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arxiv: 2605.12509 · v2 · pith:5CJCC2BEnew · submitted 2026-03-24 · 💻 cs.SI · cs.AI· cs.CE· math.CO

Representing Higher-Order Networks: A Survey of Graph-Based Frameworks

Takaaki Fujita , Florentin Smarandache This is my paper

Pith reviewed 2026-05-19 18:07 UTC · model grok-4.3

classification 💻 cs.SI cs.AIcs.CEmath.CO
keywords higher-order networksgraph formalismscomplex systemsnetwork modelingmultiway interactionssurveyunified perspectivegraph extensions
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The pith

Higher-order networks are unified through a survey of graph formalisms that extend beyond pairwise links.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper surveys mathematical notions that model higher-order networks by incorporating multiway, hierarchical, temporal, multilayer, recursive, and tensor-based interactions. It reviews foundational concepts along with various extensional frameworks and some newly introduced formalisms. The focus stays on their structural principles, relationships, and roles in representing complex systems. The central aim is to supply one perspective that lets readers compare these models and select fitting tools for theory or applications.

Core claim

By surveying foundational concepts, extensional frameworks, and newly introduced formalisms with emphasis on their structural principles, relationships, and modeling roles, the book establishes a unified perspective that helps readers compare diverse higher-order network models and identify appropriate tools for theoretical study and practical applications.

What carries the argument

Higher-order graph formalisms incorporating multiway, hierarchical, temporal, multilayer, recursive, and tensor-based interactions.

If this is right

  • Readers obtain a single viewpoint for comparing different higher-order network models.
  • Suitable tools become identifiable for specific theoretical investigations of complex systems.
  • New formalisms add options for capturing interactions that standard graphs miss.
  • Practical applications gain from clearer choices among representation methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hybrid models that mix features from several formalisms could address gaps in current representations.
  • Adoption of these frameworks might simplify the creation of analysis tools for social or biological networks.
  • Empirical tests on datasets from specific domains could show which formalisms perform best in practice.

Load-bearing premise

The collection of surveyed and newly introduced formalisms is sufficiently comprehensive and non-redundant to serve as a reliable reference for model selection across domains.

What would settle it

Identification of a real-world higher-order phenomenon that fits none of the described formalisms or that reveals contradictions in the claimed relationships among them.

Figures

Figures reproduced from arXiv: 2605.12509 by Florentin Smarandache, Takaaki Fujita.

Figure 2.1
Figure 2.1. Figure 2.1: A two-level organization chart modeled as a 1-SuperHyperGraph SHG(1) = (V, E, ∂). Teams are 1-supervertices, and e1, e2 are superedges with incidence given by ∂. v1 = {{a}, {a, b}} v2 = {{b}, {b, c}} v3 = {{c}} e1 e2 ∂(e12) = {v12, v23} [PITH_FULL_IMAGE:figures/full_fig_p013_2_1.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: A 2-SuperHyperGraph SHG(2) = (V, E, ∂) over V0 = {a, b, c}. Each 2-supervertex is a set of subsets of V0, and e1, e2 are superedges with incidence given by ∂. Let the superedge identifier set be E = {e1, e2}, and define the incidence map ∂ : E → P∗ (V ) by ∂(e1) = {v1, v2}, ∂(e2) = {v2, v3}. Then SHG(2) = (V, E, ∂) is a 2-SuperHyperGraph over V0 in the sense of Definition 2.1.4. Here e1 links the two nes… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: An illustration of the h-model in Example 2.3.2 [PITH_FULL_IMAGE:figures/full_fig_p017_2_4.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Hasse diagram of the Boolean poset on P({1, 2, 3}), highlighting the 2-chain-free middle layer A. Definition 2.4.4 (Iterated k-chain-free subset (depth r)). Let P = (X, ≤) be a finite poset and let k ≥ 2. Define recursively a sequence of posets P (r) k = [PITH_FULL_IMAGE:figures/full_fig_p020_2_5.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Illustration of an iterated power set graph of depth 2 for A = {1, 2, 3}. The figure shows a local induced fragment of Γ2(A) with example vertices X, Y, Z, W. 2.6 Johnson Graph A Johnson graph has vertices given by the w-subsets of [n], with edges joining pairs that differ by one element [85, 86, 87]. Related notions of the Johnson graph are also known, such as the generalized Johnson graph[88, 89]. Defi… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: An illustration of the Meta-Graph in Example 2.8.2: each meta-vertex is itself a dependency graph, and meta-edges encode semantic relations between them. Definition 2.8.3 (Iterated Meta-Graph (depth t)). [103] Fix (G, R) as in Definition 2.8.1. Define recursively, for t ∈ N0, a universe of level-t objects G (t) and a family of level-t relations R(t) as follows: G (0) := G, R(0) := R. Assume G (t) and R(t… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: A depth-1 Iterated Meta-Graph: the vertices M1, M2 are themselves metagraphs, and the top-level edge g is labeled by the lifted relation R ↑ api. 2.9 Meta-HyperGraph and Meta-SuperHyperGraph A meta-hypergraph is a hypergraph whose vertices are objects; each hyperedge relates finite ver￾tex sets via labeled relations (cf. [5, 108]). It can also be described as a hypergraph of hypergraphs. A meta-superhype… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p035_2.png] view at source ↗
Figure 2.9
Figure 2.9. Figure 2.9: An illustration of the MetaHyperGraph in Example 2.9.2 [PITH_FULL_IMAGE:figures/full_fig_p028_2_9.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p036_2.png] view at source ↗
Figure 2.10
Figure 2.10. Figure 2.10: A nested hypergraph where the hyperedge e2 contains another hyperedge e1. be a finite set; its elements are called n-supervertices. A nested level-n SuperHyperGraph is a triple H(n) = (Vn, En, ρ), where En is a finite set (of superhyperedges) and ρ : Vn ] En → N is a rank function such that: 1. ρ(v) = 0 for all v ∈ Vn; 2. for every e ∈ En, the superhyperedge e is a nonempty finite set satisfying e ⊆ Vn … view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p038_2.png] view at source ↗
Figure 2.11
Figure 2.11. Figure 2.11: A multi-hypergraph for repeated group interactions (Example 2.11.2) [PITH_FULL_IMAGE:figures/full_fig_p032_2_11.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p040_2.png] view at source ↗
Figure 2.12
Figure 2.12. Figure 2.12: A hierarchical superhypergraph of height 2: vertices may live on different levels and edges may cross levels. Dashed lines indicate constituent relations (coherence), while e1, e2 are superhyperedges. 2.15 Recursive HyperGraph and Recursive SuperHyperGraph A recursive hypergraph is a hypergraph-like object in which a hyperedge may contain ordinary vertices and also lower-level hyperedges as elements, th… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p045_2.png] view at source ↗
Figure 2.13
Figure 2.13. Figure 2.13: A schematic illustration of the 1-recursive hypergraph in Example 2.15.3. The recursive hyperedge e3 contains a lower-level hyperedge e1 and a vertex c. An (n, k)-recursive SuperHyperGraph combines hierarchical supervertices (via iterated power￾sets) with recursive superhyperedges of bounded depth k, allowing edges to contain supervertices and nested lower-level edges as elements. Definition 2.15.4 ((n,… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p046_2.png] view at source ↗
Figure 2.14
Figure 2.14. Figure 2.14: A depth-2 Tree-Vertex Graph for a small organization (Example 2.16.2). Leaves represent employees, internal nodes represent teams, and the root represents the department. Thus the tree groups employees into two teams {a, b} and {c, d}, and then into one department. (2) Nested labeling. Define η : N → S2 k=0 PSk (V0) \ {∅} by η(a0) = a, η(b0) = b, η(c0) = c, η(d0) = d, η(u1) = {η(a0), η(b0)} = {a, b}, η(… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p048_2.png] view at source ↗
Figure 2.15
Figure 2.15. Figure 2.15: A concrete MultiMetaGraph. Each top-level vertex is a finite nonempty family of graphs, and each directed edge is labeled by a relation on graph-families. together with source, target, and label maps by s(e1) = M1, t(e1) = M2, λ(e1) = Rcom, and s(e2) = M2, t(e2) = M3, λ(e2) = Rinc. We verify the incidence condition. Since M1 ∩ M2 = {G2} 6= ∅, we have (M1, M2) ∈ Rcom. Also, G3 ∈ M2 has 3 vertices and G4 … view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p064_2.png] view at source ↗
Figure 2.16
Figure 2.16. Figure 2.16: A concrete Transfinite SuperHyperGraph of height ω + 1. Dashed lines indi￾cate membership/containment relations used to witness downward closure, while e1, e2, e3 denote transfinite superhyperedges. Example 2.20.2 (Multi-indexed iterated powerset in two axes). Let d = 2, U1 = {a, b}, U2 = {1, 2, 3}, so the base system is U = (U1, U2). Consider the multi-index n = (1, 2). Then P (1,2)(U) = P 1 (U1) × P2 … view at source ↗
Figure 2.16
Figure 2.16. Figure 2.16 [PITH_FULL_IMAGE:figures/full_fig_p065_2_16.png] view at source ↗
Figure 2.17
Figure 2.17. Figure 2.17: A two-axis multi-indexed iterated powerset example (Example 2.20.10). The figure shows the element x ∈ P (1,2)(U), the axis-wise singleton lift Σ (1,2) 1 (x) ∈ P (2,2)(U), and the typed coordinatewise inclusion into y ∈ P (2,2)(U). Definition 2.20.3 (Axis-wise singleton lift). Let er denote the r-th standard basis vector of N d 0 . For n ∈ N d 0 , define Σ n r : P n (U) → P n+er (U) by Σ n r (x1, . . . … view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p066_2.png] view at source ↗
Figure 2.18
Figure 2.18. Figure 2.18: A HyperMatroid induced by the graphic matroid of the triangle graph K3 (Exam￾ple 2.23.3). The unique cycle of K3 gives the unique circuit. 2.24 SuperHyperMatroid A superhypermatroid is a matroid on nested-set supervertices, with supercircuits satisfying elim￾ination, capturing hierarchical dependence relations. Definition 2.24.1 (n-SuperHyperMatroid). Let V0 be a nonempty base set and fix n ∈ N0. Define… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p067_2.png] view at source ↗
Figure 2.19
Figure 2.19. Figure 2.19: A Kneser (2, 2, 2)-SuperHyperGraph in Example 2.25.2. Superedges are determined by disjoint flattened supports. Example 2.25.2 (A Kneser SuperHyperGraph). Let N = 4, n = 2, k = 2, and r = 2. Then the base set is V0 = [4] = {1, 2, 3, 4}, and 2-supervertices are elements of P 2 (V0) = P(P(V0)). Using the flattening map flat2, define the following 2-supervertices: v1 =  {1}, {2} [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p074_2.png] view at source ↗
Figure 2.20
Figure 2.20. Figure 2.20: A graded superhypergraph of height 2 in Example 2.26.2. 2.27 Hyperstructures and Superhyperstructures Many mathematical and real-world systems exhibit hierarchical organization, such as elements, groups of elements, and higher-level groupings. To model such layered interactions in a unified way, one uses hyperstructures and their iterated extensions [143, 144], called superhyperstructures. Roughly speak… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p077_2.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Coauthorship groups represented as an abstract simplicial complex: two 2-simplices sharing the edge {B, C}. 3.2 Simplicial set A simplicial set is a functor from the simplex category’s opposite to sets, encoding faces and degeneracies of abstract simplices [169, 170, 171]. This can be regarded as one of the concepts for representing “higher-order” structure in a more geometric manner. Definition 3.2.1 (S… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p078_2.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: An illustration of the nerve N(C) in Example 3.2.2 [PITH_FULL_IMAGE:figures/full_fig_p064_3_2.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p079_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: An illustration of the cell complex structure on S 1 with one 0-cell and one 1-cell (Example 3.3.4). 3.4 CW complex A CW complex is a cell complex satisfying closure-finiteness and weak topology, enabling in￾ductive construction and homotopy-friendly computations often efficient [174, 175]. This can be regarded as one of the concepts for representing “higher-order” structure in a more geometric manner. D… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p087_2.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: An illustration of the CW complex structure on S 2 in Example 3.4.2. Definition 3.5.1 (Convex polytope and face). A (convex) polytope in R d is the convex hull P = conv(S) of a finite set S ⊂ R d . A (proper) face of P is a subset of the form F = P ∩ {x ∈ R d : `(x) = max y∈P `(y)} for some linear functional ` : R d → R, with F 6= ∅; the whole polytope P is also regarded as a face. Example 3.5.2 (A conve… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: A B C D {A, B, C} {B, C, D} {B, C} shared face Abstract simplicial complex view Maximal simplices: {A, B, C} and {B, C, D}. All vertices and edges shown are automatically included as faces (downward closure) [PITH_FULL_IMAGE:figures/full_fig_p091_3_1.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Two triangles forming a polyhedral complex (Example 3.5.4). 3.6 Dowker Complex A Dowker complex constructs simplices from a binary relation, turning shared relational neigh￾borhoods into topological higher order structure for analysis [179, 180, 181]. Definition 3.6.1 (Dowker complex). Let X and Y be finite nonempty sets, and let R ⊆ X × Y be a binary relation. For each x ∈ X, define its R-neighborhood i… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p091_3.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: A 2-dimensional cubical complex formed by two adjacent unit squares. The red segment is the common 1-face Q1 ∩ Q2 = {1} × [0, 1]. 1. every face of Q1 and Q2 belongs to K, so K is closed under taking faces; 2. the intersection of any two cubes in K is either empty or a face of both cubes. In particular, Q1 ∩ Q2 = {1} × [0, 1] is a 1-dimensional face of both Q1 and Q2. Therefore K is a 2-dimensional cubica… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p092_3.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: A directed graph generating a path complex. Allowed higher-order paths are deter￾mined by directed consecutive edges. 1. a k-vector space F(σ) for each cell σ ∈ X, called the stalk of F over σ; 2. for each incidence relation σ ≤ τ , a linear map Fσ,τ : F(σ) → F(τ ), called the restriction map; such that Fσ,σ = idF(σ) for every cell σ, and Fτ,η ◦ Fσ,τ = Fσ,η whenever σ ≤ τ ≤ η. Definition 3.9.2 (Global se… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p093_3.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: A cellular sheaf on a graph with two vertices and one edge. The edge stalk stores a shared scalar compatibility value determined by linear maps from the vertex stalks. Since the only non-identity face relations are u ≤ e and v ≤ e, the functorial conditions are automatically satisfied, and hence F is a cellular sheaf. A global section of F is a triple [PITH_FULL_IMAGE:figures/full_fig_p073_3_8.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p094_3.png] view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: A 1-SuperHypercomplex built from teams of individuals (Example 3.11.2). The filled super-triangle represents the 2-dimensional superface σ = {v1, v2, v3}, and dashed links indicate team membership at the base level. Elements of K are called n-superfaces (or super-simplices). If |σ| = k + 1, then σ is a k￾dimensional superface and we set dim(σ) := k. The dimension of SuHyC(n) is dim SuHyC(n)  := sup{dim(… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p095_3.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: A Tanner hypergraph for the parity-check matrix in Example 4.3.3. The two hyper￾edges correspond to the row supports supp(H1) = {1, 2, 4} and supp(H2) = {2, 3, 4}. 4.4 Tanner SuperHyperGraph A Tanner SuperHyperGraph extends Tanner hypergraphs by allowing hierarchical supervertices and superhyperedges, modeling nested coding constraints, multilevel parity relations, and higher￾order dependency structures … view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p098_3.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: A temporal communication network over discrete time T = {1, 2, 3, 4}. Each dashed box represents one time slice, and the directed edge inside it indicates the message sent at that time. Example 4.6.2 (A temporal communication network over discrete time). Let the node set be V = {A, B, C} and let the time set be the discrete set T = {1, 2, 3, 4} ⊂ Z. Suppose that at time t = 1 user A messages B, at time t… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p100_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: An ATN with pairwise and triple interactions on V = {1, 2, 3}. The left panel visu￾alizes nonzero entries of A(2), and the right panel visualizes the nonzero symmetric triple interaction in A(3) . Example 4.8.2 (An ATN with pairwise and triple interactions on V = {1, 2, 3}). Let V = {1, 2, 3}, so n = 3, and take K = 3. Define an Adjacency-Tensor Network A = [PITH_FULL_IMAGE:figures/full_fig_p088_4_4.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p101_3.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: A concrete Heterogeneous 1-SuperHyperGraph. The supervertices are typed subsets of the base set, and the superhyperedges are nonempty subsets of the common su￾pervertex set equipped with edge-types. 5.2 Knowledge Graph, HyperGraph, and SuperHyperGraph A knowledge graph represents entities and their binary relations as structured facts, enabling semantic reasoning, retrieval, and integration across domain… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p104_3.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: A rank-3 combinatorial map on K4. Each vertex is incident with exactly one edge of each color 0, 1, 2. Then each vertex is incident with exactly one edge of color 0, one edge of color 1, and one edge of color 2. For example, v1 is incident with {v1, v2} of color 0, {v1, v4} of color 1, {v1, v3} of color 2. The same property holds for v2, v3, v4. Hence G = (G, τ ) is a combinatorial map of rank 3. Its 0-f… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p108_4.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: A concrete Multimodal 1-SuperHyperGraph. The same set of 1-supervertices is shared by two modalities: communication (solid blue) and resource sharing (dashed red). 5.10 Operadic Interaction Graph (OIG) An Operadic Interaction Graph is a colored operad whose operations represent typed multi-input interactions; operad composition models gluing interactions, encoding higher-order, composi￾tional network str… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p110_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: An Operadic Interaction Graph for a typed workflow. The composite workflow is the operadic substitution h = γ(g; f, f). Example 5.10.2 (An Operadic Interaction Graph for typed workflows). Let the color set be C = {Raw, Clean, Model}, interpreted as data/product types in a workflow. Define a symmetric C-colored operad O by specifying the following generating operations: f ∈ O(Raw; Clean) (cleaning), g ∈ O… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p112_4.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: A Symmetric Monoidal Wiring Graph for a simple digital circuit. The circuit copies an input bit, negates one copy, and combines the two signals via AND. represents the circuit that takes an input bit x, copies it to (x, x), negates the first copy to (¬x, x), and then applies AND, producing ¬x ∧ x. Hence (C, ⊗, G) is a Symmetric Monoidal Wiring Graph in the sense of Definition 5.11.1. An overview diagram … view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p115_4.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: A Relational-Arity Graph (RAG) for Example 5.12.2. Solid arrows represent the binary relation RG 2 , and dashed arrows from tuple-nodes τ1, τ2 encode the ordered triples in RG 3 . is a Σ-structure on a vertex set V , i.e. R G α ⊆ V kα for all α ∈ A. Thus each relation symbol Rα encodes kα-way interactions (ordered tuples). Example 5.12.2 (A Relational-Arity Graph with pairwise and triple interactions). L… view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p122_5.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: A Closure-Implication Graph generated by the rules {a, b} ⇒ c and {c} ⇒ d. Define cl : P(V ) → P(V ) by letting cl(S) be the smallest subset of V that contains S and is closed under the above rules; equivalently, cl(S) is obtained by repeatedly adding c whenever {a, b} ⊆ S and adding d whenever c ∈ S, until no new elements can be added. For example, cl({a}) = {a}, cl({a, b}) = {a, b, c, d}, cl({b, c}) = … view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p126_5.png] view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: A Coalgebraic Nested-Neighborhood Graph with 2-nested neighborhoods on V = {1, 2, 3}. Theorem 5.14.3 (Well-definedness of the r-nested CNNG construction). Fix an integer r ≥ 1. Let Pf : Set → Set denote the finite-powerset functor, i.e. Pf (V ) = {S ⊆ V | S is finite}, and for a map f : V → W, Pf (f) : Pf (V ) → Pf (W), Pf (f)(S) = f[S] = {f(x) | x ∈ S}. Then the r-fold iterate Fr := P r f : Set → Set is… view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p132_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p135_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p137_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p139_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p141_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p143_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p145_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p165_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p167_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p170_5.png] view at source ↗
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Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p178_6.png] view at source ↗
read the original abstract

Many real-world phenomena are naturally modeled by graphs and networks. However, classical graph models are often limited to pairwise interactions and may not adequately capture the richer structures that arise in practice. Higher-order graph formalisms extend this framework by incorporating multiway, hierarchical, temporal, multilayer, recursive, and tensor-based interactions, thereby providing more expressive representations of complex systems. This book presents a comprehensive overview of mathematical notions that can be used to model higher-order networks. It surveys foundational concepts, extensional frameworks, and newly introduced formalisms, with an emphasis on their structural principles, relationships, and modeling roles. The aim is to provide a unified perspective that helps readers compare diverse higher-order network models and identify appropriate tools for theoretical study and practical applications. This book is Edition 2.0. It mainly includes the addition of several concepts, as well as corrections and improvements of typographical errors and explanations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript surveys mathematical notions for modeling higher-order networks beyond classical pairwise graphs. It covers foundational concepts, extensional frameworks, and newly introduced formalisms, organized by structural principles and relationships, with the goal of providing a unified perspective to compare models and select appropriate tools for theoretical study and applications. This is Edition 2.0, incorporating additions plus corrections to prior typographical errors and explanations.

Significance. If the coverage is accurate and the relationships among formalisms are clearly articulated, the survey would offer a useful reference in network science and complex systems research. Explicit credit is due for the edition-2.0 updates that expand the collection and for the focus on structural relationships rather than isolated descriptions, which supports the aim of model comparison across domains.

major comments (1)
  1. [Introduction / Scope] The central claim that the surveyed and newly introduced formalisms form a sufficiently comprehensive and non-redundant set for model comparison and selection (abstract and reader's strongest claim) is load-bearing yet unsupported by any explicit inclusion/exclusion protocol, exhaustive taxonomy of interaction types (pairwise vs. multiway, static vs. temporal, etc.), or gap analysis against the broader literature. Without these, readers cannot verify representativeness versus curation, directly affecting utility as a reference.
minor comments (1)
  1. [Throughout] Notation for newly introduced formalisms could be cross-referenced more explicitly to prior sections to aid comparison; a short table summarizing key structural distinctions would improve clarity without altering content.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on the manuscript's scope and central claims. We address the major comment below and will revise the introduction to strengthen the justification for our selection of formalisms.

read point-by-point responses

  1. Referee: [Introduction / Scope] The central claim that the surveyed and newly introduced formalisms form a sufficiently comprehensive and non-redundant set for model comparison and selection (abstract and reader's strongest claim) is load-bearing yet unsupported by any explicit inclusion/exclusion protocol, exhaustive taxonomy of interaction types (pairwise vs. multiway, static vs. temporal, etc.), or gap analysis against the broader literature. Without these, readers cannot verify representativeness versus curation, directly affecting utility as a reference.

    Authors: We acknowledge the value of making the selection process more transparent. The survey is organized around structural principles to emphasize relationships and minimize redundancy rather than attempting an exhaustive catalog, which is infeasible given the rapidly expanding literature. In the revised manuscript we will add a dedicated subsection to the introduction that explicitly states our inclusion criteria (focusing on models that meaningfully extend pairwise graphs via multiway interactions, temporality, hierarchy, or tensor representations) and exclusion criteria (omitting near-duplicates or purely application-specific variants without new structural insight). We will also include a concise gap analysis noting areas such as certain dynamic hypergraph extensions and quantum-inspired formalisms that fall outside the current scope. These additions will help readers assess representativeness without altering the core comparative framework. revision: yes

Circularity Check

0 steps flagged

No circularity detected in survey of higher-order network models

full rationale

The manuscript is a survey that collects, organizes, and introduces mathematical notions for higher-order networks without presenting any derivations, predictions, or first-principles results. No equations or formal claims reduce to self-definitions, fitted inputs, or self-citation chains by construction. The central aim of providing a unified perspective rests on referential organization of external concepts rather than any internal reduction, making the work self-contained against the literature it surveys.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a survey compiling and extending concepts rather than a derivation; no free parameters, axioms, or invented entities are extractable from the abstract as load-bearing elements for a central mathematical claim.

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Works this paper leans on

299 extracted references · 299 canonical work pages · 3 internal anchors

  1. [1]

    Graph theory

    Reinhard Diestel. Graph theory. Springer (print edition); Reinhard Diestel (eBooks), 2024

    work page 2024

  2. [2]

    Graph theory and its applications

    Jonathan L Gross, Jay Yellen, and Mark Anderson. Graph theory and its applications . Chapman and Hall/CRC, 2018

    work page 2018

  3. [3]

    Hypergraph theory

    Alain Bretto. Hypergraph theory. An introduction. Mathematical Engineering. Cham: Springer , 1, 2013

    work page 2013

  4. [4]

    Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Ex- tension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-) HyperAlgebra

    Florentin Smarandache. Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Ex- tension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-) HyperAlgebra . Infinite Study, 2020

    work page 2020

  5. [5]

    T. Fujita. Metahypergraphs, metasuperhypergdynaraphs, and iterated metagraphs: Modeling graphs of graphs, hypergraphs of hypergraphs, superhypergraphs of superhypergraphs, and beyond. Current Research in Interdisciplinary Studies, 4(5):1–23, 2025

    work page 2025

  6. [6]

    What are higher-order networks? SIAM review, 65(3):686–731, 2023

    Christian Bick, Elizabeth Gross, Heather A Harrington, and Michael T Schaub. What are higher-order networks? SIAM review, 65(3):686–731, 2023

    work page 2023

  7. [7]

    Dynamics on higher-order networks: A review

    Soumen Majhi, Matjaž Perc, and Dibakar Ghosh. Dynamics on higher-order networks: A review. Journal of the Royal Society Interface, 19(188), 2022

    work page 2022

  8. [8]

    Higher-order networks representation and learning: A survey

    Hao Tian and Reza Zafarani. Higher-order networks representation and learning: A survey. ACM SIGKDD Explorations Newsletter, 26(1):1–18, 2024

    work page 2024

  9. [9]

    Triangular alignment (tame): A tensor- based approach for higher-order network alignment

    Shahin Mohammadi, David F Gleich, Tamara G Kolda, and Ananth Grama. Triangular alignment (tame): A tensor- based approach for higher-order network alignment. IEEE/ACM transactions on computational biology and bioinformatics , 14(6):1446–1458, 2016

    work page 2016

  10. [10]

    Weisfeiler and leman go neural: Higher-order graph neural networks

    Christopher Morris, Martin Ritzert, Matthias Fey, William L Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and leman go neural: Higher-order graph neural networks. In Proceedings of the AAAI conference on artificial intelligence, pages 4602–4609, 2019

    work page 2019

  11. [11]

    Efficient higher-order neural networks for classification and function approximation

    Joydeep Ghosh and Yoan Shin. Efficient higher-order neural networks for classification and function approximation. Inter- national Journal of Neural Systems , 3(04):323–350, 1992

    work page 1992

  12. [12]

    Understanding hazardous materials transportation accidents based on higher-order network theory

    Cuiping Ren, Bianbian Chen, Fengjie Xie, Xuan Zhao, Jiaqian Zhang, and Xueyan Zhou. Understanding hazardous materials transportation accidents based on higher-order network theory. International journal of environmental research and public health, 19(20):13337, 2022

    work page 2022

  13. [13]

    Actor hypernetworks and urban road hypernetworks with real-life applications

    Takaaki Fujita and Arif Mehmood. Actor hypernetworks and urban road hypernetworks with real-life applications. Neutro- sophic Computing and Machine Learning , 41:143–171, 2025

    work page 2025

  14. [14]

    River network hypergraphs and transportation network hypergraphs: A graph-theoretic approach for geo- scientific and civil applications

    Takaaki Fujita. River network hypergraphs and transportation network hypergraphs: A graph-theoretic approach for geo- scientific and civil applications. Interdisciplinary Studies on Applied Science , 2(1):19–31, 2025

    work page 2025

  15. [15]

    Research on bus route optimization based on a higher-order network model

    Fengjie Xie, Jiaxin Huang, Cuiping Ren, and Mengchan Wei. Research on bus route optimization based on a higher-order network model. In Fourth International Conference on Smart City Engineering and Public Transportation (SCEPT 2024) , volume 13160, pages 314–322. SPIE, 2024

    work page 2024

  16. [16]

    Revealing higher-order interactions through multimodal irreversibility in flood-affected transportation networks

    Xuhui Lin, Long Chen, Qiuchen Lu, Pengjun Zhao, and Tao Cheng. Revealing higher-order interactions through multimodal irreversibility in flood-affected transportation networks. Reliability Engineering & System Safety , page 111726, 2025

    work page 2025

  17. [17]

    Evolutionary dynamics of higher-order interactions in social networks

    Unai Alvarez-Rodriguez, Federico Battiston, Guilherme Ferraz de Arruda, Yamir Moreno, Matjaž Perc, and Vito Latora. Evolutionary dynamics of higher-order interactions in social networks. Nature Human Behaviour , 5(5):586–595, 2021

    work page 2021

  18. [18]

    Temporal properties of higher-order interactions in social networks

    Giulia Cencetti, Federico Battiston, Bruno Lepri, and Márton Karsai. Temporal properties of higher-order interactions in social networks. Scientific reports, 11(1):7028, 2021

    work page 2021

  19. [19]

    Fujita and F

    T. Fujita and F. Smarandache. Competition super-hypergraphs: Revealing hierarchical competition in real-world networks. Journal of Algebra and Applied Mathematics , 23(2):97–116, 2025

    work page 2025

  20. [20]

    Cognitive hypergraphs and superhypergraphs: A novel framework for complex relational modeling

    Takaaki Fujita. Cognitive hypergraphs and superhypergraphs: A novel framework for complex relational modeling. Neutro- sophic Computing & Machine Learning , 39, 2025

    work page 2025

  21. [21]

    Predicting higher order links in social interaction networks

    Yong-Jian He, Xiao-Ke Xu, and Jing Xiao. Predicting higher order links in social interaction networks. IEEE Transactions on Computational Social Systems , 11(2):2796–2806, 2023

    work page 2023

  22. [22]

    Prediction of social influence in higher-order networks

    Hao Peng, Rui Zhang, Bo Zhang, Cheng Qian, Ming Zhong, Shenghong Li, Jianmin Han, Dandan Zhao, and Wei Wang. Prediction of social influence in higher-order networks. Information Sciences , page 123075, 2026

    work page 2026

  23. [23]

    The emergence of higher-order structure in scientific and technological knowledge networks

    Thomas Gebhart and Russell James Funk. The emergence of higher-order structure in scientific and technological knowledge networks. In Academy of Management Proceedings . Academy of Management Briarcliff Manor, NY 10510, 2023

    work page 2023

  24. [24]

    Learning directed knowledge using higher-ordered neural networks: Building a predictive framework

    Yousra Moh Ousellam, Bikram Pratim Bhuyan, Rachida Fissoune, Galina Ivanova, and Amar Ramdane-Cherif. Learning directed knowledge using higher-ordered neural networks: Building a predictive framework. Applied Sciences, 15(20):11085, 2025

    work page 2025

  25. [25]

    Topology shapes dynamics of higher-order networks

    Ana P Millán, Hanlin Sun, Lorenzo Giambagli, Riccardo Muolo, Timoteo Carletti, Joaquín J Torres, Filippo Radicchi, Jürgen Kurths, and Ginestra Bianconi. Topology shapes dynamics of higher-order networks. Nature Physics , 21(3):353–361, 2025

    work page 2025

  26. [26]

    The physics of higher-order interactions in complex systems

    Federico Battiston, Enrico Amico, Alain Barrat, Ginestra Bianconi, Guilherme Ferraz de Arruda, Benedetta Franceschiello, Iacopo Iacopini, Sonia Kéfi, Vito Latora, Yamir Moreno, et al. The physics of higher-order interactions in complex systems. Nature physics , 17(10):1093–1098, 2021. 231 Bibliography 232

    work page 2021

  27. [27]

    The mass of simple and higher-order networks

    Ginestra Bianconi. The mass of simple and higher-order networks. Journal of Physics A: Mathematical and Theoretical , 57(1):015001, 2024

    work page 2024

  28. [28]

    Higher-order molecular organization as a source of biological function

    Thomas Gaudelet, Noel Malod-Dognin, and Nataša Pržulj. Higher-order molecular organization as a source of biological function. Bioinformatics, 34(17):i944–i953, 2018

    work page 2018

  29. [29]

    Unifying grain boundary networks and crystal graphs: A hypergraph and superhypergraph perspective in material sciences

    Takaaki Fujita. Unifying grain boundary networks and crystal graphs: A hypergraph and superhypergraph perspective in material sciences. Asian Journal of Advanced Research and Reports , 19(5):344–379, 2025

    work page 2025

  30. [30]

    Higher order structures in chemistry: hypergraphs reshape the molecule and the reaction

    Guillermo Restrepo. Higher order structures in chemistry: hypergraphs reshape the molecule and the reaction. Digital Discovery, 2026

    work page 2026

  31. [31]

    Toward a unified theory of brain hypergraphs and symptom hypernetworks in medicine and neuroscience

    Takaaki Fujita and Arkan A Ghaib. Toward a unified theory of brain hypergraphs and symptom hypernetworks in medicine and neuroscience. Advances in Research, 26(3):522–565, 2025

    work page 2025

  32. [32]

    Characterizing Dynamics on and of Networks via Higher-Order Interactions: Applications in Compu- tational Neuroscience

    Bengier Ülgen Kilic. Characterizing Dynamics on and of Networks via Higher-Order Interactions: Applications in Compu- tational Neuroscience. State University of New York at Buffalo, 2023

    work page 2023

  33. [33]

    Generalization of back propagation to recurrent and higher order neural networks

    Fernando Pineda. Generalization of back propagation to recurrent and higher order neural networks. In Neural information processing systems, 1987

    work page 1987

  34. [34]

    Hypergraph neural networks

    Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. Hypergraph neural networks. In Proceedings of the AAAI conference on artificial intelligence , pages 3558–3565, 2019

    work page 2019

  35. [35]

    Higher-order explanations of graph neural networks via relevant walks

    Thomas Schnake, Oliver Eberle, Jonas Lederer, Shinichi Nakajima, Kristof T Schütt, Klaus-Robert Müller, and Grégoire Montavon. Higher-order explanations of graph neural networks via relevant walks. IEEE transactions on pattern analysis and machine intelligence , 44(11):7581–7596, 2021

    work page 2021

  36. [36]

    Learning, invariance, and generalization in high-order neural networks

    C Lee Giles and Tom Maxwell. Learning, invariance, and generalization in high-order neural networks. Applied optics , 26(23):4972–4978, 1987

    work page 1987

  37. [37]

    Multi-order hypergraph convolutional neural network for dynamic social recommendation system

    Yu Wang and Qilong Zhao. Multi-order hypergraph convolutional neural network for dynamic social recommendation system. IEEE Access, 10:87639–87649, 2022

    work page 2022

  38. [38]

    Recommendation model based on higher-order semantics and node attention in heterogeneous graph neural networks

    Siyue Li, Tian Jin, Hao Luo, Erfan Wang, and Ranting Tao. Recommendation model based on higher-order semantics and node attention in heterogeneous graph neural networks. Mathematics, 13(9):1479, 2025

    work page 2025

  39. [39]

    Hypergraph diffusion for high-order recommender systems

    Darnbi Sakong, Thanh Trung Huynh, and Jun Jo. Hypergraph diffusion for high-order recommender systems. arXiv preprint arXiv:2501.16722, 2025

    work page arXiv 2025

  40. [40]

    Resilience inference for supply chains with hypergraph neural network

    Zetian Shen, Hongjun Wang, Jiyuan Chen, and Xuan Song. Resilience inference for supply chains with hypergraph neural network. arXiv preprint arXiv:2511.06208 , 2025

    work page arXiv 2025

  41. [41]

    Telecommunications hypernetwork and telecommunications superhypernetwork

    Takaaki Fujita. Telecommunications hypernetwork and telecommunications superhypernetwork. Intelligence Modeling in Electromechanical Systems, 2(1):16–31, 2025

    work page 2025

  42. [42]

    Higher-order decision theory

    Jules Hedges, Paulo Oliva, Evguenia Shprits, Viktor Winschel, and Philipp Zahn. Higher-order decision theory. In Interna- tional Conference on Algorithmic Decision Theory , pages 241–254. Springer, 2017

    work page 2017

  43. [43]

    Cardiac decision making using higher order spectra

    Roshan Joy Martis, U Rajendra Acharya, KM Mandana, Ajoy Kumar Ray, and Chandan Chakraborty. Cardiac decision making using higher order spectra. Biomedical Signal Processing and Control , 8(2):193–203, 2013

    work page 2013

  44. [44]

    Decision Making Based on Valued Fuzzy Super- hypergraphs

    Mohammad Hamidi, Florentin Smarandache, and Mohadeseh Taghinezhad. Decision Making Based on Valued Fuzzy Super- hypergraphs. Infinite Study, 2023

    work page 2023

  45. [45]

    Technical decision making with higher order structure data: perspectives on higher order structure characterization from the biopharmaceutical industry

    William F Weiss IV, John P Gabrielson, Wasfi Al-Azzam, Guodong Chen, Darryl L Davis, Tapan K Das, David B Hayes, Damian Houde, and Satish K Singh. Technical decision making with higher order structure data: perspectives on higher order structure characterization from the biopharmaceutical industry. Journal of Pharmaceutical Sciences , 105(12):3465– 3470, 2016

    work page 2016

  46. [46]

    Application of Superhypergraphs-Based Domination Number in Real World

    Mohammad Hamidi and Mohadeseh Taghinezhad. Application of Superhypergraphs-Based Domination Number in Real World. Infinite Study, 2023

    work page 2023

  47. [47]

    An application of fuzzy hypergraphs and hypergraphs in granular computing

    Qian Wang and Zengtai Gong. An application of fuzzy hypergraphs and hypergraphs in granular computing. Inf. Sci. , 429:296–314, 2018

    work page 2018

  48. [48]

    Fuzzy hypergraph network for recommending top-k profitable stocks

    Xiang Ma, Tianlong Zhao, Qiang Guo, Xue mei Li, and Caiming Zhang. Fuzzy hypergraph network for recommending top-k profitable stocks. Inf. Sci. , 613:239–255, 2022

    work page 2022

  49. [49]

    Fuzzy graphs and fuzzy hypergraphs , volume 46

    John N Mordeson and Premchand S Nair. Fuzzy graphs and fuzzy hypergraphs , volume 46. Physica, 2012

    work page 2012

  50. [50]

    Complex neutrosophic hypergraphs: New social network models

    Anam Luqman, Muhammad Akram, and Florentin Smarandache. Complex neutrosophic hypergraphs: New social network models. Algorithms, 12:234, 2019

    work page 2019

  51. [51]

    Single-valued neutrosophic hypergraphs

    Muhammad Akram, Sundas Shahzadi, and AB Saeid. Single-valued neutrosophic hypergraphs. TWMS Journal of Applied and Engineering Mathematics , 8(1):122–135, 2018

    work page 2018

  52. [52]

    Intuitionistic single-valued neutrosophic hypergraphs

    Muhammad Akram and Anam Luqman. Intuitionistic single-valued neutrosophic hypergraphs. OPSEARCH, 54:799 – 815, 2017

    work page 2017

  53. [53]

    Concentric plithogenic hypergraph based on plithogenic hypersoft sets - a novel outlook

    Nivetha Martin and Florentin Smarandache. Concentric plithogenic hypergraph based on plithogenic hypersoft sets - a novel outlook. Neutrosophic Sets and Systems , 33:5, 2020

    work page 2020

  54. [54]

    Neutrosophic super-hypergraph fusion for proactive cyber- attack countermeasures: A soft computing framework

    Ehab Roshdy, Marwa Khashaba, and Mariam Emad Ahmed Ali. Neutrosophic super-hypergraph fusion for proactive cyber- attack countermeasures: A soft computing framework. Neutrosophic Sets and Systems , 94:232–252, 2025

    work page 2025

  55. [55]

    Neutrosophic soft n-super-hypergraphs with real-world applications

    Takaaki Fujita and Florentin Smarandache. Neutrosophic soft n-super-hypergraphs with real-world applications . Infinite Study, 2025

    work page 2025

  56. [56]

    Multineutrosophic analysis of the relationship between survival and business growth in the manufacturing sector of azuay province, 2020–2023, using plithogenic n-superhypergraphs

    Eduardo Martín Campoverde Valencia, Jessica Paola Chuisaca Vásquez, and Francisco Ángel Becerra Lois. Multineutrosophic analysis of the relationship between survival and business growth in the manufacturing sector of azuay province, 2020–2023, using plithogenic n-superhypergraphs. Neutrosophic Sets and Systems , 84:341–355, 2025. 233 Bibliography

    work page 2020

  57. [57]

    Integrating smed and industry 4.0 to optimize processes with plithogenic n-superhypergraphs

    Julio Cesar Méndez Bravo, Claudia Jeaneth Bolanos Piedrahita, Manuel Alberto Méndez Bravo, and Luis Manuel Pilacuan- Bonete. Integrating smed and industry 4.0 to optimize processes with plithogenic n-superhypergraphs. Neutrosophic Sets and Systems , 84:328–340, 2025

    work page 2025

  58. [58]

    Using plithogenic n-superhypergraphs to assess the degree of relationship between information skills and digital competencies

    Berrocal Villegas Salomón Marcos, Montalvo Fritas Willner, Berrocal Villegas Carmen Rosa, Flores Fuentes Rivera María Yissel, Espejo Rivera Roberto, Laura Daysi Bautista Puma, and Dante Manuel Macazana Fernández. Using plithogenic n-superhypergraphs to assess the degree of relationship between information skills and digital competencies. Neutrosophic Sets...

    work page 2025

  59. [59]

    A new decision-making method based on bipolar neutrosophic directed hypergraphs

    Muhammad Akram and Anam Luqman. A new decision-making method based on bipolar neutrosophic directed hypergraphs. Journal of Applied Mathematics and Computing , 57:547 – 575, 2017

    work page 2017

  60. [60]

    Finding hypernetworks in directed hypergraphs

    Daniele Pretolani. Finding hypernetworks in directed hypergraphs. European Journal of Operational Research , 230(2):226– 230, 2013

    work page 2013

  61. [61]

    Review of some superhypergraph classes: Directed, bidirected, soft, and rough

    Takaaki Fujita. Review of some superhypergraph classes: Directed, bidirected, soft, and rough. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond (Second Volume), 2024

    work page 2024

  62. [62]

    Directed superhypertree-width, mixed tree-width, and bidirected tree-width

    Takaaki Fujita. Directed superhypertree-width, mixed tree-width, and bidirected tree-width. SuperHyperTopologies and SuperHyper Structures with their Applications , page 114, 2025

    work page 2025

  63. [63]

    Soft directed n-superhypergraphs with some real-world applications

    Takaaki Fujita and Florentin Smarandache. Soft directed n-superhypergraphs with some real-world applications. European Journal of Pure and Applied Mathematics , 18(4):6643–6643, 2025

    work page 2025

  64. [64]

    Oriented hypergraphs: balanceability

    Lucas J Rusnak, Selena Li, Brian Xu, Eric Yan, and Shirley Zhu. Oriented hypergraphs: balanceability. Discrete Mathematics, 345(6):112832, 2022

    work page 2022

  65. [65]

    Spectral Properties of Oriented Hypergraphs

    Nathan Reff. Spectral properties of oriented hypergraphs. arXiv preprint arXiv:1506.05054 , 2015

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  66. [66]

    Lower bounds for the laplacian spectral radius of an oriented hypergraph

    Ouail Kitouni and Nathan Reff. Lower bounds for the laplacian spectral radius of an oriented hypergraph. Australas. J Comb., 74:408–422, 2019

    work page 2019

  67. [67]

    Spectra of cycle and path families of oriented hypergraphs

    Luke Duttweiler and Nathan Reff. Spectra of cycle and path families of oriented hypergraphs. Linear Algebra and its Applications, 578:251–271, 2019

    work page 2019

  68. [68]

    Extensions of multidirected graphs: Fuzzy, neutrosophic, plithogenic, rough, soft, hypergraph, and superhy- pergraph variants

    Takaaki Fujita. Extensions of multidirected graphs: Fuzzy, neutrosophic, plithogenic, rough, soft, hypergraph, and superhy- pergraph variants. International journal of topology , 2(3):11, 2025

    work page 2025

  69. [69]

    HyperGraph and SuperHyperGraph Theory with Applications

    Takaaki Fujita and Florentin Smarandache. HyperGraph and SuperHyperGraph Theory with Applications . Neutrosophic Science International Association (NSIA) Publishing House, 2026

    work page 2026

  70. [70]

    The cardinal of the m-powerset of a set of n elements used in the superhyperstructures and neutro- sophic superhyperstructures

    Florentin Smarandache. The cardinal of the m-powerset of a set of n elements used in the superhyperstructures and neutro- sophic superhyperstructures. Systems Assessment and Engineering Management , 2:19–22, 2024

    work page 2024

  71. [71]

    Hypergraphs: combinatorics of finite sets , volume 45

    Claude Berge. Hypergraphs: combinatorics of finite sets , volume 45. Elsevier, 1984

    work page 1984

  72. [72]

    Introduction to the n-SuperHyperGraph-the most general form of graph today

    Florentin Smarandache. Introduction to the n-SuperHyperGraph-the most general form of graph today . Infinite Study, 2022

    work page 2022

  73. [73]

    A linear mathematical model of the vocational training problem in a company using neutrosophic logic, hyperfunctions, and superhyperfunction

    Maissam Jdid, Florentin Smarandache, and Takaaki Fujita. A linear mathematical model of the vocational training problem in a company using neutrosophic logic, hyperfunctions, and superhyperfunction. Neutrosophic Sets and Systems , 87:1–11, 2025

    work page 2025

  74. [74]

    SuperHyperFunction, SuperHyperStructure, Neutrosophic SuperHyperFunction and Neutrosophic SuperHyperStructure: Current understanding and future directions

    Florentin Smarandache. SuperHyperFunction, SuperHyperStructure, Neutrosophic SuperHyperFunction and Neutrosophic SuperHyperStructure: Current understanding and future directions . Infinite Study, 2023

    work page 2023

  75. [75]

    Subset Vertex Multigraphs and Neutrosophic Multi- graphs for Social Multi Networks

    WB Vasantha Kandasamy, K Ilanthenral, and Florentin Smarandache. Subset Vertex Multigraphs and Neutrosophic Multi- graphs for Social Multi Networks . Infinite Study, 2019

    work page 2019

  76. [76]

    Cycle covers of cubic multigraphs

    Brian Alspach and Cun-Quan Zhang. Cycle covers of cubic multigraphs. Discrete mathematics , 111(1-3):11–17, 1993

    work page 1993

  77. [77]

    Convolutional learning on multigraphs

    Landon Butler, Alejandro Parada-Mayorga, and Alejandro Ribeiro. Convolutional learning on multigraphs. IEEE Transac- tions on Signal Processing , 71:933–946, 2023

    work page 2023

  78. [78]

    Intuitionistic fuzzy shortest path in a multigraph

    Siddhartha Sankar Biswas, Bashir Alam, and MN Doja. Intuitionistic fuzzy shortest path in a multigraph. In International Conference on Recent Developments in Science, Engineering and Technology , pages 533–540. Springer, 2017

    work page 2017

  79. [79]

    Inverse fuzzy multigraphs and planarity with application in decision-making

    Rajab Ali Borzooei and R Almallah. Inverse fuzzy multigraphs and planarity with application in decision-making. Soft Computing, 26(4):1531–1539, 2022

    work page 2022

  80. [80]

    On intuitionistic fuzzy multigraphs and their index matrix interpretations

    Anthony Shannon and Krassimir Atanassov. On intuitionistic fuzzy multigraphs and their index matrix interpretations. In 2004 2nd International IEEE Conference on ’Intelligent Systems’ . Proceedings (IEEE Cat. No. 04EX791) , volume 2, pages 440–443. IEEE, 2004

    work page 2004

Showing first 80 references.