A finite hypergraph extends the classical graph model by allowing hyperedges that may join any nonempty subset of vertices [1, 2, 3]. Building on this idea, a finite SuperHyperGraph is obtained by iterating the powerset construction so as to form nested families of vertex and edge sets that encode multi-layer relations [4, 6, 7, 8, 5]. Such models have proved useful in, for example, molecular design, complex-network analysis, and advanced signal-processing workflows [9, 10]. Unless noted otherwise, the integer n in Pn(·) or in an n–SuperHyperGraph is assumed to be nonnegative.

Definition 1.1 (Base Set). A base set S is the underlying universe of discourse:

S = { x | x belongs to the context under consideration }.

All objects appearing in P(S) or in any iterated powerset Pn(S) are, by definition, subsets ultimately drawn from S.

Definition 1.2 (Powerset). (cf. [11, 12]) For a set S, the powerset P(S) is the family of all subsets of S: P(S) = { A ⊆ S }.

In particular, both S and the empty set ∅ are members of P(S).

Definition 1.3 (Hypergraph). [13, 14] A hypergraph is an ordered pair H = (V, E) where • V is a finite set of vertices, and • E is a finite collection of nonempty subsets of V ; elements of E are the hyperedges.

Hypergraphs naturally capture interactions among more than two participants.

Example 1.4 (Hypergraph: Sepsis Initial Management Bundle). Consider the finite vertex set V = {BC, ABX, IVF, LAC, VP}, where BC = blood cultures, ABX = broad-spectrum antibiotic, IVF = intravenous fluids, LAC = serum lactate, VP = vasopressor. A hypergraph H = (V, E) can encode clinical bundles via hyperedges that represent multi-way order sets:

E = {BC, ABX, LAC, IVF}, {IVF, VP} .

The first hyperedge models an initial “sepsis bundle” (obtain cultures, give antibiotics, draw lactate, start fluids) and the second models the joint requirement of fluids and vasopressor for persistent hypotension. By Definition 1.3, E is a finite family of nonempty subsets of V , hence H is a finite hypergraph.

Definition 1.5 (n-th Powerset). [15, 16] Let X be a set. Define P1(X) = P(X). For n ≥ 1, Pn+1(X) = P Pn(X) .

When the empty set is excluded, we write Pn ∗(X) = Pn(X) \ {∅}.

Example 1.6 (n-th Powerset: Diagnostic Imaging Panels). Let X = {CT, MRI, US} denote three imaging modalities (computed tomography, magnetic resonance imaging, ultrasound). The nonempty first powerset P1 ∗(X) = P(X) \ {∅} lists all admissible single-visit “panels” of modalities; explicitly, P1 ∗(X) = {CT}, {MRI}, {US}, {CT, MRI}, {CT, US}, {MRI, US}, {CT, MRI, US} , so |P1 ∗(X)| = 23 − 1 = 7. Elements of the second powerset P2(X) = P P1(X) are collections of such panels; for instance, T1 = {CT}, {MRI, US} , T2 = {CT, MRI}, {US} are two distinct members of P2 ∗(X). Here T1 can represent two alternative workups (CT alone vs. MRI+US), while T2 encodes a staged plan (CT+MRI followed by US). This illustrates how Pn organizes increasingly rich combinatorial choices of medical tests as n grows.

Definition 1.7 (n–SuperHyperGraph). (cf. [17, 18, 19]) Fix a finite, nonempty base set V0 and define the iterated powerset by P0(V0) := V0, P k+1(V0) := P P k(V0) (k ∈ N).

For an integer n ≥ 0, an n–SuperHyperGraph on V0 is a pair SHG(n) = (V, E) with V ⊆ P n(V0) and E ⊆ P(V ) \ {∅}.

Elements of V are the n–supervertices, and elements of E are the n–superedges (each n–superedge is a nonempty subset of V ).

Example 1.8 (n–SuperHyperGraph (here n = 2): Two-Tier Care Pathway). Let the base set of atomic actions be V0 = {ABX, BC, LAC, IVF, VP}.

Elements of P2(V0) are collections of nonempty subsets of V0, which we use to represent stage-wise groupings of actions. Define three 2–supervertices A1 = {ABX, BC}, {LAC} , A2 = {IVF}, {VP} , A3 = {ABX}, {IVF, VP} , and set V = {A1, A2, A3} ⊆ P2(V0). Form the 2–superedges as collections of these stage groups, E = {A1, A2}, {A2, A3} ⊆ P(V ) \ {∅}.

Then SHG(2) = (V, E) is a 2–SuperHyperGraph (Definition 1.7). Intuitively, A1 captures the “diagnos-tic/antibiotic initiation” stage, A2 the “hemodynamic support” stage, and A3 an alternative consolidation stage; the superedges encode permissible stage transitions in a care pathway.

A Knowledge Graph is a graph-based representation that encodes facts as entities and their relations, supporting reasoning, semantic search, and knowledge-driven applications [20, 21, 22, 23, 24]. Extended concepts such as the Knowledge HyperGraph [25, 26, 27, 28] and the Knowledge SuperHyperGraph [29] have also been proposed in the literature. A Medical Knowledge Graph adapts this paradigm to the medical domain, modeling entities such as diseases, symptoms, drugs, and procedures, and linking them through clinically meaningful relations to support clinical reasoning and decision making [30, 31, 32, 33, 34, 35].

Definition 1.9 (Medical Knowledge Graph). (cf.[30, 31, 32]) A Medical Knowledge Graph (MKG) is a typed, directed graph that represents medical knowledge as triples of the form ⟨h, r, t⟩ ∈ E × R × E, where h, t ∈ E are entities (e.g., Disease, Symptom, Drug/MedicinalProduct, Procedure, LaboratoryExam, Gene, MedicalDepartment) and r ∈ R is a typed relation (e.g., has_symptom, treats, contraindicated_with, diagnosed_by, associated_gene). An MKG thus encodes domain facts as a set T ⊆ E × R × E and may retain provenance (data source) for downstream clinical or research use.

Example 1.10 (Disease–Symptom–Therapy Subgraph). Let entities Gallstone ∈ Disease, Abdominal Pain ∈ Symptom, Ultrasound ∈ Radiographic Exam, Ursodeoxycholic Acid ∈ MedicinalProduct.

A small MKG fragment can be recorded as the triple set ⟨Gallstone, has_symptom, Abdominal Pain⟩, ⟨Gallstone, diagnosed_by, Ultrasound⟩, ⟨Ursodeoxycholic Acid, treats, Gallstone⟩ .

Such fragments support tasks like clinical decision support or knowledge-enhanced NER/RE.

Example 1.11 (Drug–Interaction and Lab Marker). Let Warfarin, Aspirin ∈ MedicinalProduct, INR ∈ LaboratoryExam.

An MKG can encode pharmacology knowledge as ⟨Warfarin, interacts_with, Aspirin⟩, ⟨Warfarin, monitored_by, INR⟩ , which is useful for medication safety alerts and therapy monitoring.

A Medical Knowledge HyperGraph further generalizes the framework by allowing hyperedges that connect multiple medical entities at once, thereby capturing complex clinical relationships that cannot be expressed through simple triples.

Definition 2.1 (Medical Knowledge HyperGraph (MKHG)). Let E be a finite set of medical entities (e.g., diseases, symptoms, drugs, procedures, laboratory exams) and let C be a set of entity categories. A typing map typ : E → C assigns a category to each entity.

Let R be a finite set of relation types equipped with • an arity function ar : R → N≥1, and • a signature function sig : R → k≥1 C , such that if ar(r) = k then sig(r) = (c1, . . . , ck) ∈ Ck specifies the required category for each argument position. A Medical Knowledge HyperGraph (MKHG) is a tuple K = (E, R, H, typ, ar, sig) where H is a finite set of directed, typed hyperedges of the form h = (r, (e1, . . . , ek)) with r ∈ R, k = ar(r), ei ∈ E, subject to the typing constraint typ(ei) = sig(r)i for each i = 1, . . . , k. The tuple order encodes direc-tion/argument roles; when k = 2 this reduces to a directed, typed edge.

Example 2.2 (A multi-entity clinical rule as a hyperedge). Let E contain the entities Sepsis ∈ Disease, Fever, Hypotension ∈ Symptom, SerumLactate ∈ LaboratoryExam.

Define a relation type r = diagnostic_criterion with ar(r) = 4, sig(r) = (Disease, Symptom, Symptom, LaboratoryExam).

Then the hyperedge h = diagnostic_criterion, (Sepsis, Fever, Hypotension, SerumLactate) ∈ H encodes, in a single typed object, a rule template that simultaneously relates a disease to multiple symptoms and a lab marker. Such k > 2 relations cannot be represented by a single binary triple without factorization.

Example 2.3 (Pharmacogenomic Dosing Rule (Drug–Gene–Disease, k=3)). Let the entity set and categories be E = {Clopidogrel, Warfarin, ACS, AF, CYP2C19_LOF, VKORC1_A}, C = {Drug, Disease, GeneVariant}. Define the typing map Clopidogrel, Warfarin ∈ Drug, ACS (acute coronary syndrome), AF (atrial fibrillation) ∈ Disease, CYP2C19_LOF, VKORC1_A ∈ GeneVariant.

Introduce a relation type r = requires_genotype_guided_dosing with ar(r) = 3, sig(r) = (Drug, GeneVariant, Disease).

Then the hyperedge set H = r, (Clopidogrel, CYP2C19_LOF, ACS) , r, (Warfarin, VKORC1_A, AF) is admissible because, for each argument position i = 1, 2, 3, the type typ(ei) matches the required category sig(r)i. The tuple order encodes direction/roles: (drug, genotype, indication). This k>2 structure captures pharmacogenomic dosing rules that cannot be represented by a single binary triple without factorization.

Example 2.4 (Acute Stroke Therapy Eligibility (Disease–Imaging–TimeWindow–Procedure, k=4)). Let E = {IschemicStroke, NoncontrastCT, CTA, Within4_5h, Within24h, IVT_Alteplase, EVT_Thrombectomy}. Use categories C = {Disease, Imaging, TimeWindow, Procedure}, and typing IschemicStroke ∈ Disease, NoncontrastCT, CTA ∈ Imaging, Within4_5h, Within24h ∈ TimeWindow, IVT_Alteplase, EVT_Thrombectomy ∈ Procedure.

Define the relation r = therapy_eligibility by ar(r) = 4, sig(r) = (Disease, Imaging, TimeWindow, Procedure).

Consider the hyperedges h1 = r, (IschemicStroke, NoncontrastCT, Within4_5h, IVT_Alteplase) , h2 = r, (IschemicStroke, CTA, Within24h, EVT_Thrombectomy) .

Both h1 and h2 satisfy the typing constraint typ(ei) = sig(r)i for all i = 1, . . . , 4. Thus an MKHG K = (E, {r}, {h1, h2}, typ, ar, sig) encodes two clinically meaningful 4–ary relations linking a disease to an imaging prerequisite, a time window, and a specific reperfusion procedure.

Theorem 2.5 (Expressiveness: MKHG generalizes hypergraphs and medical knowledge graphs).

(i) (Hypergraph embedding) Every finite hypergraph H = (V, E) can be represented by an MKHG KH so that H is recovered from KH by forgetting types, labels, and argument order.

(ii) (Medical Knowledge Graph embedding) Every Medical Knowledge Graph G = (E, R, T ) consisting of typed binary triples ⟨h, r, t⟩ is the k=2 fragment of an MKHG KG obtained by taking ar(r) = 2 and mapping each triple to a typed hyperedge.

Proof : (i) Hypergraph embedding. Let H = (V, E) be a finite hypergraph. Set E = V and choose a single category C = {⋆} with typ(v) = ⋆ for all v ∈ V . For each hyperedge S ∈ E with |S| = k ≥ 1, assign a distinct relation label rS with ar(rS) = k, sig(rS) = (⋆, . . . , ⋆).

| {z }

k times

Fix any total order ≺ on V and list the elements of S as (v1, . . . , vk) in increasing ≺-order. Define the hyperedge hS = rS, (v1, . . . , vk) ∈ H.

Let KH = (E, R, H, typ, ar, sig) with R = {rS | S ∈ E}. By construction each hS satisfies the typing constraint, hence KH is an MKHG.

To recover H from KH, apply the forgetful map that (a) discards labels and types, (b) replaces each ordered tuple by its underlying set. This yields the edge set { {v1, . . . , vk} | (rS, (v1, . . . , vk)) ∈ H } = E on vertex set V , i.e., the original hypergraph (up to the harmless choice of ≺).

(ii) Medical Knowledge Graph embedding. Let G = (E, R, T ) be an MKG of typed binary triples. Retain the same entity set E and their categories C with the typing map typ. For each r ∈ R set ar(r) = 2, sig(r) = (chead, ctail), matching the declared domain/codomain categories of r in G. Define the hyperedge set H = r, (h, t) ⟨h, r, t⟩ ∈ T .

Then KG = (E, R, H, typ, ar, sig) is an MKHG in which every triple is a length-2 hyperedge obeying the signature constraint. Conversely, the binary slice of any MKHG (all hyperedges with k = 2) reproduces a medical knowledge graph by reading (r, (h, t)) as the triple ⟨h, r, t⟩. □

A Medical Knowledge SuperHyperGraph introduces hierarchical layers via iterated powersets, enabling the representation of multi-level medical relations and generalizing graphs, hypergraphs, and typed medical knowledge models.

Definition 2.6 (Flattening of supervertices). Let E0 be a finite base set (of atoms). For n ∈ N0 and A ∈ P n(E0), define the flattening map Flat0(e) := {e} (e ∈ E0), Flatn+1(A) := Flatn(B).

Thus Flatn(A) ⊆ E0 is the set of all atoms occurring inside A at level n.

Definition 2.7 (Medical Knowledge SuperHyperGraph (MKSHG)). Let • E0 be a finite set of medical atoms (e.g., diseases, symptoms, drugs, procedures, laboratory exams); • C be a finite set of categories and typ : E0 → C a typing map; • S R a finite set of relation types equipped with an arity map ar : R → N≥1 and a signature map sig : R → k≥1 Ck so that, for r ∈ R with k = ar(r), we have sig(r) = (c1, . . . , ck).

Fix n ∈ N0. A Medical Knowledge SuperHyperGraph of level n is a tuple K(n) = (E0, C, typ, R, ar, sig, V, H) where V ⊆ P n(E0) is a finite set of n–supervertices, and H is a finite set of directed, typed superhyperedges of the form h = r, (A1, . . . , Ak) with r ∈ R, k = ar(r), Ai ∈ V, subject to the admissibility constraint ∀i ∈ {1, . . . , k} ∀e ∈ Flatn(Ai) : typ(e) = sig(r)i.

When k = 2, a superhyperedge is a typed directed edge (r, (A1, A2)); when k > 2, it encodes a k–ary clinical relation among supervertices.

Example 2.8 (A level–2 clinical pathway constraint). Let E0 contain Sepsis ∈ Disease, Fever, Hypotension ∈ Symptom, SerumLactate ∈ LaboratoryExam, IVFluids, Vasopressor ∈ Procedure.

At level n = 2 take the supervertices A = {Sepsis} , B = {Fever, Hypotension} , C = {SerumLactate} , D = {IVFluids, Vasopressor} ∈ P2(E0).

Define r = pathway_step with ar(r) = 4 and sig(r) = (Disease, Symptom, LaboratoryExam, Procedure).

Then h = (r, (A, B, C, D)) is an admissible superhyperedge because Flat2(A) = {Sepsis}, Flat2(B) = {Fever, Hypotension}, Flat2(C) = {SerumLactate}, Flat2(D) = {IVFluids, Vasopressor}, and the types match the signature positions.

Example 2.9 (Level-1 MKSHG: Multimodal COVID-19 Triage Rule). Let the finite set of medical atoms be E0 = {COVID19, Influenza, Fever, Cough, Dyspnea, PCR, Antigen, ChestXray, ChestCT}.

Assume the typing map typ : E0 → C with categories C = {Disease, Symptom, LaboratoryExam, Imaging} given by COVID19, Influenza ∈ Disease, Fever, Cough, Dyspnea ∈ Symptom, PCR, Antigen ∈ LaboratoryExam, ChestXray, ChestCT ∈ Imaging.

Fix n = 1 and define the set of supervertices V ⊆ P1(E0) = P(E0) by A = {COVID19}, B = {Fever, Cough, Dyspnea}, C = {PCR}, D = {ChestXray, ChestCT}.

Let the relation type r = triage_criterion have arity and signature ar(r) = 4, sig(r) = (Disease, Symptom, LaboratoryExam, Imaging).

The superhyperedge h = r, (A, B, C, D) is admissible since Flat1(A) = {COVID19} ⊆ Disease, Flat1(B) = {Fever, Cough, Dyspnea} ⊆ Symptom, Flat1(C) = {PCR} ⊆ LaboratoryExam, Flat1(D) = {ChestXray, ChestCT} ⊆ Imaging.

Interpretation: a single typed rule links a target disease to a set of symptoms, a lab test choice, and an imaging panel. This k>2 relation cannot be captured by one binary triple without factorization.

Example 2.10 (Level-3 MKSHG: Precision Oncology Regimen Selection). Let the atom set include E0 = {NSCLC, EGFR_Ex19del, ALK_fusion, Osimertinib, Alectinib, Carboplatin, Pemetrexed}.

Use categories C = {Disease, GeneVariant, Drug}, and typing NSCLC ∈ Disease, EGFR_Ex19del, ALK_fusion ∈ GeneVariant, Osimertinib, Alectinib, Carboplatin, Pemetrexed ∈ Drug.

Fix n = 3. Recall that P3(E0) = P P2(E0) ; its elements are collections of level-2 supervertices. Define three level-3 supervertices V ⊆ P3(E0):

A = {{NSCLC}} , B = {{EGFR_Ex19del}}, {{ALK_fusion}} , C = {{Osimertinib}}, {{Alectinib}}, {{Carboplatin, Pemetrexed}} .

Let r = personalized_regimen with ar(r) = 3, sig(r) = (Disease, GeneVariant, Drug).

Then the superhyperedge h = r, (A, B, C) ∈ H satisfies the admissibility constraint because Flat3(A) = {NSCLC} ⊆ Disease, Flat3(B) = {EGFR_Ex19del, ALK_fusion} ⊆ GeneVariant, Flat3(C) = {Osimertinib, Alectinib, Carboplatin, Pemetrexed} ⊆ Drug.

Interpretation: a hierarchical object C captures alternative regimen sets (single agents or combinations), linked to a tumor class A and biomarker group B. Using n = 3 permits packaging alternative, stageable regimens while preserving clean typing at the atom level.

Theorem 2.11 (Expressiveness). For any fixed n ∈ N0, the class of level–n Medical Knowledge SuperHyperGraphs (MKSHGs) strictly generalizes:

(i) finite n–SuperHyperGraphs; (ii) Medical Knowledge HyperGraphs (MKHGs); (iii) Medical Knowledge Graphs (MKGs) of typed binary triples.

Proof : (i) SuperHyperGraph ⇒ MKSHG. Let SHG(n) = (V, E) be an n–SuperHyperGraph on the finite base V0, so V ⊆ P n(V0) and E ⊆ P(V ) \ {∅}. Construct an MKSHG Φ(SHG(n)) as follows:

E0 := V0, C := {⋆}, typ(e) ≡ ⋆, V as given.

For each S = {A1, . . . , Ak} ∈ E (choose any fixed ordering of its elements), introduce a fresh relation rS with ar(rS) = k, sig(rS) = (⋆, . . . , ⋆).

| {z }

Add the superhyperedge hS = (rS, (A1, . . . , Ak)) to H. By construction, the admissibility constraint holds trivially because typ ≡ ⋆. Define Φ(SHG(n)) = (E0, C, typ, {rS}S∈E, ar, sig, V, H).

The forgetful projection U that erases relation labels and tuple order, U : (r, (A1, . . . , Ak)) 7−→ {A1, . . . , Ak}, sends Φ(SHG(n)) back to (V, E) because each rS is unique to S. Hence every n–SuperHyperGraph embeds in a level–n MKSHG.

(ii) MKHG ⇒ MKSHG. Let K = (E, R, H, typ, ar, sig) be a Medical Knowledge HyperGraph (typed directed hypergraph on atoms). Take n = 0, E0 := E, V := E ⊆ P0(E0), and keep (R, ar, sig) unchanged. Each hyperedge (r, (e1, . . . , ek)) ∈ H is already an admissible superhyperedge because Flat0(ei) = {ei} and typ(ei) = sig(r)i. Thus MKHGs are precisely the level–0 fragment of MKSHGs.

(iii) MKG ⇒ MKSHG. Let G = (E, R, T ) be a Medical Knowledge Graph of typed binary triples ⟨h, r, t⟩ with declared domain/codomain categories (c1, c2) for each r ∈ R. Again set n = 0, E0 := E, V := E, and keep the same typing map. Define ar(r) := 2 and sig(r) := (c1, c2). Map each triple to the superhyperedge (r, (h, t)). Admissibility is exactly the MKG type constraint. Hence every MKG embeds as the binary slice of a level–0 MKSHG.

Since the embeddings in (i)–(iii) are injective on objects (up to the harmless choice of an ordering in (i)), MKSHGs strictly generalize the three structures. □