Graph Hyperembedding and Graph SuperHyperembedding: Extensions via HyperVector and SuperHyperVector Spaces

1. Preliminaries

This section summarizes the terminology and notation used in the sequel. Unless explicitly stated otherwise, every set and every structure discussed in this paper is assumed to be finite.

1.1. Classical Structures, Hyperstructures, and n-Superhyperstructures

We briefly recall three nested levels of abstraction: classical structures, hyperstructures, and n-superhyperstructures. Informally, hyperstructures are obtained by applying the powerset construction once to a base set [1,2,3,4,5,6,7], whereas n-superhyperstructures use the n^th iterated powerset [8,9]. Related concepts within the family of superhyperstructures include SuperHyperGraphs [10,11,12], chemical hyperstructure [13,14], and SuperHyperAlgebra [15].

Definition 1.1

(Base Set). A base set S is the ground collection on which all subsequent (higher-order) constructions are formed:

$$S=\{x\mid x\mathrm{is}\mathrm{an}\mathrm{element}\mathrm{of}\mathrm{the}\mathrm{domain}\}.$$

In particular, every object in $\mathcal{P}\left(S\right)$ and in ${\mathcal{P}}_n\left(S\right)$ is ultimately built from elements of S.

Definition 1.2

(Powerset). [16] For a set S, the powerset$\mathcal{P}\left(S\right)$ is the collection of all subsets of S, including ∅:

$$\mathcal{P}\left(S\right)=\{A\mid A\subseteq S\}.$$

Definition 1.3

(n-th Powerset). [17] Let H be a nonempty set. The n-th powerset of H, denoted ${\mathcal{P}}_n\left(H\right)$, is defined recursively by

$${\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right),{\mathcal{P}}_{k+1}\left(H\right)=\mathcal{P}\left({\mathcal{P}}_k\left(H\right)\right)(k\ge 1).$$

Likewise, the n-th nonempty powerset${\mathcal{P}}_n^{*}\left(H\right)$ is given by

$${\mathcal{P}}_1^{*}\left(H\right)={\mathcal{P}}^{*}\left(H\right),{\mathcal{P}}_{k+1}^{*}\left(H\right)={\mathcal{P}}^{*}\left({\mathcal{P}}_k^{*}\left(H\right)\right),$$

where ${\mathcal{P}}^{*}\left(H\right)=\mathcal{P}\left(H\right)\setminus \{\varnothing \}$ denotes the family of all nonempty subsets of H.

To set a precise basis for hyperstructures and superhyperstructures, we next record the core definitions.

Definition 1.4

(Classical Structure). (cf. [17,18]) A classical structure consists of a nonempty set H together with one or more classical operations satisfying specified axioms.

A classical operation is a map

$${\#}_0:H^m\to H,$$

where $m\ge 1$ and $H^m$ is the m-fold Cartesian product of H. Standard examples are addition and multiplication in groups, rings, and fields.

Definition 1.5

(Hyperoperation). (cf. [19,20]) Let S be a set. A hyperoperation on S is a map

$$\circ :S\times S\to \mathcal{P}\left(S\right),$$

so that the combination of two elements produces a subset of S rather than a single element.

Definition 1.6

(Hyperstructure). (cf. [4,17]) A hyperstructure is a structure obtained by working on the powerset of a base set. Concretely, it is a pair

$$\mathcal{H}=\left(\mathcal{P}\right(S),\circ ),$$

where S is a base set and ∘ is an operation acting on elements of $\mathcal{P}\left(S\right)$. In this way, operations may be applied to collections of elements, not only to individual elements.

Definition 1.7

(SuperHyperOperations). (cf. [17]) Let H be a nonempty set. Define the iterated powersets by

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

A SuperHyperOperation of order $(m,n)$ is an m-ary map

$${\circ }^{(m,n)}:H^m\to {\mathcal{P}}_{*}^n\left(H\right),$$

where ${\mathcal{P}}_{*}^n\left(H\right)$ denotes the n-th iterated powerset of H, with the convention that it may exclude or include ∅, depending on the setting:

• If ∅ is excluded from the codomain, then ${\circ }^{(m,n)}$ is called a classical-type $(m,n)$-SuperHyperOperation.
• If ∅ is allowed in the codomain, then ${\circ }^{(m,n)}$ is called a Neutrosophic $(m,n)$-SuperHyperOperation.

Thus, SuperHyperOperations extend hyperoperations by permitting higher-order (iterated powerset) outputs.

An $(m,n)$-superhyperstructure extends the hyperstructure viewpoint by introducing an m-ary operation on the hierarchical set ${\mathcal{P}}_n\left(S\right)$, the n^th iterated powerset of a finite base set.

Definition 1.8

($(m,n)$-Superhyperstructure). [17,21] Let S be a nonempty set and let ${\mathcal{P}}_n\left(S\right)$ be its n^th iterated powerset (Definition 1.3). Fix an integer $m\ge 1$. An $(m,n)$-superhyperstructure is a pair

$${\mathcal{SH}}_{m,n}=\left({\mathcal{P}}_n\left(S\right),{\circ }^{(m,n)}\right),$$

where

$${\circ }^{(m,n)}:{\left({\mathcal{P}}_n\left(S\right)\right)}^m⟶{\mathcal{P}}_n\left(S\right)$$

is an m-ary SuperHyperOperation of order $(m,n)$. That is, for any $\mathbf{X}=(X_1,\dots ,X_m)\in {\left({\mathcal{P}}_n\left(S\right)\right)}^m$, the value ${\circ }^{(m,n)}\left(\mathbf{X}\right)$ is a subset of ${\mathcal{P}}_n\left(S\right)$ satisfying whatever axioms (e.g. associativity or distributivity) are required in the intended application.

1.2. HyperVector Space

A vector space is a set V equipped with an addition + and a scalar multiplication by a field K, and it satisfies the usual linearity axioms [22,23]. Classical generalizations include fuzzy vector spaces [24,25] and neutrosophic vector spaces [26,27]. Vector spaces are known to have numerous real-world applications and play a central role in fields such as machine learning.

A hypervector space extends the vector-space paradigm by replacing the single-valued scalar multiplication with a set-valued scalar action (an external hyperoperation)

$$\circ :K\times V⟶{\mathcal{P}}^{*}\left(V\right),$$

so that each pair $(a,x)\in K\times V$ is assigned a nonempty subset $a\circ x\subseteq V$, while distributive and associative behaviors are retained (cf. [28,29]).

Definition 1.9

(Hypervector Space). (cf. [28,29]) Let K be a field and let $(V,+)$ be an abelian group. Write ${\mathcal{P}}^{*}\left(V\right)$ for the family of all nonempty subsets of V. A hypervector space over K is a quadruple $(V,+,\circ ,K)$ in which

$$\circ :K\times V⟶{\mathcal{P}}^{*}\left(V\right)$$

is an external hyperoperation satisfying, for all $a,b\in K$ and $x,y\in V$, the axioms:

(H1)

$a\circ (x+y)\subseteq (a\circ x)+(a\circ y)$, (right distributivity)

(H2)

$(a+b)\circ x\subseteq (a\circ x)+(b\circ x)$, (left distributivity)

(H3)

$a\circ \left(b\circ x\right)=\left(ab\right)\circ x$, (associativity)

(H4)

$a\circ (-x)=(-a)\circ x=-(a\circ x)$,

(H5)

$x\in 1\circ x$.

Here, for subsets $A,B\subseteq V$,

$$A+B=\{u+v\mid u\in A,v\in B\},$$

and the expression $a\circ \left(b\circ x\right)$ is interpreted as

$$a\circ \left(b\circ x\right)=\bigcup _{y\in b\circ x}(a\circ y).$$

1.3. $(m,n)$-SuperhyperVector Space

An $(m,n)$-SuperhyperVector Space extends the hypervector-space framework by allowing an m-ary scalar SuperHyperOperation whose values lie in the n-th iterated (nonempty) powerset of the underlying abelian group [30,31].

Definition 1.10

($(m,n)$-SuperhyperVector Space). [30,31] Let K be a field, let $(V,+)$ be an abelian group, and fix integers $m,n\ge 1$. Let ${\mathcal{P}}_n^{*}\left(V\right)$ denote the family of all nonempty elements of the n-th iterated powerset ${\mathcal{P}}_n\left(V\right)$. An $(m,n)$-SuperhyperVector Space over K is a quadruple

$$\left(V,+,{\circ }^{(m,n)},K\right),$$

where

$${\circ }^{(m,n)}:K^m\times V⟶{\mathcal{P}}_n^{*}\left(V\right)$$

is an m-ary SuperHyperOperation such that, for all $\mathbf{a}=(a_1,\dots ,a_m)$, $\mathbf{b}=(b_1,\dots ,b_m)\in K^m$ and all $x,y\in V$,

(SV1)

${\circ }^{(m,n)}(\mathbf{a},x+y)\subseteq {\circ }^{(m,n)}(\mathbf{a},x)+{\circ }^{(m,n)}(\mathbf{a},y)$,

(SV2)

${\circ }^{(m,n)}(\mathbf{a}+\mathbf{b},x)\subseteq {\circ }^{(m,n)}(\mathbf{a},x)+{\circ }^{(m,n)}(\mathbf{b},x)$,

(SV3)

${\circ }^{(m,n)}(\mathbf{a},{\circ }^{(m,n)}(\mathbf{b},x))={\circ }^{(m,n)}(\mathbf{a}·\mathbf{b},x)$,

(SV4)

${\circ }^{(m,n)}(-\mathbf{a},x)=-{\circ }^{(m,n)}(\mathbf{a},x)$,

(SV5)

$x\in {\circ }^{(m,n)}((1,\dots ,1),x)$.

Here,

$$\mathbf{a}+\mathbf{b}=(a_1+b_1,\dots ,a_m+b_m),\mathbf{a}·\mathbf{b}=(a_1{b}_1,\dots ,a_m{b}_m),$$

and for subsets $A,B\subseteq V$,

$$A+B=\{u+v\mid u\in A,v\in B\}.$$

For reference, a comparison of Vector Space, HyperVector Space, and $(m,n)$-SuperHyperVector Space is provided in Table 1.

1.4. Graph Embedding

Graph embedding represents vertices (or entire graphs) by vectors in a relatively low-dimensional space, with the aim of retaining adjacency information and broader structural similarity. Such representations enable efficient downstream tasks such as prediction and retrieval [32,33,34,35].

Definition 1.11

(Graph embedding in a vector space). [34,36,37] 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 $d\ge 1$.

A d-dimensional graph embedding consists of a map

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

together with

$$s:V\times V\to \mathbb{R}(\mathrm{a}\mathrm{chosen}\mathrm{graph}\mathrm{proximity}/\mathrm{relation}),$$

and

$$\mathrm{sim}:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R}(\mathrm{a}\mathrm{chosen}\mathrm{similarity}\mathrm{in}\mathrm{the}\mathrm{embedding}\mathrm{space}),$$

such that, for the vertex pairs of interest $(u,v)$, the similarity score $\mathrm{sim}\left(\phi \right(u),\phi (v\left)\right)$ provides an approximation to $s(u,v)$ (or at least preserves its ranking).

A standard way to express this requirement is to select $\phi$ by solving an optimization problem of the form

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

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

2. Results

This section presents the results.

2.1. Graph Hyperembedding

Graph Hyperembedding maps vertices to nonempty sets of hypervectors in a hypervector space, preserving graph proximities via aggregated similarity.

Definition 2.1

(Aggregation operator). An aggregation operator is a map

$$\mathrm{Agg}:{\mathcal{P}}^{*}\left(\mathbb{R}\right)⟶\mathbb{R}$$

satisfying the singleton normalization:

$$\mathrm{Agg}\left(\right\{r\left\}\right)=r(\forall r\in \mathbb{R}).$$

Typical examples include $\mathrm{Agg}\left(X\right)=maxX$, $\mathrm{Agg}\left(X\right)=minX$, or $\mathrm{Agg}\left(X\right)={\displaystyle \frac{1}{\left|X\right|}}{\sum }_{x\in X}x$ for finite X.

Definition 2.2

(Induced similarity between nonempty subsets). Let $(H,+,\circ ,K)$ be a hypervector space and let

$$\mathrm{sim}:H\times H\to \mathbb{R}$$

be a prescribed similarity function on (hyper)vectors. Let $\mathrm{Agg}$ be an aggregation operator (Definition 2.1). For $A,B\in {\mathcal{P}}^{*}\left(H\right)$, define the induced similarity

$$\mathrm{SIM}(A,B)=\mathrm{Agg}\left(\left\{\mathrm{sim}\right(x,y)\mid x\in A,y\in B\}\right).$$

Definition 2.3

(Graph hyperembedding in a hypervector space). Let $G=(\mathcal{V},\mathcal{E},w)$ be a (possibly weighted) simple graph with $\mathcal{V}=\{v_1,\dots ,v_n\}$ and $\mathcal{E}\subseteq \left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{V}}{2}}\right)$. Fix a hypervector space $(H,+,\circ ,K)$. Fix a graph proximity function $s:\mathcal{V}\times \mathcal{V}\to \mathbb{R}$, a similarity $\mathrm{sim}:H\times H\to \mathbb{R}$, and an aggregation operator $\mathrm{Agg}$.

A graph hyperembedding of G into $(H,+,\circ ,K)$ is a set-valued map

$$\mathsf{\Phi }:\mathcal{V}⟶{\mathcal{P}}^{*}\left(H\right),$$

together with the induced similarity $\mathrm{SIM}$ from Definition 2.2, such that $\mathrm{SIM}\left(\mathsf{\Phi }\right(u),\mathsf{\Phi }(v\left)\right)$ approximates (or preserves the ordering of) $s(u,v)$ for relevant pairs $(u,v)$.

As in ordinary graph embedding, one common specification is to define $\mathsf{\Phi }$ as a minimizer of

$$\underset{\mathsf{\Phi }:\mathcal{V}\to {\mathcal{P}}^{*}\left(H\right)}{min}\sum _{(u,v)\in \mathsf{\Omega }}L\left(s(u,v),\mathrm{SIM}\left(\mathsf{\Phi }\right(u),\mathsf{\Phi }(v\left)\right)\right)+\lambda \mathcal{R}\left(\mathsf{\Phi }\right),$$

where $\mathsf{\Omega }\subseteq \mathcal{V}\times \mathcal{V}$, L is a loss function, $\mathcal{R}$ is a regularizer, and $\lambda \ge 0$.

Remark 2.4

(Deterministic (singleton) case). If $\mathsf{\Phi }\left(v\right)$ is a singleton for every $v\in \mathcal{V}$, i.e., $\mathsf{\Phi }\left(v\right)=\left\{h_v\right\}$ with $h_v\in H$, then

$$\mathrm{SIM}(\mathsf{\Phi }\left(u\right),\mathsf{\Phi }\left(v\right))=\mathrm{Agg}\left(\left\{\mathrm{sim}(h_u,h_v)\right\}\right)=\mathrm{sim}(h_u,h_v)$$

by Definition 2.1. Thus the hyperembedding reduces to an ordinary point embedding into H.

Example 2.5

(A concrete graph hyperembedding with set-valued vertex representations). Let $G=(V,E)$ be the path on three vertices

$$V=\{v_1,v_2,v_3\},E=\{\{v_1,v_2\},\{v_2,v_3\}\}.$$

(Target hypervector space.) Let $K=\mathbb{R}$ and let $H={\mathbb{R}}^2$ with the usual vector addition +. Define an external hyperoperation $\circ :K\times H\to {\mathcal{P}}^{*}\left(H\right)$ by

$$a\circ x=\left\{ax\right\}(a\in \mathbb{R},x\in {\mathbb{R}}^2).$$

Then $(H,+,\circ ,K)$ is a hypervector space.

(Proximity, similarity, and aggregation.) Define the graph proximity

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

and define a similarity on H by the negative squared Euclidean distance:

$$\mathrm{sim}(x,y)={-\parallel x-y\parallel }_2^2.$$

Choose the aggregation operator $\mathrm{Agg}:{\mathcal{P}}^{*}\left(\mathbb{R}\right)\to \mathbb{R}$ as

$$\mathrm{Agg}\left(X\right)=maxX.$$

Hence the induced similarity for nonempty sets $A,B\subseteq H$ is

$$\mathrm{SIM}(A,B)=max\left\{\mathrm{sim}\right(x,y)\mid x\in A,y\in B\}.$$

(Set-valued embedding.) Define $\mathsf{\Phi }:V\to {\mathcal{P}}^{*}\left(H\right)$ by

$$\mathsf{\Phi }\left(v_1\right)=\{(0,0),(0.1,0)\},\mathsf{\Phi }\left(v_2\right)=\{(1,0),(1.1,0)\},\mathsf{\Phi }\left(v_3\right)=\{(2,0),(2.1,0)\}.$$

This $\mathsf{\Phi }$ is a graph hyperembedding because adjacent vertices obtain higher induced similarity than the non-adjacent pair, as shown by the explicit computations below.

(Verification by direct calculation.) For $\{v_1,v_2\}\in E$, compute all four squared distances:

$${\parallel (0,0)-(1,0)\parallel }_2^2=1,{\parallel (0,0)-(1.1,0)\parallel }_2^2=1.21,$$

$${\parallel (0.1,0)-(1,0)\parallel }_2^2=0.81,{\parallel (0.1,0)-(1.1,0)\parallel }_2^2=1.$$

Thus the corresponding similarities are

$$-1,-1.21,-0.81,-1,$$

and therefore

$$\mathrm{SIM}(\mathsf{\Phi }\left(v_1\right),\mathsf{\Phi }\left(v_2\right))=max\{-1,-1.21,-0.81,-1\}=-0.81.$$

Similarly, for $\{v_2,v_3\}\in E$ we obtain

$$\mathrm{SIM}(\mathsf{\Phi }\left(v_2\right),\mathsf{\Phi }\left(v_3\right))=-0.81$$

(by the same distance pattern shifted by $+1$ in the first coordinate).

For the non-edge pair $\{v_1,v_3\}\notin E$, compute:

$${\parallel (0,0)-(2,0)\parallel }_2^2=4,{\parallel (0,0)-(2.1,0)\parallel }_2^2=4.41,$$

$${\parallel (0.1,0)-(2,0)\parallel }_2^2=3.61,{\parallel (0.1,0)-(2.1,0)\parallel }_2^2=4,$$

so the similarities are

$$-4,-4.41,-3.61,-4,$$

and hence

$$\mathrm{SIM}(\mathsf{\Phi }\left(v_1\right),\mathsf{\Phi }\left(v_3\right))=max\{-4,-4.41,-3.61,-4\}=-3.61.$$

Therefore,

$$\mathrm{SIM}(\mathsf{\Phi }\left(v_1\right),\mathsf{\Phi }\left(v_2\right))=\mathrm{SIM}(\mathsf{\Phi }\left(v_2\right),\mathsf{\Phi }\left(v_3\right))=-0.81>-3.61=\mathrm{SIM}(\mathsf{\Phi }\left(v_1\right),\mathsf{\Phi }\left(v_3\right)),$$

which matches the intended proximity ordering $s(v_1,v_2)=s(v_2,v_3)=1>0=s(v_1,v_3)$.

Theorem 2.6

(Graph hyperembedding generalizes graph embedding). Let $(W,+,·,K)$ be a (classical) vector space over a field K. Define a hyperoperation $\circ :K\times W\to {\mathcal{P}}^{*}\left(W\right)$ by

$$a\circ x=\{a·x\}(a\in K,x\in W).$$

Then $(W,+,\circ ,K)$ is a hypervector space.

Moreover, let $G=(\mathcal{V},\mathcal{E},w)$ be a graph, let $\phi :\mathcal{V}\to W$ be a (classical) graph embedding, and let $s:\mathcal{V}\times \mathcal{V}\to \mathbb{R}$ and $\mathrm{sim}:W\times W\to \mathbb{R}$ be the proximity and similarity used for φ. Choose any aggregation operator $\mathrm{Agg}$. Define the singleton lift

$$\mathsf{\Phi }:\mathcal{V}\to {\mathcal{P}}^{*}\left(W\right),\mathsf{\Phi }\left(v\right)=\left\{\phi \left(v\right)\right\}.$$

Then Φ is a graph hyperembedding of G into $(W,+,\circ ,K)$ in the sense of Definition 2.3, and it reproduces the same pairwise similarities:

$$\mathrm{SIM}\left(\mathsf{\Phi }\right(u),\mathsf{\Phi }(v\left)\right)=\mathrm{sim}\left(\phi \right(u),\phi (v\left)\right)(\forall u,v\in \mathcal{V}).$$

In particular, Graph Hyperembedding is a genuine generalization of Graph Embedding.

Proof. 

Step 1 (Hypervector space axioms). We verify the axioms (H1)–(H5) of the Hypervector Space definition for $(W,+,\circ ,K)$ with $a\circ x=\{a·x\}$. Recall that for subsets $A,B\subseteq W$,

$$A+B=\{u+v\mid u\in A,v\in B\}.$$

(H1) Right distributivity:

$$a\circ (x+y)=\{a·(x+y\left)\right\}=\{a·x+a·y\}\subseteq \{a·x\}+\{a·y\}=(a\circ x)+(a\circ y).$$

(H2) Left distributivity:

$$(a+b)\circ x=\left\{\right(a+b)·x\}=\{a·x+b·x\}\subseteq \{a·x\}+\{b·x\}=(a\circ x)+(b\circ x).$$

(H3) Associativity: First note $b\circ x=\{b·x\}$. Hence, using the standard interpretation

$$a\circ (b\circ x)=\bigcup _{y\in b\circ x}(a\circ y),$$

we compute

$$a\circ (b\circ x)=\bigcup _{y\in \{b·x\}}\{a·y\}=\{a·(b·x)\}=\{\left(ab\right)·x\}=\left(ab\right)\circ x.$$

(H4) Compatibility with additive inverses:

$$a\circ (-x)=\{a·(-x\left)\right\}=\{-(a·x\left)\right\}=-\left(\right\{a·x\left\}\right)=-(a\circ x),$$

and similarly

$$(-a)\circ x=\left\{\right(-a)·x\}=\{-(a·x\left)\right\}=-(a\circ x).$$

(H5) Unit condition:

$$1\circ x=\{1·x\}=\left\{x\right\},$$

so $x\in 1\circ x$.

Therefore $(W,+,\circ ,K)$ is a hypervector space.

Step 2 (Reduction of similarities). Let $\mathsf{\Phi }\left(v\right)=\left\{\phi \right(v\left)\right\}$. By Definition 2.2,

$$\mathrm{SIM}(\mathsf{\Phi }\left(u\right),\mathsf{\Phi }\left(v\right))=\mathrm{Agg}\left(\left\{\mathrm{sim}\right(x,y)\mid x\in \{\phi \left(u\right)\},y\in \{\phi \left(v\right)\left\}\right\}\right)=\mathrm{Agg}\left(\left\{\mathrm{sim}(\phi \left(u\right),\phi \left(v\right))\right\}\right).$$

By the singleton normalization in Definition 2.1,

$$\mathrm{Agg}\left(\right\{\mathrm{sim}\left(\phi \right(u),\phi (v\left)\right)\left\}\right)=\mathrm{sim}\left(\phi \right(u),\phi (v\left)\right).$$

Hence $\mathrm{SIM}\left(\mathsf{\Phi }\right(u),\mathsf{\Phi }(v\left)\right)=\mathrm{sim}\left(\phi \right(u),\phi (v\left)\right)$ for all $u,v$, so $\mathsf{\Phi }$ satisfies the same proximity-preservation requirement as $\phi$. Thus $\mathsf{\Phi }$ is a graph hyperembedding, and Graph Hyperembedding generalizes Graph Embedding. □

2.2. Graph SuperHyperembedding

Graph SuperHyperembedding maps graph vertices to n-th iterated nonempty powerset elements in an (m,n)-SuperhyperVector Space, preserving proximities.

Definition 2.7

(Lifted Minkowski sum on ${\mathcal{P}}_k^{*}\left(U\right)$). Let $(U,+)$ be an abelian group. Define binary operations ${\oplus }_k$ on ${\mathcal{P}}_k^{*}\left(U\right)$ recursively as follows.

For $k=0$, set ${\oplus }_0:=+$ on U.

For $k\ge 0$ and $X,Y\in {\mathcal{P}}_{k+1}^{*}\left(U\right)={\mathcal{P}}^{*}\left({\mathcal{P}}_k^{*}\left(U\right)\right)$, define

$$X{\oplus }_{k+1}Y=\left\{x{\oplus }_{k}y|x\in X,y\in Y\right\}.$$

In particular, for $k=0$ and nonempty subsets $A,B\subseteq U$,

$$A{\oplus }_{1}B=\{a+b\mid a\in A,b\in B\},$$

which is the usual Minkowski sum.

Definition 2.8

(Singleton tower operator). Let U be a nonempty set. Define maps ${\iota }_k:{\mathcal{P}}_0^{*}\left(U\right)\to {\mathcal{P}}_k^{*}\left(U\right)$ by

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

More generally, for $X\in {\mathcal{P}}_j^{*}\left(U\right)$ and $n\ge j$, define ${\iota }_{j\to n}\left(X\right)\in {\mathcal{P}}_n^{*}\left(U\right)$ by

$${\iota }_{j\to j}\left(X\right)=X,{\iota }_{j\to (t+1)}\left(X\right)=\left\{{\iota }_{j\to t}\left(X\right)\right\}(t\ge j).$$

Thus ${\iota }_{1\to n}\left(A\right)=\left\{\{\dots \left\{A\right\}\dots \}\right\}$ (with $n-1$ additional braces) for $A\in {\mathcal{P}}_1^{*}\left(U\right)$.

Lemma 2.9

(Compatibility of ${\iota }_n$ with lifted sum). Let $(U,+)$ be an abelian group and define ${\oplus }_k$ as in Definition 2.7. Then for all $n\ge 1$ and all $u,v\in U$,

$${\iota }_n\left(u\right){\oplus }_n{\iota }_n\left(v\right)={\iota }_n(u+v).$$

Proof. 

We prove by induction on n.

Base case $n=1$:

$${\iota }_1\left(u\right){\oplus }_1{\iota }_1\left(v\right)=\left\{u\right\}{\oplus }_1\left\{v\right\}=\{u+v\}={\iota }_1(u+v).$$

Induction step: assume ${\iota }_n\left(u\right){\oplus }_n{\iota }_n\left(v\right)={\iota }_n(u+v)$. Then using Definition 2.7,

$${\iota }_{n+1}\left(u\right){\oplus }_{n+1}{\iota }_{n+1}\left(v\right)=\left\{{\iota }_n\left(u\right)\right\}{\oplus }_{n+1}\left\{{\iota }_n\left(v\right)\right\}=\left\{{\iota }_n\left(u\right){\oplus }_n{\iota }_n\left(v\right)\right\}.$$

By the induction hypothesis, ${\iota }_n\left(u\right){\oplus }_n{\iota }_n\left(v\right)={\iota }_n(u+v)$, hence

$${\iota }_{n+1}\left(u\right){\oplus }_{n+1}{\iota }_{n+1}\left(v\right)=\left\{{\iota }_n(u+v)\right\}={\iota }_{n+1}(u+v).$$

□

Definition 2.10

(Aggregation operator). An aggregation operator is a map

$$\mathrm{Agg}:{\mathcal{P}}^{*}\left(\mathbb{R}\right)\to \mathbb{R}$$

satisfying

$$\mathrm{Agg}\left(\right\{r\left\}\right)=r(\forall r\in \mathbb{R}).$$

Definition 2.11

(Recursive induced similarity on ${\mathcal{P}}_n^{*}\left(U\right)$). Let U be a nonempty set and fix a base similarity

$${\mathrm{sim}}_0:U\times U\to \mathbb{R}.$$

Fix an aggregation operator $\mathrm{Agg}$ (Definition 2.10). Define similarities ${\mathrm{SIM}}_k:{\mathcal{P}}_k^{*}\left(U\right)\times {\mathcal{P}}_k^{*}\left(U\right)\to \mathbb{R}$ recursively by

$${\mathrm{SIM}}_0={\mathrm{sim}}_0,$$

and for $k\ge 0$, for $X,Y\in {\mathcal{P}}_{k+1}^{*}\left(U\right)$,

$${\mathrm{SIM}}_{k+1}(X,Y)=\mathrm{Agg}\left(\{{\mathrm{SIM}}_k(x,y)\mid x\in X,y\in Y\}\right).$$

Lemma 2.12

(Singleton-collapse of the recursive similarity). With ${\mathrm{SIM}}_k$ defined as in Definition 2.11, for any $k\ge 0$ and any $X,Y\in {\mathcal{P}}_k^{*}\left(U\right)$,

$${\mathrm{SIM}}_{k+1}(\left\{X\right\},\left\{Y\right\})={\mathrm{SIM}}_k(X,Y).$$

Consequently, for $n\ge 1$ and any $A,B\in {\mathcal{P}}_1^{*}\left(U\right)$,

$${\mathrm{SIM}}_n\left({\iota }_{1\to n}\left(A\right),{\iota }_{1\to n}\left(B\right)\right)={\mathrm{SIM}}_1(A,B).$$

Proof. 

First claim:

$${\mathrm{SIM}}_{k+1}(\left\{X\right\},\left\{Y\right\})=\mathrm{Agg}\left(\{{\mathrm{SIM}}_k(x,y)\mid x\in \left\{X\right\},y\in \left\{Y\right\}\}\right)=\mathrm{Agg}\left(\left\{{\mathrm{SIM}}_k(X,Y)\right\}\right)={\mathrm{SIM}}_k(X,Y),$$

using the singleton normalization $\mathrm{Agg}\left(\right\{r\left\}\right)=r$.

Second claim follows by iterating the first claim $n-1$ times:

$${\mathrm{SIM}}_n({\iota }_{1\to n}\left(A\right),{\iota }_{1\to n}\left(B\right))={\mathrm{SIM}}_{n-1}({\iota }_{1\to (n-1)}\left(A\right),{\iota }_{1\to (n-1)}\left(B\right))=\dots ={\mathrm{SIM}}_1(A,B).$$

□

Definition 2.13

(Graph superhyperembedding in an $(m,n)$-SuperhyperVector Space). Let $G=(\mathcal{V},\mathcal{E},w)$ be a (possibly weighted) simple graph with $\mathcal{V}=\{v_1,\dots ,v_N\}$ and $\mathcal{E}\subseteq \left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{V}}{2}}\right)$. Fix an $(m,n)$-SuperhyperVector Space

$$\left(U,+,{\circ }^{(m,n)},K\right)$$

(in particular $(U,+)$ is an abelian group), and fix a graph proximity function

$$s:\mathcal{V}\times \mathcal{V}\to \mathbb{R}.$$

Fix also a base similarity ${\mathrm{sim}}_0:U\times U\to \mathbb{R}$ and an aggregation operator $\mathrm{Agg}$. Let ${\mathrm{SIM}}_n$ be the induced similarity on ${\mathcal{P}}_n^{*}\left(U\right)$ from Definition 2.11.

A graph superhyperembedding of G into $\left(U,+,{\circ }^{(m,n)},K\right)$ is a map

$$\mathsf{\Psi }:\mathcal{V}⟶{\mathcal{P}}_n^{*}\left(U\right)$$

such that ${\mathrm{SIM}}_n(\mathsf{\Psi }\left(u\right),\mathsf{\Psi }\left(v\right))$ approximates (or preserves the ordering of) $s(u,v)$ for relevant pairs $(u,v)$.

One common specification is to define $\mathsf{\Psi }$ as a minimizer of

$$\underset{\mathsf{\Psi }:\mathcal{V}\to {\mathcal{P}}_n^{*}\left(U\right)}{min}\sum _{(u,v)\in \mathsf{\Omega }}L\left(s(u,v),{\mathrm{SIM}}_n(\mathsf{\Psi }\left(u\right),\mathsf{\Psi }\left(v\right))\right)+\lambda \mathcal{R}\left(\mathsf{\Psi }\right),$$

where $\mathsf{\Omega }\subseteq \mathcal{V}\times \mathcal{V}$, L is a loss function, $\mathcal{R}$ a regularizer, and $\lambda \ge 0$.

Example 2.14

(A level-$n=2$ Graph SuperHyperembedding on a short path). Let $G=(\mathcal{V},\mathcal{E})$ be the path on three vertices

$$\mathcal{V}=\{v_1,v_2,v_3\},\mathcal{E}=\left\{\{v_1,v_2\},\{v_2,v_3\}\right\}.$$

Define the proximity

$$s(u,v)=\left\{\begin{array}{cc}1,\hfill & \{u,v\}\in \mathcal{E},\hfill \\ 0,\hfill & \{u,v\}\notin \mathcal{E}.\hfill \end{array}\right.$$

(Target $(m,n)$-SuperhyperVector Space.) Let $K=\mathbb{R}$ and let $U={\mathbb{R}}^2$ with the usual addition +. Fix $(m,n)=(2,2)$ and define

$${\circ }^{(2,2)}:K^2\times U\to {\mathcal{P}}_2^{*}\left(U\right),{\circ }^{(2,2)}\left((a_1,a_2),x\right)=\left\{\left\{\left(a_1{a}_2\right)x\right\}\right\}={\iota }_2\left(\left(a_1{a}_2\right)x\right).$$

This is the canonical singleton-tower lifting of ordinary scalar multiplication; hence the axioms (SV1)–(SV5) follow from the usual vector-space identities exactly as in the singleton reduction arguments used for hypervector spaces.

(Base similarity and aggregation.) Let

$${\mathrm{sim}}_0(x,y)=-{\parallel x-y\parallel }_2^2(x,y\in U),$$

and choose the aggregation operator $\mathrm{Agg}\left(X\right)=maxX$. Let ${\mathrm{SIM}}_2$ be the induced similarity on ${\mathcal{P}}_2^{*}\left(U\right)$ constructed recursively (Definition 2.11).

(Definition of the superhyperembedding.) Define $\mathsf{\Psi }:\mathcal{V}\to {\mathcal{P}}_2^{*}\left(U\right)$ by the following nonempty families of nonempty subsets:

$$\mathsf{\Psi }\left(v_1\right)=\{A_1,A_2\},\mathsf{\Psi }\left(v_2\right)=\{B_1,B_2\},\mathsf{\Psi }\left(v_3\right)=\{C_1,C_2\},$$

where

$$A_1=\{(0,0),(0.2,0)\},A_2=\{(0,0.1),(0.2,0.1)\},$$

$$B_1=\{(1,0),(1.2,0)\},B_2=\{(1,0.1),(1.2,0.1)\},$$

$$C_1=\{(3,0),(3.2,0)\},C_2=\{(3,0.1),(3.2,0.1)\}.$$

Thus each $\mathsf{\Psi }\left(v_i\right)$ is an element of ${\mathcal{P}}_2^{*}\left(U\right)={\mathcal{P}}^{*}\left({\mathcal{P}}^{*}\left(U\right)\right)$.

(Verification: edge pairs have larger induced similarity than the non-edge pair.) First compute ${\mathrm{SIM}}_1(A_1,B_1)$. We list the four squared distances:

$${\parallel (0,0)-(1,0)\parallel }_2^2=1,{\parallel (0,0)-(1.2,0)\parallel }_2^2=1.44,$$

$${\parallel (0.2,0)-(1,0)\parallel }_2^2=0.64,{\parallel (0.2,0)-(1.2,0)\parallel }_2^2=1.$$

Hence the four similarities ${\mathrm{sim}}_0$ are

$$-1,-1.44,-0.64,-1,$$

so (since $\mathrm{Agg}=max$)

$${\mathrm{SIM}}_1(A_1,B_1)=max\{-1,-1.44,-0.64,-1\}=-0.64.$$

By the same computation (shifted in the second coordinate),

$${\mathrm{SIM}}_1(A_2,B_2)=-0.64.$$

Therefore, using the level-2 recursion ${\mathrm{SIM}}_2(\{A_1,A_2\},\{B_1,B_2\})={max}_{i,j}{\mathrm{SIM}}_1(A_i,B_j),$ we obtain

$${\mathrm{SIM}}_2(\mathsf{\Psi }\left(v_1\right),\mathsf{\Psi }\left(v_2\right))=-0.64.$$

Next, for the edge $\{v_2,v_3\}\in \mathcal{E}$, compute ${\mathrm{SIM}}_1(B_1,C_1)$. The closest pair is $(1.2,0)\in B_1$ and $(3,0)\in C_1$, giving

$${\parallel (1.2,0)-(3,0)\parallel }_2^2={\left(1.8\right)}^2=3.24\Rightarrow {\mathrm{SIM}}_1(B_1,C_1)\ge -3.24.$$

A direct enumeration of all four pairs in $B_1\times C_1$ shows the maximum similarity is indeed $-3.24$, so ${\mathrm{SIM}}_1(B_1,C_1)=-3.24$, and similarly ${\mathrm{SIM}}_1(B_2,C_2)=-3.24$. Hence

$${\mathrm{SIM}}_2(\mathsf{\Psi }\left(v_2\right),\mathsf{\Psi }\left(v_3\right))=-3.24.$$

Finally, for the non-edge $\{v_1,v_3\}\notin \mathcal{E}$, the closest pair is $(0.2,0)\in A_1$ and $(3,0)\in C_1$, giving

$${\parallel (0.2,0)-(3,0)\parallel }_2^2={\left(2.8\right)}^2=7.84\Rightarrow {\mathrm{SIM}}_2(\mathsf{\Psi }\left(v_1\right),\mathsf{\Psi }\left(v_3\right))=-7.84.$$

Consequently,

$$min\left\{{\mathrm{SIM}}_2(\mathsf{\Psi }\left(v_1\right),\mathsf{\Psi }\left(v_2\right)),{\mathrm{SIM}}_2(\mathsf{\Psi }\left(v_2\right),\mathsf{\Psi }\left(v_3\right))\right\}=min\{-0.64,-3.24\}=-3.24>-7.84={\mathrm{SIM}}_2(\mathsf{\Psi }\left(v_1\right),\mathsf{\Psi }\left(v_3\right)),$$

which matches the intended proximity ordering $1>0$ for edges versus the non-edge. Thus $\mathsf{\Psi }$ is a concrete Graph SuperHyperembedding of G at level $n=2$.

Example 2.15

(A level-$n=3$ Graph SuperHyperembedding on a 4-cycle). Let $G=(\mathcal{V},\mathcal{E})$ be the cycle $C_4$ with

$$\mathcal{V}=\{a,b,c,d\},\mathcal{E}=\left\{\{a,b\},\{b,c\},\{c,d\},\{d,a\}\right\}.$$

Define the proximity s by $s(u,v)=1$ for $\{u,v\}\in \mathcal{E}$ and $s(u,v)=0$ otherwise.

(Target $(m,n)$-SuperhyperVector Space.) Let $K=\mathbb{R}$, $U={\mathbb{R}}^2$ with usual addition, and fix $(m,n)=(3,3)$. Define the canonical singleton-tower scalar SuperHyperOperation

$${\circ }^{(3,3)}:K^3\times U\to {\mathcal{P}}_3^{*}\left(U\right),{\circ }^{(3,3)}\left((\alpha ,\beta ,\gamma ),x\right)={\iota }_3\left(\left(\alpha \beta \gamma \right)x\right).$$

(Base similarity and aggregation.) Let

$${\mathrm{sim}}_0(x,y)=-{\parallel x-y\parallel }_2^2,\mathrm{Agg}\left(X\right)=maxX,$$

and let ${\mathrm{SIM}}_3$ be the induced similarity on ${\mathcal{P}}_3^{*}\left(U\right)$ obtained from Definition 2.11.

(Mode construction in ${\mathcal{P}}_3^{*}\left(U\right)$.) We first define nonempty subsets of U (level 1 objects):

$$A_1=\{(0,0),(0,0.1)\},A_2=\{(0.1,0),(0.1,0.1)\},$$

$$B_1=\{(1,0),(1,0.1)\},B_2=\{(1.1,0),(1.1,0.1)\},$$

$$C_1=\{(1,1),(1,1.1)\},C_2=\{(1.1,1),(1.1,1.1)\},$$

$$D_1=\{(0,1),(0,1.1)\},D_2=\{(0.1,1),(0.1,1.1)\}.$$

Next define level 2 objects (nonempty sets of the above subsets):

$$X_a^{\left(1\right)}=\{A_1,A_2\},X_a^{\left(2\right)}=\left\{A_1\right\},$$

$$X_b^{\left(1\right)}=\{B_1,B_2\},X_b^{\left(2\right)}=\left\{B_1\right\},$$

$$X_c^{\left(1\right)}=\{C_1,C_2\},X_c^{\left(2\right)}=\left\{C_1\right\},$$

$$X_d^{\left(1\right)}=\{D_1,D_2\},X_d^{\left(2\right)}=\left\{D_1\right\}.$$

Finally, define the level 3 embedding $\mathsf{\Psi }:\mathcal{V}\to {\mathcal{P}}_3^{*}\left(U\right)$ by

$$\mathsf{\Psi }\left(a\right)=\{X_a^{\left(1\right)},X_a^{\left(2\right)}\},\mathsf{\Psi }\left(b\right)=\{X_b^{\left(1\right)},X_b^{\left(2\right)}\},\mathsf{\Psi }\left(c\right)=\{X_c^{\left(1\right)},X_c^{\left(2\right)}\},\mathsf{\Psi }\left(d\right)=\{X_d^{\left(1\right)},X_d^{\left(2\right)}\}.$$

Each $\mathsf{\Psi }(·)$ is a nonempty set of level-2 objects, hence an element of ${\mathcal{P}}_3^{*}\left(U\right)$.

(Verification: an edge similarity dominates a diagonal similarity.) Consider first the edge $\{a,b\}\in \mathcal{E}$. Compute the level-1 induced similarity ${\mathrm{SIM}}_1(A_2,B_1)$. The closest pair is $(0.1,0)\in A_2$ and $(1,0)\in B_1$:

$${\parallel (0.1,0)-(1,0)\parallel }_2^2={\left(0.9\right)}^2=0.81\Rightarrow {\mathrm{SIM}}_1(A_2,B_1)=-0.81.$$

One checks similarly that all other pairs among $A_i$ and $B_j$ yield similarities $\le -0.81$, so

$${\mathrm{SIM}}_2\left(X_a^{\left(1\right)},X_b^{\left(1\right)}\right)=\underset{i\in \{1,2\},j\in \{1,2\}}{max}{\mathrm{SIM}}_1(A_i,B_j)=-0.81.$$

Because ${\mathrm{SIM}}_3$ is the max-aggregation over mode pairs,

$${\mathrm{SIM}}_3\left(\mathsf{\Psi }\left(a\right),\mathsf{\Psi }\left(b\right)\right)=\underset{p\in \{1,2\},q\in \{1,2\}}{max}{\mathrm{SIM}}_2\left(X_a^{\left(p\right)},X_b^{\left(q\right)}\right)\ge {\mathrm{SIM}}_2\left(X_a^{\left(1\right)},X_b^{\left(1\right)}\right)=-0.81.$$

In fact, the other mode combinations give values $\le -0.81$, so

$${\mathrm{SIM}}_3\left(\mathsf{\Psi }\left(a\right),\mathsf{\Psi }\left(b\right)\right)=-0.81.$$

Now consider the diagonal non-edge $\{a,c\}\notin \mathcal{E}$. Compute ${\mathrm{SIM}}_1(A_2,C_1)$ using the closest pair $(0.1,0.1)\in A_2$ and $(1,1)\in C_1$:

$${\parallel (0.1,0.1)-(1,1)\parallel }_2^2={\left(0.9\right)}^2+{\left(0.9\right)}^2=0.81+0.81=1.62\Rightarrow {\mathrm{SIM}}_1(A_2,C_1)=-1.62.$$

Enumerating the remaining pairs among $A_i$ and $C_j$ gives similarities $\le -1.62$, hence

$${\mathrm{SIM}}_2\left(X_a^{\left(1\right)},X_c^{\left(1\right)}\right)=-1.62.$$

The other mode combinations yield values $\le -1.62$ (for instance, ${\mathrm{SIM}}_2(X_a^{\left(2\right)},X_c^{\left(2\right)})={\mathrm{SIM}}_1(A_1,C_1)=-1.81$), so

$${\mathrm{SIM}}_3\left(\mathsf{\Psi }\left(a\right),\mathsf{\Psi }\left(c\right)\right)=\underset{p,q\in \{1,2\}}{max}{\mathrm{SIM}}_2\left(X_a^{\left(p\right)},X_c^{\left(q\right)}\right)=-1.62.$$

Therefore,

$${\mathrm{SIM}}_3\left(\mathsf{\Psi }\left(a\right),\mathsf{\Psi }\left(b\right)\right)=-0.81>-1.62={\mathrm{SIM}}_3\left(\mathsf{\Psi }\left(a\right),\mathsf{\Psi }\left(c\right)\right),$$

which matches $s(a,b)=1>0=s(a,c)$. By the symmetric construction of the four corners, the same edge-versus-diagonal ordering holds for $\{b,c\},\{c,d\},\{d,a\}$ versus the non-edges $\{a,c\},\{b,d\}$. Hence $\mathsf{\Psi }$ is a concrete Graph SuperHyperembedding of $C_4$ at level $n=3$.

Theorem 2.16

(Graph SuperHyperembedding generalizes Graph Hyperembedding and Graph Embedding). Let $G=(\mathcal{V},\mathcal{E},w)$ be a graph, $s:\mathcal{V}\times \mathcal{V}\to \mathbb{R}$ a proximity, ${\mathrm{sim}}_0:U\times U\to \mathbb{R}$ a base similarity, and $\mathrm{Agg}$ an aggregation operator.

(I)

(Generalizes Graph Hyperembedding) Assume $n\ge 1$. Let $\mathsf{\Phi }:\mathcal{V}\to {\mathcal{P}}_1^{*}\left(U\right)={\mathcal{P}}^{*}\left(U\right)$ be a graph hyperembedding in the sense of Definition 2.3 (with induced ${\mathrm{SIM}}_1$). Define

$$\mathsf{\Psi }\left(v\right)={\iota }_{1\to n}\left(\mathsf{\Phi }\left(v\right)\right)\in {\mathcal{P}}_n^{*}\left(U\right)(v\in \mathcal{V}).$$

Then Ψ is a graph superhyperembedding (Definition 2.13), and for all $u,v\in \mathcal{V}$,

$${\mathrm{SIM}}_n(\mathsf{\Psi }\left(u\right),\mathsf{\Psi }\left(v\right))={\mathrm{SIM}}_1(\mathsf{\Phi }\left(u\right),\mathsf{\Phi }\left(v\right)).$$

(II)

(Generalizes Graph Embedding) Let $(W,+,·,K)$ be a classical vector space. Fix ${\mathrm{sim}}_0:W\times W\to \mathbb{R}$ and $\mathrm{Agg}$. Let $\phi :\mathcal{V}\to W$ be a classical graph embedding (Definition: Graph embedding in a vector space). Define a lifted map

$$\mathsf{\Phi }\left(v\right)=\left\{\phi \left(v\right)\right\}\in {\mathcal{P}}_1^{*}\left(W\right),\mathsf{\Psi }\left(v\right)={\iota }_{1\to n}\left(\mathsf{\Phi }\left(v\right)\right)\in {\mathcal{P}}_n^{*}\left(W\right).$$

Then Ψ is a graph superhyperembedding into ${\mathcal{P}}_n^{*}\left(W\right)$, and for all $u,v\in \mathcal{V}$,

$${\mathrm{SIM}}_n(\mathsf{\Psi }\left(u\right),\mathsf{\Psi }\left(v\right))={\mathrm{sim}}_0(\phi \left(u\right),\phi \left(v\right)).$$

In particular, Graph SuperHyperembedding strictly contains the usual Graph Embedding as the singleton case.

Proof. (I) Let $n\ge 1$ and define $\mathsf{\Psi }\left(v\right)={\iota }_{1\to n}\left(\mathsf{\Phi }\left(v\right)\right)$. For any $u,v\in \mathcal{V}$, by Lemma 2.12,

$${\mathrm{SIM}}_n(\mathsf{\Psi }\left(u\right),\mathsf{\Psi }\left(v\right))={\mathrm{SIM}}_n\left({\iota }_{1\to n}\left(\mathsf{\Phi }\left(u\right)\right),{\iota }_{1\to n}\left(\mathsf{\Phi }\left(v\right)\right)\right)={\mathrm{SIM}}_1(\mathsf{\Phi }\left(u\right),\mathsf{\Phi }\left(v\right)).$$

Therefore the proximity-preservation property for $\mathsf{\Psi }$ at level n is identical to that for $\mathsf{\Phi }$ at level 1. Hence $\mathsf{\Psi }$ is a graph superhyperembedding.

(II) Let $\phi :\mathcal{V}\to W$ be a classical graph embedding and define $\mathsf{\Phi }\left(v\right)=\left\{\phi \right(v\left)\right\}$ and $\mathsf{\Psi }\left(v\right)={\iota }_{1\to n}\left(\mathsf{\Phi }\left(v\right)\right)$. First compute ${\mathrm{SIM}}_1$ on singletons:

$${\mathrm{SIM}}_1(\mathsf{\Phi }\left(u\right),\mathsf{\Phi }\left(v\right))=\mathrm{Agg}\left(\{{\mathrm{SIM}}_0(x,y)\mid x\in \left\{\phi \left(u\right)\right\},y\in \left\{\phi \left(v\right)\right\}\}\right)=\mathrm{Agg}\left(\left\{{\mathrm{sim}}_0(\phi \left(u\right),\phi \left(v\right))\right\}\right)={\mathrm{sim}}_0(\phi \left(u\right),\phi \left(v\right)).$$

Next apply Lemma 2.12 to lift from level 1 to level n:

$${\mathrm{SIM}}_n(\mathsf{\Psi }\left(u\right),\mathsf{\Psi }\left(v\right))={\mathrm{SIM}}_n\left({\iota }_{1\to n}\left(\mathsf{\Phi }\left(u\right)\right),{\iota }_{1\to n}\left(\mathsf{\Phi }\left(v\right)\right)\right)={\mathrm{SIM}}_1(\mathsf{\Phi }\left(u\right),\mathsf{\Phi }\left(v\right))={\mathrm{sim}}_0(\phi \left(u\right),\phi \left(v\right)).$$

Thus the induced similarities in the superhyperembedding match the classical similarities of $\phi$, so $\mathsf{\Psi }$ preserves the same proximity information and is a graph superhyperembedding. □

3. Conclusion

In this paper, we introduced two extensions of classical graph embedding by employing HyperVector and SuperHyperVector structures: namely, Graph Hyperembedding and Graph SuperHyperembedding. For reference, we summarize Graph embedding, Graph Hyperembedding, and Graph SuperHyperembedding in a comparative Table 2.

In future work, we aim to investigate extended frameworks based on Fuzzy Sets[38], Neutrosophic Sets[39,40], Uncertain Sets[41,42], Plithogenic Sets[43,44], and related generalizations.