impose the additional condition $V=V_0$, then the allowable edge families are exactly $\mathcal{E}\subseteq \mathcal{P}\left(V_0\right)\setminus \{\varnothing \}$. Hence the correspondence

$$(V_0,\mathcal{E})\in \mathcal{H}\left(V_0\right)⟷{\mathrm{SHG}}^{\left(0\right)}=(V_0,\mathcal{E})\in \mathcal{S}{\mathcal{H}}^{\left(0\right)}\left(V_0\right)$$

is a bijection onto the subset $\{{\mathrm{SHG}}^{\left(0\right)}\in \mathcal{S}{\mathcal{H}}^{\left(0\right)}\left(V_0\right)\mid V=V_0\}$.

Now let ${\mathsf{Gen}}_H=(\mathcal{Z},{\pi }_{\theta },h_{\theta })$ be a hypergraph generator, so $h_{\theta }:\mathcal{Z}\times \mathcal{C}\to \mathcal{H}\left(V_0\right)$. Define $g_{\theta }^{\left(0\right)}:\mathcal{Z}\times \mathcal{C}\to \mathcal{S}{\mathcal{H}}^{\left(0\right)}\left(V_0\right)$ by

$$g_{\theta }^{\left(0\right)}(z,c):=h_{\theta }(z,c),$$

viewing $h_{\theta }(z,c)=(V_0,\mathcal{E})$ as the 0-SuperHypergraph $(V_0,\mathcal{E})$. Then $g_{\theta }^{\left(0\right)}$ is measurable whenever $h_{\theta }$ is measurable, and the output distributions coincide. Therefore ${\mathsf{Gen}}_H$ is a special case of ${\mathsf{Gen}}_0$. □

Theorem 4 

(Graph generation is a special case of n-SuperHypergraph generation). Let $V_0$ be a finite vertex set and let $\mathcal{G}\left(V_0\right)$ be the set of simple graphs on $V_0$:

$$\mathcal{G}\left(V_0\right)=\{(V_0,E_G)\mid E_G\subseteq \left({\displaystyle \genfrac{}{}{0pt}{}{V_0}{2}}\right)\}.$$

Then $\mathcal{G}\left(V_0\right)$ is naturally identified with the subset of $\mathcal{S}{\mathcal{H}}^{\left(0\right)}\left(V_0\right)$ consisting of those 0-SuperHyperGraphs $(V,E)$ with $V=V_0$ and $E\subseteq \left({\displaystyle \genfrac{}{}{0pt}{}{V_0}{2}}\right)$. Consequently, every graph generator is an instance of a 0-SuperHypergraph generator.

Proof. 

As in the proof of Theorem 3, a 0-SuperHypergraph with $V=V_0$ is of the form $(V_0,E)$ where $E\subseteq \mathcal{P}\left(V_0\right)\setminus \{\varnothing \}$. If we additionally require $E\subseteq \left({\displaystyle \genfrac{}{}{0pt}{}{V_0}{2}}\right)$, then each edge is a 2-element subset of $V_0$, so $(V_0,E)$ is precisely a simple undirected graph on $V_0$. Thus the map

$$(V_0,E_G)\in \mathcal{G}\left(V_0\right)⟼(V_0,E_G)\in \mathcal{S}{\mathcal{H}}^{\left(0\right)}\left(V_0\right)$$

is a bijection onto $\{{\mathrm{SHG}}^{\left(0\right)}\in \mathcal{S}{\mathcal{H}}^{\left(0\right)}\left(V_0\right)\mid V=V_0,E\subseteq \left({\displaystyle \genfrac{}{}{0pt}{}{V_0}{2}}\right)\}$.

Given any graph generator $(\mathcal{Z},{\pi }_{\theta },{\gamma }_{\theta })$ with ${\gamma }_{\theta }:\mathcal{Z}\times \mathcal{C}\to \mathcal{G}\left(V_0\right)$, define $g_{\theta }^{\left(0\right)}$ by the same output map viewed in $\mathcal{S}{\mathcal{H}}^{\left(0\right)}\left(V_0\right)$:

$$g_{\theta }^{\left(0\right)}(z,c):={\gamma }_{\theta }(z,c).$$

Then the induced output distributions coincide, so graph generation is a special case of 0-SuperHypergraph generation. □

6. Review and Results: Causal SuperHyperGraph

A causal graph is a directed graph that represents direct causal dependencies among variables, usually derived from structural equations and used to define do-interventions and identify causal effects[111,112,113,114,115]. A causal hypergraph extends this idea by using directed hyperedges so that a set of variables can jointly act as a single causal mechanism on a target variable[116,117,118,119]. A causal superhypergraph further extends causal hypergraphs by allowing vertices to be nested set-objects obtained via iterated powersets, enabling hierarchical multiway causal mechanisms across multiple abstraction levels.

Table 5. Concise comparison: causal graph vs. causal hypergraph vs. causal superhypergraph.

Aspect	Causal Graph	Causal HyperGraph	Causal SuperHyperGraph
Underlying object	Directed graph $G=(V,E)$ with $E\subseteq V\times V$	Hypergraph $H=(V,\mathcal{E})$ with $\mathcal{E}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}$	n–SuperHyperGraph ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$ with $V\subseteq {\mathcal{P}}^n\left(V_0\right)$ and $E\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}$
Causal primitive	Edge $j\to i$ (direct cause)	Hyperedge $e\in \mathcal{E}$ (multiway relation), optionally oriented $e\Rightarrow v$	Superedge $\epsilon \in E$ among supervertices (nested groups), optionally oriented across levels
Structural equation form	$X_i=f_i\left({\left(X_j\right)}_{j\in \mathrm{Pa}\left(i\right)},U_i\right)$	$X_v=f_v\left({\left(X_u\right)}_{u\in N_H\left(v\right)},U_v\right)$ or $X_{e\Rightarrow v}=f_{e\Rightarrow v}({\left(X_u\right)}_{u\in e},U_{e\Rightarrow v})$	$Y_s=F_s\left({\left(Y_t\right)}_{t\in N_{\mathrm{SHG}}\left(s\right)},U_s\right)$ for supervertices $s\in V$
Independence reading	d-separation on G	Separation via hyperedge-based paths / factorization over $\mathcal{E}$ (model-dependent)	Separation on superpaths and level-wise neighborhoods; supports hierarchical blocking (model-dependent)
What it represents	Pairwise directed influence	Multiway (group) influence in one step	Multiway influence with explicit nesting / hierarchy of groups and relations
Reduction to simpler model	—	If all hyperedges have size 2 and are oriented, reduces to a causal graph	If $n=0$ and supervertices are singletons, reduces to a causal hypergraph; if also edges are size 2, reduces to a causal graph

Table 5. Concise comparison: causal graph vs. causal hypergraph vs. causal superhypergraph.

Aspect	Causal Graph	Causal HyperGraph	Causal SuperHyperGraph
Underlying object	Directed graph $G=(V,E)$ with $E\subseteq V\times V$	Hypergraph $H=(V,\mathcal{E})$ with $\mathcal{E}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}$	n–SuperHyperGraph ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$ with $V\subseteq {\mathcal{P}}^n\left(V_0\right)$ and $E\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}$
Causal primitive	Edge $j\to i$ (direct cause)	Hyperedge $e\in \mathcal{E}$ (multiway relation), optionally oriented $e\Rightarrow v$	Superedge $\epsilon \in E$ among supervertices (nested groups), optionally oriented across levels
Structural equation form	$X_i=f_i\left({\left(X_j\right)}_{j\in \mathrm{Pa}\left(i\right)},U_i\right)$	$X_v=f_v\left({\left(X_u\right)}_{u\in N_H\left(v\right)},U_v\right)$ or $X_{e\Rightarrow v}=f_{e\Rightarrow v}({\left(X_u\right)}_{u\in e},U_{e\Rightarrow v})$	$Y_s=F_s\left({\left(Y_t\right)}_{t\in N_{\mathrm{SHG}}\left(s\right)},U_s\right)$ for supervertices $s\in V$
Independence reading	d-separation on G	Separation via hyperedge-based paths / factorization over $\mathcal{E}$ (model-dependent)	Separation on superpaths and level-wise neighborhoods; supports hierarchical blocking (model-dependent)
What it represents	Pairwise directed influence	Multiway (group) influence in one step	Multiway influence with explicit nesting / hierarchy of groups and relations
Reduction to simpler model	—	If all hyperedges have size 2 and are oriented, reduces to a causal graph	If $n=0$ and supervertices are singletons, reduces to a causal hypergraph; if also edges are size 2, reduces to a causal graph

Definition 17  (Structural Causal Model (SCM)).

Fix a finite index set of endogenous variables $V=\{1,\dots ,n\}$. For each $i\in V$, let ${\mathcal{X}}_i$ be the state space of $X_i$. Let $U=\{1,\dots ,n\}$ index exogenous noises, with spaces ${\mathcal{U}}_i$.

A structural causal model is a quadruple

$$M=(\mathcal{U},\mathcal{X},F,P_U),$$

where

• $\mathcal{U}={\prod }_{i\in V}{\mathcal{U}}_i$ and $\mathcal{X}={\prod }_{i\in V}{\mathcal{X}}_i$;
• $P_U$ is a probability distribution on $\mathcal{U}$;
• $F={\left\{f_i\right\}}_{i\in V}$ is a family of measurable maps

  $$f_i:\left(\prod _{j\in Pa\left(i\right)}{\mathcal{X}}_j\right)\times {\mathcal{U}}_i⟶{\mathcal{X}}_i,$$
  with some parent set $Pa\left(i\right)\subseteq V\setminus \left\{i\right\}$.

The endogenous variables $X={\left(X_i\right)}_{i\in V}$ are determined by the structural equations

$$X_i=f_i\left({\left(X_j\right)}_{j\in Pa\left(i\right)},U_i\right)(i\in V),$$

where $U={\left(U_i\right)}_{i\in V}\sim P_U$.

Definition 18 

(Causal graph induced by an SCM). Let $M=(\mathcal{U},\mathcal{X},F,P_U)$ be an SCM with endogenous index set V. The causal graph (also called thefunctional graph) of M is the directed graph

$$G_M=(V,E)$$

whose edge set is

$$E=\left\{\right(j,i)\in V\times V\mid j\in Pa(i\left)\right\}.$$

We write $j\to i$ for $(j,i)\in E$ and interpret $j\to i$ as a direct causal dependence of $X_i$ on $X_j$ in the model M. If $G_M$ is acyclic, then M is called recursive (or acyclic).

Definition 19  (Intervention (do-operator)).

Let $I\subseteq V$ and fix values $x_I\in {\prod }_{i\in I}{\mathcal{X}}_i$. The intervention $\mathrm{do}(X_I=x_I)$ produces the intervened SCM

$$M^{\mathrm{do}(I=x_I)}=(\mathcal{U},\mathcal{X},F^{\prime },P_U),$$

where

$$f_i^{\prime }=\left\{\begin{array}{cc}(·)\mapsto x_i,\hfill & i\in I,\hfill \\ f_i,\hfill & i\notin I.\hfill \end{array}\right.$$

The corresponding interventional distribution $P(X\mid \mathrm{do}(X_I=x_I))$ is the distribution of the unique solution X (when it exists) under the modified equations and $U\sim P_U$.

Definition 20 

(Directed hypergraph). A directed hypergraph is a pair

$$H=(V,\mathcal{E}),$$

where V is a finite node set and $\mathcal{E}$ is a finite family of directed hyperedges. Each $e\in \mathcal{E}$ is an ordered pair

$$e=\left(T\left(e\right)\to h\left(e\right)\right),$$

where the tail$T\left(e\right)\subseteq V\setminus \left\{h\right(e\left)\right\}$ is nonempty and the head$h\left(e\right)\in V$.

Definition 21 

(Causal hypergraph model). Fix state spaces ${\left\{{\mathcal{X}}_i\right\}}_{i\in V}$ and noise spaces ${\left\{{\mathcal{U}}_i\right\}}_{i\in V}$. A causal hypergraph model is a tuple

$$\mathfrak{M}=(\mathcal{U},\mathcal{X},H,\mathrm{\Phi },G,P_U),$$

where

• $\mathcal{U}={\prod }_{i\in V}{\mathcal{U}}_i$, $\mathcal{X}={\prod }_{i\in V}{\mathcal{X}}_i$, and $P_U$ is a distribution on $\mathcal{U}$;
• $H=(V,\mathcal{E})$ is a directed hypergraph;
• $\mathrm{\Phi }={\left\{{\phi }_e\right\}}_{e\in \mathcal{E}}$ is a family ofhyperedge mechanisms

  $${\phi }_e:\prod _{j\in T\left(e\right)}{\mathcal{X}}_j⟶{\mathcal{Z}}_e,$$
  for some intermediate spaces ${\mathcal{Z}}_e$;
• $G={\left\{g_i\right\}}_{i\in V}$ is a family ofaggregation functions

  $$g_i:\left(\prod _{e\in {\mathcal{E}}_i}{\mathcal{Z}}_e\right)\times {\mathcal{U}}_i⟶{\mathcal{X}}_i,{\mathcal{E}}_i:=\{e\in \mathcal{E}\mid h\left(e\right)=i\}.$$

The structural equations of $\mathfrak{M}$ are

$$X_i=g_i\left({\left({\phi }_e\left(X_{T\left(e\right)}\right)\right)}_{e\in {\mathcal{E}}_i},U_i\right)(i\in V),$$

where $X_{T\left(e\right)}:={\left(X_j\right)}_{j\in T\left(e\right)}$ and $U\sim P_U$. Intuitively, each hyperedge $T\left(e\right)\to i$ represents a joint causal mechanism acting on the tuple $X_{T\left(e\right)}$.

Definition 22 

(Graph induced by a causal hypergraph). Let $H=(V,\mathcal{E})$ be a directed hypergraph. Its 2-section causal graph (edge-expansion) is the directed graph

$$\Gamma \left(H\right)=(V,E_{\Gamma }),E_{\Gamma }=\{(j,i)\mid \exists e\in \mathcal{E}\mathrm{s}.\mathrm{t}.i=h\left(e\right)\mathrm{and}j\in T\left(e\right)\}.$$

Thus every hyperedge $T\to i$ is expanded into ordinary directed edges $j\to i$ for all $j\in T$.

Theorem 5 

(Causal graphs are a special case of causal hypergraphs). Let $G=(V,E)$ be a directed graph. Define a directed hypergraph $H_G=(V,{\mathcal{E}}_G)$ by

$${\mathcal{E}}_G=\{(\left\{j\right\}\to i)\mid (j,i)\in E\}.$$

Then:

• $\Gamma \left(H_G\right)=G$.
• For any SCM M with causal graph $G_M=G$, there exists a causal hypergraph model $\mathfrak{M}$ on $H_G$ whose induced edge-expansion graph equals G, and whose structural equations coincide with those of M.

Proof.  (1) By construction, a directed edge $(j,i)\in E$ corresponds to the hyperedge $\left\{j\right\}\to i\in {\mathcal{E}}_G$. Hence $(j,i)\in E_{\Gamma }$ if and only if $(j,i)\in E$, so $\Gamma \left(H_G\right)=G$.

(2) Let $M=(\mathcal{U},\mathcal{X},F,P_U)$ be an SCM with

$$X_i=f_i\left({\left(X_j\right)}_{j\in Pa\left(i\right)},U_i\right),Pa\left(i\right)=\{j\mid (j,i)\in E\}.$$

Define $\mathfrak{M}$ on $H_G$ as follows. For each hyperedge $e=\left(\right\{j\}\to i)$, set ${\mathcal{Z}}_e:={\mathcal{X}}_j$ and

$${\phi }_{\left(\right\{j\}\to i)}\left(x_j\right):=x_j.$$

For each $i\in V$, list the incoming hyperedges as ${\mathcal{E}}_i=\{(\left\{j\right\}\to i)\mid j\in Pa\left(i\right)\}$, identify ${\prod }_{e\in {\mathcal{E}}_i}{\mathcal{Z}}_e\cong {\prod }_{j\in Pa\left(i\right)}{\mathcal{X}}_j$, and set

$$g_i\left({\left(z_e\right)}_{e\in {\mathcal{E}}_i},u_i\right):=f_i\left({\left(z_{\left(\right\{j\}\to i)}\right)}_{j\in Pa\left(i\right)},u_i\right).$$

Then the structural equation of $\mathfrak{M}$ becomes

$$X_i=g_i\left({\left({\phi }_{\left(\right\{j\}\to i)}\left(X_j\right)\right)}_{j\in Pa\left(i\right)},U_i\right)=g_i\left({\left(X_j\right)}_{j\in Pa\left(i\right)},U_i\right)=f_i\left({\left(X_j\right)}_{j\in Pa\left(i\right)},U_i\right),$$

which is exactly the SCM equation. The induced graph statement follows from (1). □

Definition 23  (Granger-causal hypergraph (time-series causal hypergraph)).

Let $Y_t=(Y_t^{\left(1\right)},\dots ,Y_t^{\left(d\right)})$ be a multivariate time series and fix a maximum lag $K\in \mathbb{N}$. Form a node set of lagged variables

$$V=\{Y_{t-k}^{\left(j\right)}\mid j\in \{1,\dots ,d\},k\in \{1,\dots ,K\}\}.$$

For a target component $Y_t^{\left(i\right)}$, let $C_i\subseteq V$ be a nonempty set of lagged predictors. We say $C_i$ Granger-causes$Y_t^{\left(i\right)}$ (relative to the history V) if, in a chosen model class, using $C_i$ improves the prediction of $Y_t^{\left(i\right)}$ beyond what is achievable without $C_i$ (e.g., via likelihood-ratio tests, information criteria, or out-of-sample error reduction).

A Granger-causal hypergraph is a directed hypergraph $H=(V,\mathcal{E})$ whose hyperedges are

$$e_i=(C_i\to Y_t^{\left(i\right)}),$$

interpreted as a multi-source temporal influence relation. (Granger causality is predictive/temporal and does not, by itself, assert interventional causality.)

Definition 24 

(Directed n-SuperHyperGraph). Fix $n\ge 0$ and a finite, nonempty base set $V_0$. A directed n-SuperHyperGraph on $V_0$ is a pair

$${\overrightarrow{\mathrm{SHG}}}^{\left(n\right)}=(V,\mathcal{E})$$

such that

$$V\subseteq {\mathrm{PS}}_{*}^n\left(V_0\right),\mathcal{E}\subseteq \left(\mathrm{PS}\left(V\right)\setminus \{\varnothing \}\right)\times V.$$

Each directed superhyperedge $e\in \mathcal{E}$ is written as

$$e=\left(\mathrm{Tail}\left(e\right)\to \mathrm{Head}\left(e\right)\right),$$

where $\mathrm{Tail}\left(e\right)\subseteq V\setminus \left\{\mathrm{Head}\right(e\left)\right\}$ is nonempty and $\mathrm{Head}\left(e\right)\in V$. Elements of V aren-supervertices. The pair $\left(\mathrm{Tail}\right(e),\mathrm{Head}(e\left)\right)$ encodes a multi-source directed relation.

Remark 3 

(Use of the SuperHyperGraph idea). Definition 24 uses the SuperHyperGraph viewpoint by allowing vertices to be nested objects $v\in {\mathrm{PS}}^n\left(V_0\right)$ (iterated powerset level n). For $n=0$, one recovers ordinary vertices $V\subseteq V_0$.

Definition 25 

(Causal n-SuperHyperGraph model). Fix $n\ge 0$ and a directed n-SuperHyperGraph ${\overrightarrow{\mathrm{SHG}}}^{\left(n\right)}=(V,\mathcal{E})$ on $V_0$. For each $v\in V$, let ${\mathcal{X}}_v$ be a measurable state space and ${\mathcal{U}}_v$ a measurable noise space. Set $\mathcal{X}:={\prod }_{v\in V}{\mathcal{X}}_v$ and $\mathcal{U}:={\prod }_{v\in V}{\mathcal{U}}_v$.

A causal n-SuperHyperGraph model is a tuple

$${\mathfrak{M}}^{\left(n\right)}=\left(\mathcal{U},\mathcal{X},{\overrightarrow{\mathrm{SHG}}}^{\left(n\right)},\mathrm{\Phi },G,P_U\right),$$

where:

(i) 

$P_U$ is a probability distribution on $\mathcal{U}$.

(ii) 

$\mathrm{\Phi }={\left\{{\phi }_e\right\}}_{e\in \mathcal{E}}$ is a family of superhyperedge mechanisms such that for each $e=\left(\mathrm{Tail}\right(e)\to \mathrm{Head}(e\left)\right)\in \mathcal{E}$ there exists a measurable space ${\mathcal{Z}}_e$ and a measurable map

$${\phi }_e:\prod _{u\in \mathrm{Tail}\left(e\right)}{\mathcal{X}}_u⟶{\mathcal{Z}}_e.$$

(iii) 

$G={\left\{g_v\right\}}_{v\in V}$ is a family of aggregation functions where, for each $v\in V$, letting

$${\mathcal{E}}_v:=\{e\in \mathcal{E}\mid \mathrm{Head}\left(e\right)=v\},$$

we require a measurable map

$$g_v:\left(\prod _{e\in {\mathcal{E}}_v}{\mathcal{Z}}_e\right)\times {\mathcal{U}}_v⟶{\mathcal{X}}_v.$$

The endogenous variables $X={\left(X_v\right)}_{v\in V}$ are determined by the structural equations

$$X_v=g_v\left({\left({\phi }_e\left(X_{\mathrm{Tail}\left(e\right)}\right)\right)}_{e\in {\mathcal{E}}_v},U_v\right)(v\in V),$$

where $X_{\mathrm{Tail}\left(e\right)}:={\left(X_u\right)}_{u\in \mathrm{Tail}\left(e\right)}$ and $U={\left(U_v\right)}_{v\in V}\sim P_U$.

Definition 26  (Intervention (do-operator) on a causal n-SuperHyperGraph).

Let ${\mathfrak{M}}^{\left(n\right)}$ be as in Definition 25. Fix a nonempty index set $I\subseteq V$ and values $x_I\in {\prod }_{v\in I}{\mathcal{X}}_v$. The intervention $\mathrm{do}(X_I=x_I)$ produces the intervened model

$${\left({\mathfrak{M}}^{\left(n\right)}\right)}^{\mathrm{do}(I=x_I)}=\left(\mathcal{U},\mathcal{X},{\overrightarrow{\mathrm{SHG}}}^{\left(n\right)},\mathrm{\Phi },G^{\prime },P_U\right),$$

where $g_v^{\prime }$ is defined by

$$g_v^{\prime }=\left\{\begin{array}{cc}(·)\mapsto x_v,\hfill & v\in I,\hfill \\ g_v,\hfill & v\notin I.\hfill \end{array}\right.$$

The interventional distribution $P(X\mid \mathrm{do}(X_I=x_I))$ is the distribution of the solution under $G^{\prime }$ and $U\sim P_U$, when a (measurable) solution exists.

Theorem 6  (Causal hypergraphs are a special case($n=0$)).

Let $H=(V_0,{\mathcal{E}}_H)$ be a directed hypergraph whose hyperedges are pairs $(T\to i)$ with $T\subseteq V_0\setminus \left\{i\right\}$, $T\ne \varnothing$, and $i\in V_0$. Consider any causal hypergraph model on H given by structural equations

$$X_i=g_i\left({\left({\phi }_e\left(X_{\mathrm{Tail}\left(e\right)}\right)\right)}_{e\in {\mathcal{E}}_{H,i}},U_i\right),{\mathcal{E}}_{H,i}:=\{e\in {\mathcal{E}}_H\mid \mathrm{Head}\left(e\right)=i\}.$$

Then it is a special case of a causal 0-SuperHyperGraph model (Definition 25) by taking $V:=V_0\subseteq {\mathrm{PS}}_{*}^0\left(V_0\right)$ and $\mathcal{E}:={\mathcal{E}}_H$.

Proof. 

Since ${\mathrm{PS}}^0\left(V_0\right)=V_0$ and $V_0\ne \varnothing$, we have ${\mathrm{PS}}_{*}^0\left(V_0\right)=V_0$. Hence $V:=V_0$ satisfies $V\subseteq {\mathrm{PS}}_{*}^0\left(V_0\right)$, so $(V,{\mathcal{E}}_H)$ is a directed 0-SuperHyperGraph in the sense of Definition 24. Keeping the same state spaces ${\mathcal{X}}_i$, noise spaces ${\mathcal{U}}_i$, distribution $P_U$, mechanisms ${\phi }_e$, and aggregators $g_i$, the structural equations in Definition 25 become

$$X_i=g_i\left({\left({\phi }_e\left(X_{\mathrm{Tail}\left(e\right)}\right)\right)}_{e\in {\mathcal{E}}_i},U_i\right)\mathrm{with}{\mathcal{E}}_i=\{e\in {\mathcal{E}}_H\mid \mathrm{Head}\left(e\right)=i\},$$

which is exactly the given causal hypergraph model. Therefore the causal hypergraph model is a special case. □

Theorem 7  (Causal graphs (SCMs) are a special case ($n=0$, singleton tails)).

Let $G=(V_0,E_G)$ be a directed graph and let an SCM be given by

$$X_i=f_i\left({\left(X_j\right)}_{j\in Pa\left(i\right)},U_i\right),Pa\left(i\right):=\{j\in V_0\mid (j,i)\in E_G\}.$$

Define a directed hypergraph $H_G=(V_0,{\mathcal{E}}_G)$ by

$${\mathcal{E}}_G:=\{(\left\{j\right\}\to i)\mid (j,i)\in E_G\}.$$

Then the SCM is a special case of a causal 0-SuperHyperGraph model on ${\overrightarrow{\mathrm{SHG}}}^{\left(0\right)}=(V_0,{\mathcal{E}}_G)$.

Proof. 

By ${\mathrm{PS}}_{*}^0\left(V_0\right)=V_0$, the pair $(V_0,{\mathcal{E}}_G)$ is a directed 0-SuperHyperGraph. We now construct $\mathrm{\Phi }=\left\{{\phi }_e\right\}$ and $G=\left\{g_i\right\}$ so that the 0-SuperHyperGraph equations coincide with the SCM equations.

For each edge $(j,i)\in E_G$, let $e=(\left\{j\right\}\to i)\in {\mathcal{E}}_G$ and set ${\mathcal{Z}}_e:={\mathcal{X}}_j$ and

$${\phi }_e\left(x_j\right):=x_j(x_j\in {\mathcal{X}}_j).$$

For a fixed $i\in V_0$, the incoming hyperedges are

$${\mathcal{E}}_i=\{(\left\{j\right\}\to i)\mid j\in Pa\left(i\right)\}.$$

Hence

$$\prod _{e\in {\mathcal{E}}_i}{\mathcal{Z}}_e=\prod _{j\in Pa\left(i\right)}{\mathcal{X}}_j.$$

Define $g_i$ by

$$g_i\left({\left(z_e\right)}_{e\in {\mathcal{E}}_i},u_i\right):=f_i\left({\left(z_{\left(\right\{j\}\to i)}\right)}_{j\in Pa\left(i\right)},u_i\right).$$

Then the structural equation in Definition 25 becomes

$$\begin{array}{cc}\hfill X_i& =g_i\left({\left({\phi }_{\left(\right\{j\}\to i)}\left(X_j\right)\right)}_{j\in Pa\left(i\right)},U_i\right)\hfill \\ \hfill & =g_i\left({\left(X_j\right)}_{j\in Pa\left(i\right)},U_i\right)\hfill \\ \hfill & =f_i\left({\left(X_j\right)}_{j\in Pa\left(i\right)},U_i\right),\hfill \end{array}$$

which is exactly the SCM equation. Therefore the causal graph (SCM) is a special case of a causal 0-SuperHyperGraph model. □

7. Review and Results: SuperHyperGraph Convolutional Network

GCN aggregates each node’s features from its neighbors via normalized adjacency, then applies a learnable linear transform and nonlinearity per layer [120,121,122,123,124]. HGCN aggregates vertex features through hyperedges via an incidence-based diffusion operator, then applies learnable transformations to model multiway interactions [69,71,76,125]. SHGCN propagates features on supervertices using super-incidence across nested superedges, enabling hierarchical multiway diffusion and generalizing both GCN and HGCN. Table 6 presents the comparison among GCN, HGCN, and SHGCN.

Definition 27 

(Graph and node features). Let $G=(V,E)$ be a finite (undirected) graph with $\left|V\right|=n$. Let $A\in {\{0,1\}}^{n\times n}$ be its adjacency matrix, and let $X\in {\mathbb{R}}^{n\times d_0}$ be a node-feature matrix whose ith row $x_i^{\top }$ is the feature vector of vertex $v_i\in V$. Define the self-loop augmented adjacency

$$\tilde{A}=A+I_n,$$

and its degree matrix $\tilde{D}\in {\mathbb{R}}^{n\times n}$ by

$${\tilde{D}}_{ii}=\sum _{j=1}^n{\tilde{A}}_{ij}.$$

Definition 28 

(One GCN layer). Fix an activation function $\sigma :\mathbb{R}\to \mathbb{R}$ applied entrywise. A graph convolutional layer maps $X^{(\ell )}\in {\mathbb{R}}^{n\times d_{\ell }}$ to $X^{(\ell +1)}\in {\mathbb{R}}^{n\times d_{\ell +1}}$ by

$$X^{(\ell +1)}=\sigma \left({\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2}{X}^{(\ell )}{W}^{(\ell )}\right),$$

where $W^{(\ell )}\in {\mathbb{R}}^{d_{\ell }\times d_{\ell +1}}$ is a learnable weight matrix. AGCNis a composition of such layers (optionally followed by task-specific readout).

Remark 4 

(Interpretation). The operator ${\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2}$ performs degree-normalized neighborhood averaging (including self-loops), so each layer mixes information along edges and then applies a learnable linear map.

Definition 29 

(Weighted hypergraph and incidence matrix). A (finite) weighted hypergraph is a triple $H=(V,\mathcal{E},w)$ where $V=\{v_1,\dots ,v_n\}$ is a finite vertex set, $\mathcal{E}=\{e_1,\dots ,e_m\}$ is a finite family of nonempty subsets of V (hyperedges), and $w:\mathcal{E}\to {\mathbb{R}}_{>0}$ assigns positive weights. The incidence matrix is $H_{\mathrm{inc}}\in {\{0,1\}}^{n\times m}$ defined by

$${\left(H_{\mathrm{inc}}\right)}_{i\alpha }=\left\{\begin{array}{cc}1,\hfill & v_i\in e_{\alpha },\hfill \\ 0,\hfill & v_i\notin e_{\alpha }.\hfill \end{array}\right.$$

Let $W\in {\mathbb{R}}^{m\times m}$ be diagonal with $W_{\alpha \alpha }=w\left(e_{\alpha }\right)$. Define the vertex-degree matrix $D_v\in {\mathbb{R}}^{n\times n}$ and the hyperedge-degree matrix $D_e\in {\mathbb{R}}^{m\times m}$ by

$${\left(D_v\right)}_{ii}=\sum _{\alpha =1}^m{W}_{\alpha \alpha }{\left(H_{\mathrm{inc}}\right)}_{i\alpha },{\left(D_e\right)}_{\alpha \alpha }=\sum _{i=1}^n{\left(H_{\mathrm{inc}}\right)}_{i\alpha }=\left|e_{\alpha }\right|.$$

Definition 30  (One HGCN layer (spectral/message-passing form)).

Let $X^{(\ell )}\in {\mathbb{R}}^{n\times d_{\ell }}$ be vertex features at layer ℓ. Ahypergraph convolutional layermaps $X^{(\ell )}$ to $X^{(\ell +1)}$ by

$$X^{(\ell +1)}=\sigma \left(D_v^{-1/2}{H}_{\mathrm{inc}}W{D}_e^{-1}{H}_{\mathrm{inc}}^{\top }{D}_v^{-1/2}{X}^{(\ell )}{\Theta }^{(\ell )}\right),$$

where ${\Theta }^{(\ell )}\in {\mathbb{R}}^{d_{\ell }\times d_{\ell +1}}$ is learnable and σ is applied entrywise. An HGCN is a composition of such hypergraph convolutional layers (optionally with readout).

Remark 5 

(Two-stage aggregation: vertex→hyperedge→vertex). The matrix product $H_{\mathrm{inc}}^{\top }{X}^{(\ell )}$ aggregates vertex features to hyperedges, and then $H_{\mathrm{inc}}(·)$ propagates hyperedge information back to vertices. Thus one layer performs a normalized diffusion through hyperedges, capturing multiway interactions.

Remark 6 

(Row-normalized variant). A commonly used simplification replaces the symmetric normalization by

$$X^{(\ell +1)}=\sigma \left(D_v^{-1}{H}_{\mathrm{inc}}W{D}_e^{-1}{H}_{\mathrm{inc}}^{\top }{X}^{(\ell )}{\Theta }^{(\ell )}\right),$$

which still implements the vertex→hyperedge→vertex propagation with degree control.

Definition 31 

(Iterated powersets and flattening). Let $V_0$ be a finite nonempty set. Define

$${\mathcal{P}}^0\left(V_0\right):=V_0,{\mathcal{P}}^{k+1}\left(V_0\right):=\mathcal{P}\left({\mathcal{P}}^k\left(V_0\right)\right)(k\ge 0).$$

Define the flattening maps ${\mathrm{Flat}}_n:{\mathcal{P}}^n\left(V_0\right)\to \mathcal{P}\left(V_0\right)$ recursively by

$${\mathrm{Flat}}_0\left(x\right):=\left\{x\right\}(x\in V_0),{\mathrm{Flat}}_{n+1}\left(X\right):=\bigcup _{Y\in X}{\mathrm{Flat}}_n\left(Y\right)(X\in {\mathcal{P}}^{n+1}\left(V_0\right)).$$

Definition 32 

(n-SuperHyperGraph and super-incidence). Fix $n\ge 0$ and a base set $V_0$. Ann-SuperHyperGraph is a pair

$${\mathrm{SHG}}^{\left(n\right)}=(V,E),V\subseteq {\mathcal{P}}^n\left(V_0\right),E\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}.$$

Write $V=\{v_1,\dots ,v_N\}$ and $E=\{e_1,\dots ,e_M\}$.

The super-incidence matrix$H^{\left(n\right)}\in {\{0,1\}}^{N\times M}$ is defined by

$$H_{i\alpha }^{\left(n\right)}:=\left\{\begin{array}{cc}1,\hfill & v_i\in e_{\alpha },\hfill \\ 0,\hfill & v_i\notin e_{\alpha }.\hfill \end{array}\right.$$

Let $W^{\left(n\right)}=\mathrm{diag}(w_1,\dots ,w_M)$ be a diagonal matrix of positive superedge weights.

Define degree matrices $D_v^{\left(n\right)}\in {\mathbb{R}}^{N\times N}$ and $D_e^{\left(n\right)}\in {\mathbb{R}}^{M\times M}$ by

$${\left(D_v^{\left(n\right)}\right)}_{ii}:=\sum _{\alpha =1}^M{w}_{\alpha }{H}_{i\alpha }^{\left(n\right)},{\left(D_e^{\left(n\right)}\right)}_{\alpha \alpha }:=\sum _{i=1}^N{H}_{i\alpha }^{\left(n\right)}=\left|e_{\alpha }\right|.$$

Two canonical propagation operators on supervertices are:

(1)Superhypergraph diffusion operator(incidence-based):

$$S_{\mathrm{diff}}^{\left(n\right)}:={\left(D_v^{\left(n\right)}\right)}^{-1/2}{H}^{\left(n\right)}{W}^{\left(n\right)}{\left(D_e^{\left(n\right)}\right)}^{-1}{\left(H^{\left(n\right)}\right)}^{\mathsf{T}}{\left(D_v^{\left(n\right)}\right)}^{-1/2}.$$

(2)Supervertex 2-section (clique-expansion) operator: Let $A_2^{\left(n\right)}\in {\{0,1\}}^{N\times N}$ be the adjacency matrix of the 2-section graph on V, i.e., for $i\ne j$,

$${\left(A_2^{\left(n\right)}\right)}_{ij}:=\left\{\begin{array}{cc}1,\hfill & \exists \alpha \in \{1,\dots ,M\}\mathrm{such}\mathrm{that}{v}_i\in e_{\alpha }\mathrm{and}{v}_j\in e_{\alpha },\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.{\left(A_2^{\left(n\right)}\right)}_{ii}:=0.$$

Define ${\tilde{A}}_2^{\left(n\right)}:=A_2^{\left(n\right)}+I_N$ and ${\tilde{D}}_2^{\left(n\right)}$ by ${\tilde{D}}_2^{\left(n\right)}(i,i):={\sum }_{j=1}^N{\tilde{A}}_2^{\left(n\right)}(i,j)$. Then set

$$S_{2-\sec }^{\left(n\right)}:={\left({\tilde{D}}_2^{\left(n\right)}\right)}^{-1/2}{\tilde{A}}_2^{\left(n\right)}{\left({\tilde{D}}_2^{\left(n\right)}\right)}^{-1/2}.$$

Definition 33 

(Lifting base features to n-supervertices). Let ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$ be on base set $V_0=\{x_1,\dots ,x_{n_0}\}$. Let $X^{\left(0\right)}\in {\mathbb{R}}^{n_0\times d_0}$ be base vertex features, where row $X_{p,:}^{\left(0\right)}$ corresponds to $x_p$.Define the support of a supervertex $v\in V$ by

$$B\left(v\right):={\mathrm{Flat}}_n\left(v\right)\subseteq V_0.$$

Define lifted supervertex features $X^{(n,0)}\in {\mathbb{R}}^{\left|V\right|\times d_0}$ by mean aggregation: for $v_i\in V$,

$$X_{i,:}^{(n,0)}:={\displaystyle \frac{1}{\left|B\right(v_i\left)\right|}}\sum _{x_p\in B\left(v_i\right)}{X}_{p,:}^{\left(0\right)}.$$

Definition 34 

(One n-SuperHyperGraph convolutional layer). Let ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$ and let $S^{\left(n\right)}$ be a chosen propagation operator on V (e.g., $S_{\mathrm{diff}}^{\left(n\right)}$ or $S_{2-\sec }^{\left(n\right)}$ from Definition 32). Let $X^{(\ell )}\in {\mathbb{R}}^{\left|V\right|\times d_{\ell }}$ be supervertex features at layer ℓ.

Ann-SuperHyperGraph convolutional layer is the map

$$X^{(\ell +1)}=\sigma \left(S^{\left(n\right)}{X}^{(\ell )}{\Theta }^{(\ell )}\right),$$

where ${\Theta }^{(\ell )}\in {\mathbb{R}}^{d_{\ell }\times d_{\ell +1}}$ is learnable and σ is applied entrywise.

Definition 35 

(n-SuperHyperGraph Convolutional Network (n-SHGCN)). Let ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$ be an n-SuperHyperGraph on base set $V_0$. Given base features $X^{\left(0\right)}\in {\mathbb{R}}^{|V_0|\times d_0}$, form lifted features $X^{(n,0)}$ by Definition 33. Fix a propagation operator $S^{\left(n\right)}$ on V.

AnL-layer n-SHGCN is a sequence

$$X^{(n,0)}\mapsto X^{(n,1)}\mapsto \dots \mapsto X^{(n,L)},$$

where each step is an n-SuperHyperGraph convolutional layer (Definition 34). Optionally, a readout map is applied for downstream tasks.

Theorem 8 

(n-SHGCN generalizes HGCN and GCN). Fix $d_0,d_1\ge 1$.

(i) 

(HGCN as a special case) Let $H=(V_0,\mathcal{E},w)$ be a weighted hypergraph. Consider the 0-SuperHyperGraph ${\mathrm{SHG}}^{\left(0\right)}=(V,E)$ defined by

$$V:=V_0,E:=\mathcal{E}.$$

Choose $S^{\left(0\right)}:=S_{\mathrm{diff}}^{\left(0\right)}$. Then a one-layer n-SHGCN update on ${\mathrm{SHG}}^{\left(0\right)}$ coincides exactly with a one-layer HGCN update on H.

(ii) 

(GCN as a special case) Let $G=(V_0,E_G)$ be a (simple) graph. Form the 2-uniform hypergraph $H_G=(V_0,{\mathcal{E}}_G)$ where ${\mathcal{E}}_G:=\{\{u,v\}\mid \{u,v\}\in E_G\}$, and view it as the 0-SuperHyperGraph ${\mathrm{SHG}}^{\left(0\right)}=(V_0,{\mathcal{E}}_G)$. Choose $S^{\left(0\right)}:=S_{2-\sec }^{\left(0\right)}$. Then a one-layer n-SHGCN update on ${\mathrm{SHG}}^{\left(0\right)}$ coincides exactly with a one-layer GCN update on G.

Proof.  (i) Let $H=(V_0,\mathcal{E},w)$ be given. Define ${\mathrm{SHG}}^{\left(0\right)}=(V,E)$ by $V=V_0$ and $E=\mathcal{E}$. Since $n=0$, the flattening satisfies ${\mathrm{Flat}}_0\left(v\right)=\left\{v\right\}$ for $v\in V_0$, hence Definition 33 gives $X^{(0,0)}=X^{\left(0\right)}$ (no change of features).

The super-incidence matrix $H^{\left(0\right)}$ (Definition 32) is precisely the usual hypergraph incidence matrix, because its entries satisfy $H_{i\alpha }^{\left(0\right)}=1$ iff $v_i\in e_{\alpha }$. The degree matrices $D_v^{\left(0\right)}$ and $D_e^{\left(0\right)}$ also match the standard hypergraph degrees:

$${\left(D_v^{\left(0\right)}\right)}_{ii}=\sum _{\alpha }{w}_{\alpha }{H}_{i\alpha }^{\left(0\right)},{\left(D_e^{\left(0\right)}\right)}_{\alpha \alpha }=\sum _i{H}_{i\alpha }^{\left(0\right)}=\left|e_{\alpha }\right|.$$

Therefore the n-SHGCN diffusion operator

$$S_{\mathrm{diff}}^{\left(0\right)}={\left(D_v^{\left(0\right)}\right)}^{-1/2}{H}^{\left(0\right)}{W}^{\left(0\right)}{\left(D_e^{\left(0\right)}\right)}^{-1}{\left(H^{\left(0\right)}\right)}^{\mathsf{T}}{\left(D_v^{\left(0\right)}\right)}^{-1/2}$$

is exactly the HGCN propagation matrix. Substituting into the layer update

$$X^{\left(1\right)}=\sigma \left(S_{\mathrm{diff}}^{\left(0\right)}{X}^{\left(0\right)}{\Theta }^{\left(0\right)}\right)$$

shows that the one-layer n-SHGCN update coincides with the one-layer HGCN update.

(ii) Let $G=(V_0,E_G)$ be a simple graph and let ${\mathcal{E}}_G=\{\{u,v\}\mid \{u,v\}\in E_G\}$. Consider ${\mathrm{SHG}}^{\left(0\right)}=(V_0,{\mathcal{E}}_G)$ and choose $S^{\left(0\right)}=S_{2-\sec }^{\left(0\right)}$.

We show that the 2-section adjacency matrix $A_2^{\left(0\right)}$ equals the usual adjacency matrix A of G. For distinct $u,v\in V_0$,

$${\left(A_2^{\left(0\right)}\right)}_{uv}=1⟺\exists e\in {\mathcal{E}}_G\mathrm{with}u\in e\mathrm{and}v\in e⟺\{u,v\}\in E_G⟺A_{uv}=1.$$

Thus $A_2^{\left(0\right)}=A$. Consequently,

$${\tilde{A}}_2^{\left(0\right)}=A_2^{\left(0\right)}+I=A+I,{\tilde{D}}_2^{\left(0\right)}=\tilde{D},$$

where $\tilde{D}$ is the degree matrix of $\tilde{A}=A+I$. Hence

$$S_{2-\sec }^{\left(0\right)}={\left({\tilde{D}}_2^{\left(0\right)}\right)}^{-1/2}{\tilde{A}}_2^{\left(0\right)}{\left({\tilde{D}}_2^{\left(0\right)}\right)}^{-1/2}={\tilde{D}}^{-1/2}(A+I){\tilde{D}}^{-1/2},$$

which is exactly the standard GCN propagation matrix. Substituting into the layer update yields

$$X^{\left(1\right)}=\sigma \left({\tilde{D}}^{-1/2}(A+I){\tilde{D}}^{-1/2}{X}^{\left(0\right)}{\Theta }^{\left(0\right)}\right),$$

so the one-layer n-SHGCN update coincides with the one-layer GCN update.

This proves both claims. □

8. SuperHyperGraph Embedding

Graph embedding maps nodes or graphs into low-dimensional vectors, preserving adjacency and structural similarity for efficient prediction and retrieval tasks [126,127,128,129]. Hypergraph embedding maps vertices and hyperedges into vectors, preserving higher-order relations among multiple vertices for robust classification and link prediction [61,130,131,132]. Superhypergraph embedding maps nested hyperedges into vectors, capturing hyperedge-of-hyperedge structure to support hierarchical reasoning, representation learning, and prediction tasks efficiently. For reference, Table 7 presents a comparison of Graph Embedding, HyperGraph Embedding, and SuperHyperGraph Embedding.

Definition 36 

(Graph embedding in a vector space). [128,133,134] Let $G=(V,E,w)$ be a (possibly weighted) simple graph with vertex set $V=\{v_1,\dots ,v_n\}$, edge set $E\subseteq \left({\displaystyle \genfrac{}{}{0pt}{}{V}{2}}\right)$, and weight function $w:E\to {\mathbb{R}}_{>0}$. Fix an integer $d\ge 1$.

Ad-dimensional graph embedding is a map

$$\phi :V⟶{\mathbb{R}}^d,\mathrm{equivalently}Z\in {\mathbb{R}}^{n\times d}\mathrm{with}\mathrm{row}{Z}_{i,:}=\phi \left(v_i\right),$$

together with a prescribed graph proximity (or relation) function

$$s:V\times V\to \mathbb{R},$$

and a prescribed embedding similarity function

$$\mathrm{sim}:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R},$$

such that $\mathrm{sim}\left(\phi \right(u),\phi (v\left)\right)$ approximates $s(u,v)$ (or preserves its ordering) for relevant pairs $(u,v)$.

Formally, one common way to specify this requirement is by defining φ as a minimizer of

$$\underset{\phi :V\to {\mathbb{R}}^d}{min}\sum _{(u,v)\in \Omega }L\left(s(u,v),\mathrm{sim}\left(\phi \right(u),\phi (v\left)\right)\right)+\lambda \mathcal{R}\left(\phi \right),$$

where $\Omega \subseteq V\times V$ is the set of pairs used for learning/evaluation, L is a loss function, $\mathcal{R}$ is a regularizer, and $\lambda \ge 0$.

Definition 37 

(Hypergraph and incidence representation). A (weighted) hypergraph is a triple $\mathcal{H}=(V,\mathcal{E},W)$ where $V=\{v_1,\dots ,v_n\}$ is a vertex set, $\mathcal{E}=\{e_1,\dots ,e_m\}$ is a set of hyperedges with $e\subseteq V$ and $\left|e\right|\ge 2$, and $W=\mathrm{diag}(w_{e_1},\dots ,w_{e_m})$ is a positive diagonal matrix of hyperedge weights.

The vertex-hyperedge membership is represented by the incidence matrix $H\in {\{0,1\}}^{n\times m}$ defined by

$$H(v,e)=\left\{\begin{array}{cc}1,\hfill & v\in e,\hfill \\ 0,\hfill & v\notin e.\hfill \end{array}\right.$$

Define diagonal degree matrices $D_v\in {\mathbb{R}}^{n\times n}$ and $D_e\in {\mathbb{R}}^{m\times m}$ by

$$D_v(i,i)=\sum _{e\in \mathcal{E}}{w}_{e}H(v_i,e),D_e(j,j)=\sum _{v\in V}H(v,e_j).$$

A normalized hypergraph operator and Laplacian can be defined by

$$\Theta :=D_v^{-1/2}HW{D}_e^{-1}{H}^{\mathsf{T}}{D}_v^{-1/2},\Delta :=I-\Theta .$$

Definition 38 

(Hypergraph embedding in a vector space). Let $\mathcal{H}=(V,\mathcal{E},W)$ be a (weighted) hypergraph with $\left|V\right|=n$ and fix $d\ge 1$.

Ad-dimensional (vertex) hypergraph embedding is a map

$$\psi :V⟶{\mathbb{R}}^d,\mathrm{equivalently}Z\in {\mathbb{R}}^{n\times d}\mathrm{with}\mathrm{row}{Z}_{i,:}=\psi \left(v_i\right),$$

together with a prescribed hyperedge proximity functional

$$F_e:{\left({\mathbb{R}}^d\right)}^{\left|e\right|}\to \mathbb{R}(e\in \mathcal{E}),$$

such that the embedding preserves higher-order relations encoded by hyperedges.

A standard formal specification is to define ψ as a minimizer of

$$\underset{\psi :V\to {\mathbb{R}}^d}{min}\sum _{e\in \mathcal{E}}{w}_e{\mathcal{L}}_e\left(F_e\left({\left(\psi \left(v\right)\right)}_{v\in e}\right)\right)+\lambda \mathcal{R}\left(\psi \right),$$

where ${\mathcal{L}}_e$ is a loss that enforces the desired within-hyperedge geometry (e.g., small dispersion, large n-wise similarity, or consistency with observed hyperedge events), and $\mathcal{R}$ is a regularizer.

One concrete (spectral) instance uses the normalized hypergraph Laplacian Δ: choose $U\in {\mathbb{R}}^{n\times d}$ whose columns are eigenvectors corresponding to the d smallest nontrivial eigenvalues of Δ, and set $\psi \left(v_i\right)=U_{i,:}$.

Definition 39 

(Hypergraph embedding for downstream tasks as a representation matrix). In many learning settings, a hypergraph embedding is presented as a representation matrix

$$Z_{\mathcal{H}}\in {\mathbb{R}}^{\left|V\right|\times C},$$

where each row represents the embedding of a vertex, and the representation is used for downstream tasks (e.g., node classification, graph classification) via a predictor built on $Z_{\mathcal{H}}$.

Definition 40 

(Nesting map and flattening map). Let $V_0$ be a finite, nonempty set, and let ${\mathrm{PS}}^0\left(V_0\right):=V_0$ and ${\mathrm{PS}}^{k+1}\left(V_0\right):=\mathrm{PS}\left({\mathrm{PS}}^k\left(V_0\right)\right)$ for $k\ge 0$.

Define then-fold nesting map ${\mathrm{Nest}}_n:V_0\to {\mathrm{PS}}^n\left(V_0\right)$ by

$${\mathrm{Nest}}_0\left(x\right):=x,{\mathrm{Nest}}_{n+1}\left(x\right):=\left\{{\mathrm{Nest}}_n\left(x\right)\right\}(x\in V_0,n\ge 0).$$

Define then-level flattening map ${\mathrm{Flat}}_n:{\mathrm{PS}}^n\left(V_0\right)\to \mathrm{PS}\left(V_0\right)$ recursively as follows:

$${\mathrm{Flat}}_0\left(x\right):=\left\{x\right\}(x\in V_0),{\mathrm{Flat}}_{n+1}\left(X\right):=\bigcup _{Y\in X}{\mathrm{Flat}}_n\left(Y\right)(X\in {\mathrm{PS}}^{n+1}\left(V_0\right),n\ge 0).$$

Lemma 1 

(Flattening a nested singleton). For every $n\ge 0$ and every $x\in V_0$,

$${\mathrm{Flat}}_n\left({\mathrm{Nest}}_n\left(x\right)\right)=\left\{x\right\}.$$

Proof. 

We prove by induction on n.

If $n=0$, then ${\mathrm{Nest}}_0\left(x\right)=x$ and ${\mathrm{Flat}}_0\left(x\right)=\left\{x\right\}$ by definition, hence ${\mathrm{Flat}}_0\left({\mathrm{Nest}}_0\left(x\right)\right)=\left\{x\right\}$.

Assume ${\mathrm{Flat}}_n\left({\mathrm{Nest}}_n\left(x\right)\right)=\left\{x\right\}$ holds for some $n\ge 0$. Then

$${\mathrm{Nest}}_{n+1}\left(x\right)=\left\{{\mathrm{Nest}}_n\left(x\right)\right\}.$$

Applying the recursive definition of ${\mathrm{Flat}}_{n+1}$ gives

$${\mathrm{Flat}}_{n+1}\left({\mathrm{Nest}}_{n+1}\left(x\right)\right)={\mathrm{Flat}}_{n+1}\left(\left\{{\mathrm{Nest}}_n\left(x\right)\right\}\right)=\bigcup _{Y\in \left\{{\mathrm{Nest}}_n\left(x\right)\right\}}{\mathrm{Flat}}_n\left(Y\right)={\mathrm{Flat}}_n\left({\mathrm{Nest}}_n\left(x\right)\right)=\left\{x\right\}.$$

Thus the statement holds for $n+1$. This completes the induction. □

Definition 41 

(n-lift of a hypergraph). Let $H=(V_0,\mathcal{E})$ be a hypergraph, i.e., $\mathcal{E}\subseteq \mathrm{PS}\left(V_0\right)\setminus \{\varnothing \}$. Fix $n\ge 0$.

Define the n-lifted vertex set and hyperedge family by

$$V_n:=\{{\mathrm{Nest}}_n\left(v\right)\mid v\in V_0\}\subseteq {\mathrm{PS}}^n\left(V_0\right),{\mathcal{E}}_n:=\left\{{\mathrm{Nest}}_n\left(e\right)\mid e\in \mathcal{E}\right\},$$

where for each hyperedge $e\subseteq V_0$ we set

$${\mathrm{Nest}}_n\left(e\right):=\{{\mathrm{Nest}}_n\left(v\right)\mid v\in e\}\subseteq V_n.$$

Then

$${\mathrm{Lift}}_n\left(H\right):=(V_n,{\mathcal{E}}_n)$$

is an n-SuperHyperGraph in the sense that $V_n\subseteq {\mathrm{PS}}^n\left(V_0\right)$ and ${\mathcal{E}}_n\subseteq \mathrm{PS}\left(V_n\right)\setminus \{\varnothing \}$.

Definition 42 

(n-SuperHypergraph embedding). Let ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$ be an n-SuperHyperGraph on a base set $V_0$, so $V\subseteq {\mathrm{PS}}^n\left(V_0\right)$ and $E\subseteq \mathrm{PS}\left(V\right)\setminus \{\varnothing \}$. Fix an integer $d\ge 1$.

Ann-SuperHypergraph embedding in ${\mathbb{R}}^d$ is a pair of maps

$$\phi :V_0\to {\mathbb{R}}^d,\psi :V\to {\mathbb{R}}^d,$$

together with (i) an aggregation rule $\mathrm{Agg}$ for finite multisets of vectors and (ii) a family of hyperedge-coherence losses ${\left({\mathcal{L}}_e\right)}_{e\in E}$, where each ${\mathcal{L}}_e$ maps the tuple ${\left(\psi \left(v\right)\right)}_{v\in e}$ to a nonnegative real number.

A concrete (canonical) choice that is widely used is:

$$\mathrm{Agg}\left(\{y_1,\dots ,y_k\}\right):={\displaystyle \frac{1}{k}}\sum _{i=1}^k{y}_i,$$

and for each $e\in E$,

$${\mathcal{L}}_e\left({\left(\psi \left(v\right)\right)}_{v\in e}\right):={\displaystyle \frac{1}{\left|e\right|}}\sum _{v\in e}{∥\psi \left(v\right)-{\mu }_e∥}^2,{\mu }_e:={\displaystyle \frac{1}{\left|e\right|}}\sum _{u\in e}\psi \left(u\right).$$

In addition, to reflect theSuperHyperGraph nesting, we enforce a hierarchical consistency term via ${\mathrm{Flat}}_n$: for each $v\in V$, define its base support $B\left(v\right):={\mathrm{Flat}}_n\left(v\right)\subseteq V_0$ and set

$${\mathrm{Anchor}}_{\phi }\left(v\right):=\mathrm{Agg}\left(\left\{\phi \right(x)\mid x\in B(v\left)\right\}\right)\in {\mathbb{R}}^d.$$

One standard way to specify that $(\phi ,\psi )$ is an embedding is to require it to minimize an objective of the form

$$J(\phi ,\psi ):=\sum _{e\in E}{w}_e{\mathcal{L}}_e\left({\left(\psi \left(v\right)\right)}_{v\in e}\right)+\beta \sum _{v\in V}{∥\psi \left(v\right)-{\mathrm{Anchor}}_{\phi }\left(v\right)∥}^2+\lambda \mathcal{R}(\phi ,\psi ),$$

where $w_e>0$ are hyperedge weights, $\beta ,\lambda \ge 0$, and $\mathcal{R}$ is a regularizer.

Theorem 9 

(SuperHypergraph embeddings generalize hypergraph and graph embeddings). Fix $d\ge 1$ and $n\ge 0$.

(i) 

(Hypergraph case) Let $H=(V_0,\mathcal{E})$ be a hypergraph and let $\mathrm{\Phi }:V_0\to {\mathbb{R}}^d$ be any vertex embedding of H (e.g., specified by minimizing a hyperedge-coherence objective over $\mathcal{E}$). Then ${\mathrm{Lift}}_n\left(H\right)$ admits an n-SuperHypergraph embedding $(\phi ,\psi )$ such that

$$\psi \left({\mathrm{Nest}}_n\left(v\right)\right)=\mathrm{\Phi }\left(v\right)\mathrm{for}\mathrm{all}v\in V_0,$$

and for the canonical loss ${\mathcal{L}}_e$ in Definition 42, the hyperedge terms on H and on ${\mathrm{Lift}}_n\left(H\right)$ coincide under this identification.

(ii)  

(Graph case) Let $G=(V_0,E_G)$ be a (simple) graph and regard it as the 2-uniform hypergraph $H_G=(V_0,{\mathcal{E}}_G)$ with ${\mathcal{E}}_G:=\{\{u,v\}\mid \{u,v\}\in E_G\}$. Then every graph embedding $\mathrm{\Phi }:V_0\to {\mathbb{R}}^d$ induces an n-SuperHypergraph embedding of ${\mathrm{Lift}}_n\left(H_G\right)$ via the same rule $\psi \left({\mathrm{Nest}}_n\left(v\right)\right)=\mathrm{\Phi }\left(v\right)$. Thus graph embeddings are contained as a special case of n-SuperHypergraph embeddings.

Proof.  (i) Consider the lifted n-SuperHyperGraph ${\mathrm{Lift}}_n\left(H\right)=(V_n,{\mathcal{E}}_n)$ from Definition 41. Define

$$\phi :=\mathrm{\Phi }\mathrm{on}{V}_0,\psi :V_n\to {\mathbb{R}}^d\mathrm{by}\psi \left({\mathrm{Nest}}_n\left(v\right)\right):=\mathrm{\Phi }\left(v\right).$$

This is well-defined because each element of $V_n$ is uniquely of the form ${\mathrm{Nest}}_n\left(v\right)$.

Let $e\in \mathcal{E}$ be a hyperedge, and let $e_n:={\mathrm{Nest}}_n\left(e\right)\in {\mathcal{E}}_n$ be its lifted hyperedge. Using the canonical coherence loss from Definition 42, we compute explicitly:

$${\mu }_{e_n}={\displaystyle \frac{1}{|e_n|}}\sum _{u^{\prime }\in e_n}\psi \left(u^{\prime }\right)={\displaystyle \frac{1}{\left|e\right|}}\sum _{u\in e}\psi \left({\mathrm{Nest}}_n\left(u\right)\right)={\displaystyle \frac{1}{\left|e\right|}}\sum _{u\in e}\mathrm{\Phi }\left(u\right)={\mu }_e,$$

where ${\mu }_e:={\displaystyle \frac{1}{\left|e\right|}}{\sum }_{u\in e}\mathrm{\Phi }\left(u\right)$ is the corresponding mean for the hyperedge e under $\mathrm{\Phi }$. Therefore,

$${\mathcal{L}}_{e_n}\left({\left(\psi \left(v^{\prime }\right)\right)}_{v^{\prime }\in e_n}\right)={\displaystyle \frac{1}{\left|e\right|}}\sum _{v\in e}{∥\psi \left({\mathrm{Nest}}_n\left(v\right)\right)-{\mu }_{e_n}∥}^2={\displaystyle \frac{1}{\left|e\right|}}\sum _{v\in e}{∥\mathrm{\Phi }\left(v\right)-{\mu }_e∥}^2={\mathcal{L}}_e\left({\left(\mathrm{\Phi }\left(v\right)\right)}_{v\in e}\right).$$

Hence the sum of hyperedge-coherence terms is preserved by the lift.

Moreover, the hierarchical consistency term is compatible with the lift. For any $v\in V_0$, Lemma 1 yields

$${\mathrm{Flat}}_n\left({\mathrm{Nest}}_n\left(v\right)\right)=\left\{v\right\}.$$

With mean aggregation,

$${\mathrm{Anchor}}_{\phi }\left({\mathrm{Nest}}_n\left(v\right)\right)=\mathrm{Agg}\left(\{\phi \left(x\right)\mid x\in {\mathrm{Flat}}_n\left({\mathrm{Nest}}_n\left(v\right)\right)\}\right)=\mathrm{Agg}\left(\left\{\phi \left(v\right)\right\}\right)=\phi \left(v\right)=\mathrm{\Phi }\left(v\right).$$

Therefore,

$${∥\psi \left({\mathrm{Nest}}_n\left(v\right)\right)-{\mathrm{Anchor}}_{\phi }\left({\mathrm{Nest}}_n\left(v\right)\right)∥}^2={\parallel \mathrm{\Phi }\left(v\right)-\mathrm{\Phi }\left(v\right)\parallel }^2=0,$$

so the hierarchical term vanishes on lifted vertices. This shows that $(\phi ,\psi )$ is an n-SuperHypergraph embedding of ${\mathrm{Lift}}_n\left(H\right)$ and that the lifted model recovers the original hypergraph embedding behavior.

(ii) A graph $G=(V_0,E_G)$ is a special case of a hypergraph by taking only 2-element hyperedges. Applying part (i) to $H_G=(V_0,{\mathcal{E}}_G)$ produces an n-SuperHypergraph embedding of ${\mathrm{Lift}}_n\left(H_G\right)$ with $\psi \left({\mathrm{Nest}}_n\left(v\right)\right)=\mathrm{\Phi }\left(v\right)$. Thus graph embeddings are included as a special case. □

9. Result and Reviews: SuperHyperGraph-Based Natural Language Processing

Natural language processing (NLP) studies learning algorithms that map text to structured representations, meanings, or decisions using corpora, representations, and task-specific objective functions [135,136,137,138,139]. Related notions such as Fuzzy Natural Language [140,141,142] and Natural HyperLanguage [143,144] are also known in the literature. Graph-based NLP encodes text as a graph of tokens, entities, or relations and learns predictors on vertices and edges via message passing and attention mechanisms [145,146,147,148,149]. Hypergraph-based NLP encodes text as a hypergraph whose hyperedges connect multiple items simultaneously, learning representations that capture higher-order, multiway linguistic interactions. Superhypergraph-based NLP encodes text as an n-level superhypergraph through iterated powerset constructions, enabling nested linguistic groupings and learning predictors that generalize both graph and hypergraph pipelines. The comparison of Graph-Based NLP, HyperGraph-Based NLP, and SuperHyperGraph-Based NLP is presented in Table 8.

Definition 43  (Graph-Based Natural Language Processing (Graph-NLP)).

Fix a finite alphabet Σ and let $\mathcal{X}\subseteq {\Sigma }^{*}$ be a set of text objects (e.g., tokens, sentences, documents, dialogue threads), and let $\mathcal{Y}$ be an output space (e.g., labels, rankings, structured outputs).

A graph construction (encoder) is a mapping

$$\Gamma :\mathcal{X}⟶\mathcal{G},$$

where $\mathcal{G}$ is a class of attributed graphs, and for each $x\in \mathcal{X}$,

$$\Gamma \left(x\right)=G_x=(V_x,E_x,{\ell }_V,{\ell }_E,w)$$

with

• $V_x$ a finite vertex set (representing linguistic units such as tokens, entities, sentences, posts),
• $E_x\subseteq V_x\times V_x$ an edge set (representing relations such as adjacency, dependency, coreference, reply-to, semantic links),
• ${\ell }_V:V_x\to {\mathcal{A}}_V$ and ${\ell }_E:E_x\to {\mathcal{A}}_E$ vertex/edge attributes (discrete or continuous),
• w optional weights on vertices/edges.

A graph-based NLP model family is a set of measurable maps

$$\mathcal{F}=\{f_{\theta }:\mathcal{G}\to \mathcal{Y}\mid \theta \in \Theta \},$$

where each $f_{\theta }$ is allowed to use the graph structure (e.g., paths, neighborhoods, walks, cuts, centrality, or message passing) in computing the output.

Given a distribution $\mathbb{P}$ over pairs $(x,y)\in \mathcal{X}\times \mathcal{Y}$ and a loss $\mathcal{L}:\mathcal{Y}\times \mathcal{Y}\to {\mathbb{R}}_{\ge 0}$,Graph-NLPis the learning/optimization problem

$$\underset{\theta \in \Theta }{min}{\mathbb{E}}_{(x,y)\sim \mathbb{P}}\left[\mathcal{L}\left(f_{\theta }\left(\Gamma \left(x\right)\right),y\right)\right].$$

Any pipeline $x\mapsto \Gamma \left(x\right)\mapsto f_{\theta }\left(\Gamma \left(x\right)\right)$ that is designed and evaluated for NLP tasks (e.g., classification, extraction, retrieval, summarization, entailment) is called agraph-based NLP method.

Definition 44  (HyperGraph-Based Natural Language Processing (HyperGraph-NLP)).

Fix a finite alphabet Σ, a set of text objects $\mathcal{X}\subseteq {\Sigma }^{*}$, and an output space $\mathcal{Y}$.

A hypergraph construction (encoder) is a mapping

$${\Gamma }_H:\mathcal{X}\to \mathcal{H},$$

where $\mathcal{H}$ is a class of finite attributed hypergraphs. For each $x\in \mathcal{X}$,

$${\Gamma }_H\left(x\right)=H_x=(V_x,{\mathcal{E}}_x,{\ell }_V,{\ell }_{\mathcal{E}},w),$$

where $V_x$ is a finite vertex set, ${\mathcal{E}}_x\subseteq \mathcal{P}\left(V_x\right)\setminus \{\varnothing \}$ is a family of hyperedges, ${\ell }_V:V_x\to {\mathcal{A}}_V$ and ${\ell }_{\mathcal{E}}:{\mathcal{E}}_x\to {\mathcal{A}}_E$ are attribute maps, and w denotes optional weights (on vertices and/or hyperedges).

A hypergraph-based NLP model family is a set of (measurable) maps

$${\mathcal{F}}_H=\{f_{\theta }:\mathcal{H}\to \mathcal{Y}\mid \theta \in \Theta \}.$$

Given a distribution $\mathbb{P}$ on $\mathcal{X}\times \mathcal{Y}$ and a loss $\mathcal{L}:\mathcal{Y}\times \mathcal{Y}\to {\mathbb{R}}_{\ge 0}$, HyperGraph-NLP is the learning problem

$$\underset{\theta \in \Theta }{min}{\mathbb{E}}_{(x,y)\sim \mathbb{P}}\left[\mathcal{L}\left(f_{\theta }\left({\Gamma }_H\left(x\right)\right),y\right)\right].$$

Definition 45  (SuperHyperGraph-Based Natural Language Processing (SuperHyperGraph-NLP)).

Fix $\Sigma ,\mathcal{X},\mathcal{Y}$ as above and fix an integer $n\ge 0$.

Ann–SuperHyperGraph construction (encoder) is a mapping

$${\Gamma }_{SH}^{\left(n\right)}:\mathcal{X}\to {\mathcal{S}}^{\left(n\right)},$$

where ${\mathcal{S}}^{\left(n\right)}$ is a class of finite attributed n–SuperHyperGraphs. For each $x\in \mathcal{X}$,

$${\Gamma }_{SH}^{\left(n\right)}\left(x\right)={\mathrm{SHG}}_x^{\left(n\right)}=(V_{0,x},V_x,E_x,{\ell }_V,{\ell }_E,w),$$

such that

$$V_{0,x}\ne \varnothing \mathrm{is}\mathrm{finite},V_x\subseteq {\mathcal{P}}^n\left(V_{0,x}\right),E_x\subseteq \mathcal{P}\left(V_x\right)\setminus \{\varnothing \}.$$

Elements of $V_x$ aren–supervertices and elements of $E_x$ aren–superedges. The maps ${\ell }_V:V_x\to {\mathcal{A}}_V$ and ${\ell }_E:E_x\to {\mathcal{A}}_E$ assign attributes, and w denotes optional weights.

A superhypergraph-based NLP model family is

$${\mathcal{F}}_{SH}^{\left(n\right)}=\{g_{\theta }:{\mathcal{S}}^{\left(n\right)}\to \mathcal{Y}\mid \theta \in \Theta \}.$$

Given $\mathbb{P}$ and $\mathcal{L}$ as above, SuperHyperGraph-NLP is

$$\underset{\theta \in \Theta }{min}{\mathbb{E}}_{(x,y)\sim \mathbb{P}}\left[\mathcal{L}\left(g_{\theta }\left({\Gamma }_{SH}^{\left(n\right)}\left(x\right)\right),y\right)\right].$$

Definition 46 

(Graph-to-hypergraph inclusion). Let $G=(V,E)$ be a (simple, undirected) graph with $E\subseteq \left\{\right\{u,v\}\subseteq V\mid u\ne v\}$. Define the associated 2-uniform hypergraph

$$\iota \left(G\right)=(V,\mathcal{E})\mathrm{by}\mathcal{E}=\left\{\right\{u,v\}\mid \{u,v\}\in E\}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}.$$

Theorem 10 

(HyperGraph-NLP generalizes Graph-NLP). Every instance of Graph-Based Natural Language Processing is a special case of HyperGraph-Based Natural Language Processing.

Proof. 

Consider any Graph-NLP pipeline consisting of a graph encoder ${\Gamma }_G:\mathcal{X}\to \mathcal{G}$ and a model family ${\mathcal{F}}_G=\{f_{\theta }:\mathcal{G}\to \mathcal{Y}\}$, where $\mathcal{G}$ is a class of finite attributed graphs.

Define a hypergraph encoder by composition:

$${\Gamma }_H=\iota \circ {\Gamma }_G:\mathcal{X}\to \mathcal{H},$$

where $\mathcal{H}$ contains (at least) all 2-uniform attributed hypergraphs obtained from graphs in $\mathcal{G}$. Define a hypergraph model family on this subclass by

$$f_{\theta }^H\left(\iota \left(G\right)\right)=f_{\theta }\left(G\right)\mathrm{for}\mathrm{every}G\in \mathcal{G}.$$

Then for every $(x,y)$,

$$\mathcal{L}\left(f_{\theta }^H\left({\Gamma }_H\left(x\right)\right),y\right)=\mathcal{L}\left(f_{\theta }^H\left(\iota \left({\Gamma }_G\left(x\right)\right)\right),y\right)=\mathcal{L}\left(f_{\theta }\left({\Gamma }_G\left(x\right)\right),y\right).$$

Taking expectation over $(x,y)\sim \mathbb{P}$ shows that the optimization objective of the constructed HyperGraph-NLP instance is identical to the original Graph-NLP objective. Hence Graph-NLP is a special case of HyperGraph-NLP. □

Definition 47 

(Hypergraph-to-superhypergraph inclusion at level $n=0$). Let $H=(V,\mathcal{E})$ be a hypergraph with $\mathcal{E}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}$. Define

$$j_0\left(H\right)={\mathrm{SHG}}^{\left(0\right)}=(V_0,V,E)\mathrm{by}{V}_0=V,V=V_0\subseteq {\mathcal{P}}^0\left(V_0\right),E=\mathcal{E}.$$

Definition 48 

(Singleton lifting of graphs/hypergraphs to level n). For $n\ge 0$, define ${\sigma }_n:V_0\to {\mathcal{P}}^n\left(V_0\right)$ recursively by

$${\sigma }_0\left(v\right)=v,{\sigma }_{k+1}\left(v\right)=\left\{{\sigma }_k\left(v\right)\right\}(k\ge 0).$$

Given a hypergraph $H=(V_0,\mathcal{E})$, define itsn-lift

$$j_n\left(H\right)={\mathrm{SHG}}^{\left(n\right)}=(V_0,V^{\left(n\right)},E^{\left(n\right)}),$$

where

$$V^{\left(n\right)}=\{{\sigma }_n\left(v\right)\mid v\in V_0\}\subseteq {\mathcal{P}}^n\left(V_0\right),E^{\left(n\right)}=\left\{\{{\sigma }_n\left(v\right)\mid v\in e\}|e\in \mathcal{E}\right\}.$$

(Attributes and weights are transported along ${\sigma }_n$ in the obvious way.)

Theorem 11 

(SuperHyperGraph-NLP generalizes Graph-NLP and HyperGraph-NLP). For every $n\ge 0$, every instance of HyperGraph-NLP is a special case of SuperHyperGraph-NLP at level n. Consequently, every instance of Graph-NLP is also a special case of SuperHyperGraph-NLP.

Proof. 

Fix $n\ge 0$.

Step 1 (HyperGraph-NLP ⇒ SuperHyperGraph-NLP). Take any HyperGraph-NLP pipeline with encoder ${\Gamma }_H:\mathcal{X}\to \mathcal{H}$ and model family ${\mathcal{F}}_H=\{f_{\theta }:\mathcal{H}\to \mathcal{Y}\}$. Define a superhypergraph encoder by

$${\Gamma }_{SH}^{\left(n\right)}=j_n\circ {\Gamma }_H:\mathcal{X}\to {\mathcal{S}}^{\left(n\right)}.$$

Define a superhypergraph model family on the image of $j_n$ by

$$g_{\theta }\left(j_n\left(H\right)\right)=f_{\theta }\left(H\right)\mathrm{for}\mathrm{every}H\in \mathcal{H}.$$

Then for every $(x,y)$,

$$\mathcal{L}\left(g_{\theta }\left({\Gamma }_{SH}^{\left(n\right)}\left(x\right)\right),y\right)=\mathcal{L}\left(g_{\theta }\left(j_n\left({\Gamma }_H\left(x\right)\right)\right),y\right)=\mathcal{L}\left(f_{\theta }\left({\Gamma }_H\left(x\right)\right),y\right).$$

Thus the expected risk objective is preserved, and HyperGraph-NLP is a special case of SuperHyperGraph-NLP.

Step 2 (Graph-NLP ⇒ SuperHyperGraph-NLP). By Theorem above, Graph-NLP is a special case of HyperGraph-NLP via the inclusion $\iota$ into 2-uniform hypergraphs. Composing with Step 1 yields a SuperHyperGraph-NLP instance (at level n) with identical objective value for all $\theta$. Therefore SuperHyperGraph-NLP generalizes both Graph-NLP and HyperGraph-NLP. □

10. Conclusion

This paper extended several core frameworks—including GCN, GraphRAG, causal graphs, graph embedding, and graph generation—by integrating the SuperHyperGraph perspective. Future work will include computational experiments and quantitative evaluation on real-world datasets. We also hope that further research will explore extensions of the concepts presented here using Fuzzy Sets [150], Rough Sets [151,152], Neutrosophic Sets [79,153,154], Uncertain Sets[78,155], and Plithogenic Sets [156,157].