EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6835
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Random n-SuperHyperGraphs: A Probabilistic Model
and Generation Algorithm for Hierarchical Networks

Takaaki Fujita1,∗, Florentin Smarandache2

1 Independent Researcher, Tokyo, Japan
2 University of New Mexico, Gallup Campus, NM 87301, USA

Abstract. Hypergraphs generalize classical graphs by allowing hyperedges to join any nonempty
subset of vertices [1]. Superhypergraphs extend this idea by iterating the powerset operation,
producing nested layers that capture hierarchical and self-referential relationships among vertex
collections [2]. While random graph and hypergraph models form edges independently with prob-
ability p, they do not account for higher-order, multi-scale, or nested dependencies often observed
in real-world networks. We introduce the random superhypergraph SuHG(n)(V0, p), defined on the
n-fold powerset of a base set V0. We give a concise mathematical formulation, derive key properties
(including expectation, variance, concentration, and substructure thresholds), and present efficient
algorithms for generation. This framework provides a unified, probabilistic model for complex
systems with layered, uncertain connectivity.

2020 Mathematics Subject Classifications: 05C65

Key Words and Phrases: Hypergraph, superhypergraph, random graph, random hypergraph,
random superhypergraph

1. Introduction

1.1. Graph, Hypergraph, and Superhypergraph Theory

Graph theory formalizes the study of structures consisting of vertices connected by
edges, serving as a fundamental tool for modeling pairwise relationships or interactions [3–
5]. Its applications span data mining, algorithm design, artificial intelligence, neural net-
works, quantum information, and chemistry (e.g., [6–10]). However, classical graphs often
have limitations in representing hierarchical network concepts and multiway interactions.
To address these challenges, the concepts of Hypergraph and Superhypergraph have been
actively studied in recent years.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6835

Email addresses: Takaaki.fujita060@gmail.com (T. Fujita),
smarand@unm.edu (F. Smarandache)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 2 of 29

A hypergraph extends this framework by allowing edges (hyperedges) to join any
nonempty subset of vertices, thereby capturing higher-order associations [11–15]. Hy-
pergraphs are known as a natural generalization of graphs and, like graphs, have been
studied in a wide range of applications [16–21]. Building on this, a Superhypergraph ap-
plies recursive powerset operations to form nested layers of vertices and edges, enabling
the representation of hierarchical or multi-level connectivity [2, 22–25]. Superhypergraphs
generalize hypergraphs. Such structures have proven valuable for modeling complex, lay-
ered systems in numerous domains [26, 27].

Table 1 highlights the principal differences between graphs, hypergraphs, and superhy-
pergraphs. Here, n is assumed to be a finite positive integer. Moreover, numerous graph
algorithms for Graphs, Hypergraphs, and Superhypergraphs have also been extensively
studied (e.g., [28–31]).

Concept Notation Edge Type Extension Mecha-
nism

Graph G = (V,E) E ⊆ {{u, v} | u, v ∈ V, u ̸= v} Connects exactly two
vertices.

Hypergraph H = (V,E) E ⊆ PS(V ) \ {∅} Joins any nonempty
subset of vertices.

Superhypergraph SuHG(n) =
(V0, V, E)

V ⊆ PSn(V ), E ⊆ PSn(V ) Applies an n-fold
powerset for nested
structure.

Table 1: Comparison of Graph, Hypergraph, and Superhypergraph

( PS(V ) denotes the powerset of V , i.e. the set of all subsets of V . PSn(V ) is the n-fold iterated

powerset, defined by PS1(V ) = PS(V ) and PSk+1(V ) = PS(PSk(V )) for k ≥ 1. )

1.2. Random Graph Theory

Graphs are also extensively used from a probabilistic perspective (cf.[32–34]). A ran-
dom graph is a graph in which each possible edge between n nodes appears independently
with a fixed probability[35–38]. A random hypergraph generalizes this by including each
possible k-element subset of nodes as a hyperedge independently with probability p[39–
43]. Studying random graph theory provides insights into the emergence of global network
properties from local randomness, informs phase transitions, percolation phenomena, and
has applications in network science, epidemiology, and algorithmic analysis, bridging prob-
ability theory and combinatorics to model complex systems(cf.[44]). Algorithmic research
on random graphs has also been conducted[45].

But random graph models assume independent edge formation, ignore higher-order
correlations, lack hierarchical structure, and cannot capture multi-scale interactions or
nested dependencies present in real-world networks.

Motivation and Our Contributions From the above, research on random graphs,
SuperHyperGraphs, and hypergraphs is of great importance. However, research on Super-



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 3 of 29

HyperGraphs is still in its early stages, and investigations into random SuperHyperGraphs
have not yet been advanced.

In this paper, we investigate the notion of a random SuperHyperGraph and provide a
concise mathematical exploration of its structure. Additionally, we present a simple algo-
rithm for generating random SuperHyperGraphs. Table 2 highlights the principal differ-
ences between Random Graph, Random k-Hypergraph, and Random n-Superhypergraph.
Unifying graphs and hypergraphs, Random n-Superhypergraphs enable modeling hierar-
chical, multi-scale structures, capture nested high-order interactions in complex systems,
and support probabilistic analysis of layered networks.

Model Notation Edge Domain Inclusion Rule

Random Graph G(n, p) All 2-element subsets of
V = {1, . . . , n}

Each {u, v} appears
independently with
probability p.

Random k-Hypergraph Hk(n, p) All k-element subsets of
V = {1, . . . , n}

Each e ⊂ V with
|e| = k appears
independently with
probability p.

Random n-SuperHyperGraph SuHG(n)(V0, p) All nonempty subsets of
V (n) = PSn(V0), i.e.
PS(V (n)) \ {∅}

Each potential hy-
peredge e ⊆ V (n),
e ̸= ∅, is included
independently with
probability p.

Table 2: Comparison of Random Graph, Random k-Hypergraph, and Random n-SuperHyperGraph

Structure of This Paper This paper is structured as follows. In Section 2, we intro-
duce the definitions and fundamental concepts of SuperHyperGraphs and random graphs.
Section 3 presents concrete examples and investigates the mathematical properties of ran-
dom SuperHyperGraphs. In Section 4, we examine algorithms for generating and analyzing
random SuperHyperGraphs. Finally, Section 5 concludes the paper and outlines directions
for future research.

2. Preliminaries

This section provides an overview of the fundamental concepts and definitions essential
for the discussions in this paper. Unless otherwise specified, all graphs considered in this
paper are assumed to be finite.

2.1. SuperHyperGraphs

A SuperHyperGraph extends the notion of a hypergraph by applying the powerset
operator repeatedly, thereby modeling hierarchical, nested, and self-referential connections
among vertices and edges [2, 46, 47]. Below we give precise definitions.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 4 of 29

Definition 1 (Base Set). Let S be a nonempty set of elements. We call S the base set.
All subsequent constructions originate from S.

Definition 2 (Powerset). [48, 49] For any set S, its powerset is

PS(S) = {A | A ⊆ S} ,

the collection of all subsets of S, including ∅ and S itself.

The n-th iterated powerset is obtained by applying the powerset operation to a set
repeatedly n times, producing nested subsets[50–52].

Definition 3 (n-th Iterated Powerset). (cf.[53–55]) Let n ∈ N. Define recursively

PS1(S) = PS(S),

PSk+1(S) = PS
(
PSk(S)

)
(k ≥ 1).

Then PSn(S) is the n-th iterated powerset of S. If one wishes to exclude the empty set
at each stage, write PS∗

1(S) = PS(S) \ {∅} and

PS∗
k+1(S) = PS

(
PS∗

k(S)
)
\ {∅}.

Example 1 (Hierarchical Meal Planning). Let

S = {Eggs, Bread, Butter, Jam}

be basic breakfast ingredients. Then:

• PS1(S) lists all single-meal combinations (e.g. {Eggs,Bread}).

• PS2(S) = PS(PS1(S)) collects weekly menus, each a set of daily combinations.

• More generally, PSn(S) for n ≥ 3 represents n-level meal schedules (e.g. monthly
plans).

This illustrates how iterated powersets encode multi-scale planning: single meals → weekly
menus → monthly schedules, and so on.

Definition 4 (Hypergraph). [1, 56] A hypergraph H = (V (H), E(H)) consists of:

• A nonempty set V (H) of vertices.

• A set E(H) of hyperedges, where each hyperedge is a nonempty subset of V (H),
thereby allowing connections among multiple vertices.

Unlike standard graphs, hypergraphs are well-suited to represent higher-order relationships.
In this paper, we restrict ourselves to the case where both V (H) and E(H) are finite.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 5 of 29

Definition 5 (n-SuperHyperGraph). (cf. [2]) Let V0 be a finite nonempty base set. For
k ∈ N0 define the iterated powerset

PS0(V0) := V0, PSk+1(V0) := PS
(
PSk(V0)

)
,

and write PS∗(X) := PS(X) \ {∅} for the collection of all nonempty subsets of X.
Fix n ∈ N. An n-SuperHyperGraph over V0 is a pair

SuHG(n) = (V,E)

such that
V ⊆ PSn(V0) and E ⊆ PS∗(V ).

Elements of V are called n-supervertices; elements e ∈ E (each a nonempty subset of V )
are called n-superedges.

Example 2 (Small company as a 2-SuperHyperGraph). Let the base set of employees be

V0 = {Hiroko,Yutaka,Tae, Shinya}.

Choose two project teams (elements of PS1(V0)):

g1 = {Hiroko,Tae}, g2 = {Yutaka,Shinya}.

Consider the following 2-supervertex set (a subset of PS2(V0) = PS(PS(V0))):

V =
{
{g1}, {g2}, {g1, g2}

}
⊆ PS2(V0).

Define superedges as nonempty subsets of V :

E =
{
{{g1}, {g1, g2}}, {{g2}, {g1, g2}}

}
⊆ PS∗(V ).

Then SuHG(2) = (V,E) is a valid 2-SuperHyperGraph. Informally, {g1} and {g2} are “sin-
gle–team” supervertices, {g1, g2} is a “department–level” supervertex, and each superedge
joins a team to the department–level node.

Example 3 (Social network as a 3-SuperHyperGraph). Let the base set of users be

V0 = {Hiroko,Yutaka,Emiko,Kenji,Takashi}.

Pick some friend groups fi ∈ PS1(V0), e.g.

f1 = {Hiroko,Yutaka}, f2 = {Yutaka,Emiko,Takashi}, f3 = {Emiko,Kenji}, f4 = {Hiroko,Takashi}.

Form communities Cj ∈ PS2(V0) = PS(PS(V0)), for instance

C1 = {f1, f2}, C2 = {f2, f3, f4}, C3 = {f1, f4}.

Select 3-supervertices in PS3(V0) = PS(PS2(V0)):

V =
{
{C1}, {C2}, {C1, C2}, {C2, C3}

}
⊆ PS3(V0).

Define superedges as nonempty subsets of V ; for example,

E =
{
{{C1, C2}, {C2, C3}}

}
⊆ PS∗(V ).

Then SuHG(3) = (V,E) is a valid 3-SuperHyperGraph: the chosen superedge links two
meta–community supervertices.



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2.2. Random graph and Random hypergraph

Random graphs and hypergraphs provide probabilistic frameworks for modeling net-
works and higher-order relationships. In the classical Erdős–Rényi model, each potential
edge between n vertices is included independently with probability p [35, 38, 57, 58]. The
hypergraph analogue extends this by treating every k-element subset of the vertex set as
a hyperedge, each included independently with probability p [39–41, 43]. The relevant
definitions are provided as follows.

Definition 6 (Erdős–Rényi Random Graph G(n, p)). (cf.[35, 38]) Let n ∈ N and p ∈
[0, 1]. The Erdős–Rényi random graph G(n, p) is the probability space whose sample space
consists of all simple undirected graphs on the vertex set V = {1, 2, . . . , n}. Each of the(
n
2

)
possible edges is included independently with probability p. Equivalently, for any fixed

graph G = (V,E),

Pr
(
G(n, p) = G

)
= p|E| (1− p)(

n
2)−|E|.

Example 4 (Erdős–Rényi Random Graph in a Social Network). Suppose we have n users
on a social platform, labeled V = {1, 2, . . . , n}. We model friendships by the random graph
G(n, p) where each of the

(
n
2

)
possible pairs {i, j} becomes a friendship edge independently

with probability p. Concretely, if

E =
{
{i, j} : Xij = 1

}
, Xij ∼ Bernoulli(p),

then
Pr

(
G(n, p) = (V,E)

)
= p|E| (1− p)(

n
2)−|E|.

Here E[|E|] = p
(
n
2

)
gives the expected number of friendships.

Definition 7 (Uniform Random Graph G(n,m)). (cf.[35, 38]) Let n,m ∈ N with 0 ≤
m ≤

(
n
2

)
. The uniform random graph G(n,m) is the probability space of all simple graphs

on V = {1, 2, . . . , n} having exactly m edges, each graph being equally likely. Thus

Pr
(
G(n,m) = G

)
=


1((n2)
m

) , |E(G)| = m,

0, otherwise.

Definition 8 (k-Uniform Random Hypergraph Hk(n, p)). Let n, k ∈ N with 1 ≤ k ≤ n
and p ∈ [0, 1]. The k-uniform random hypergraph Hk(n, p) is the probability space whose
sample space consists of all k-uniform hypergraphs on V = {1, 2, . . . , n}, where each of the(
n
k

)
possible k-element subsets is included independently with probability p. Equivalently,

for any fixed hypergraph H = (V,E) with |e| = k for all e ∈ E,

Pr
(
Hk(n, p) = H

)
= p|E| (1− p)(

n
k)−|E|.

Example 5 (k-Uniform Random Hypergraph for Research Teams). Let n researchers
form collaborative teams of size k. Label them V = {1, 2, . . . , n}. In the k-uniform random



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 7 of 29

hypergraph Hk(n, p), each possible team e ∈
(
V
k

)
is included as a hyperedge independently

with probability p. Thus for any fixed hypergraph (V,E) with |E| = m,

Pr
(
Hk(n, p) = (V,E)

)
= pm (1− p)(

n
k)−m.

This models the random formation of m research groups, each comprising exactly k col-
laborators.

Example 6 (k-Uniform Random Hypergraph in Academic CoAuthorship). Consider a
set of n researchers in a department, labeled

V = {1, 2, . . . , n}.

We wish to model the random formation of collaborative research groups of size k. In
this scenario, each possible subset {i1, i2, . . . , ik} ⊆ V of size k represents a potential co-
authorship team. In the k-uniform random hypergraph model Hk(n, p), each of the

(
n
k

)
possible k-member subsets is included independently as a hyperedge with probability p.

Concretely, let
E =

{
e ⊆ V : |e| = k and Xe = 1

}
,

Xe ∼ Bernoulli(p),

where Xe = 1 indicates that the researchers in e form a coauthored paper. Then for any
fixed hypergraph (V,E) with |e| = k for all e ∈ E,

Pr
(
Hk(n, p) = (V,E)

)
=

p |E| (1− p
)(nk)−|E|

.

Thus, the random hypergraph Hk(n, p) captures the probability that exactly |E| distinct
groups of k researchers collaborate on papers, where each group forms independently with
probability p.

3. Main Results: Random n-SuperHyperGraph

In this section, we discuss, as one of the results of this paper, the definition and
application examples of Random n-SuperHyperGraph.

3.1. Definition of Random n-SuperHyperGraph

A Random n-SuperHyperGraph is a probabilistic model selecting each element of n-th
iterated powerset as hyperedges independently with probability p. The definition of the
Random n-SuperHyperGraph is given as follows.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 8 of 29

Definition 9 (Random n-SuperHyperGraph SuHG(n)(V0, p)). Let V0 be a finite base set
with |V0| = N , let n ∈ N, and fix p ∈ [0, 1]. Define

V (n) = PSn(V0), Mn =
∣∣V (n)

∣∣,
Qn = 2Mn − 1.

We view SuHG(n) = (V (n), E) as a hypergraph whose vertex set is V (n) and whose edge
set E is a collection of nonempty subsets of V (n):

E ⊆ PS
(
V (n)

)
\ {∅}.

In the random model SuHG(n)(V0, p), each of the Qn possible nonempty subsets e ⊆ V (n)

is included in E independently with probability p. Thus for any fixed E ⊆ PS(V (n)) \ {∅},

Pr
(
SuHG(n)(V0, p) = (V (n), E)

)
= p|E| (1− p)Qn−|E| .

3.2. Real-World Examples of Random SuperHyperGraphs

We begin by exploring real-world scenarios where Random SuperHyperGraphs are
applicable. Several concrete examples are presented below.

Example 7 (Random 2-SuperHyperGraph: Company Task-Force Formation). Let the
base set of employees be

V0 = {Hiroko, Yutaka, Tae, Shinya}.

Then
PS1(V0) = {{Hiroko,Tae}, {Yutaka, Shinya}, . . . }

lists all two-person teams. Denote

TeamXXX = {Hiroko,Tae},

TeamYYY = {Yutaka, Shinya}.

Next,
PS2(V0)

= PS
(
PS1(V0)

)
= {{TeamXXX}, {TeamYYY},

{TeamXXX,TeamYYY}}.

We take this as the supervertex set:

V (2) = PS2(V0).



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 9 of 29

In our corrected random model, each potential hyperedge is a nonempty subset of V (2). Let

E = PS
(
V (2)

)
\ {∅}, Q2 = 2|V

(2)| − 1 = 7.

In SuHG(2)(V0, p), each e ∈ E is included independently with probability p. Thus for any
fixed E ⊆ E,

Pr
(
SuHG(2)(V0, p) = (V (2), E)

)
= p|E| (1− p)Q2−|E|.

For example, the edge {{TeamXXX}, {TeamXXX,TeamYYY}} connects the team TeamXXX
with the department-level supervertex {TeamXXX,TeamYYY}.

Example 8 (Random 2-SuperHyperGraph: Social Media Community Formation). On a
platform with users

V0 = {Hiroko, Yutaka, Tae, Shinya, Eve},

the level-1 powerset PS1(V0) lists all friend groups (e.g. g1 = {Hiroko,Yutaka}, g2 =
{Yutaka,Tae,Eve}, . . . ). The level-2 powerset PS2(V0) collects all candidate communities:

V (2) = PS2(V0)

= { {gi} : gi ∈ PS1(V0)} ∪

{{gi, gj} : gi, gj ∈ PS1(V0), i ̸= j} ∪ . . . .

Define the set of all nonempty hyperedges E = PS(V (2)) \ {∅}. In the random model
SuHG(2)(V0, p), each e ∈ E appears independently with probability p. Concretely, for any
chosen E ⊆ E,

Pr
(
SuHG(2)(V0, p) = (V (2), E)

)
= p|E| (1− p) |E|−|E|.

An example hyperedge { {g1}, {g1, g2}} connects a single friend group g1 with a larger
community {g1, g2}.

Example 9 (Random 3-SuperHyperGraph: Manufacturing Product Lines). Let basic
components be {Bolt,Nut,Plate,Gear}. Then PS1 lists all modules (e.g. {Bolt,Nut}),
PS2 lists all subassemblies (e.g. {Module1,Module2}), and PS3 lists all products (e.g.
{SubAsm}). Setting

V (3) = PS3(V0), E = PS(V (3)) \ {∅},

the random model SuHG(3)(V0, p) includes each e ∈ E independently with probability p:

Pr
(
SuHG(3)(V0, p) = (V (3), E)

)
= p|E| (1− p) |E|−|E|.

Here a hyperedge such as {{SubAsm}, {SubAsm,OtherAsm}} links different product vari-
ants.



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Example 10 (Social Networks: Overlapping Community Formation). Let

V0 = {Alice, Bob, Carol, Dave, Eve}.

PS1(V0) lists all friend circles (e.g. g1 = {Alice,Bob}, g2 = {Bob,Carol,Eve}, . . . ). Then

PS2(V0) = PS
(
PS1(V0)

)
collects all candidate communities, for instance

Comm1 = {g1, g2}

,
Comm2 = {g2, g3}

, etc. Set
V (2) = PS2(V0),

E = PS(V (2)) \ {∅}, Q2 = 2|V
(2)| − 1.

In the random model SuHG(2)(V0, p), each e ∈ E is included independently with probability
p. Thus, for any fixed hyperedge set E ⊆ E,

Pr
(
SuHG(2)(V0, p) = (V (2), E)

)
= p|E|(1− p)Q2−|E|.

Here each selected e represents an active overlapping community of friend circles at a given
time.

Example 11 (Biology: Protein Complexes, Pathways, and Systems). Let

V0 = {A, B, C, D}

be proteins. Then
PS1(V0) = {{A,B}, {B,C}, {C,D}, . . . }

lists all protein complexes, e.g. Cpx1 = {A,B}, Cpx2 = {B,C}. Next,

PS2(V0) = PS
(
PS1(V0)

)
lists all pathways (collections of complexes), e.g. Path1 = {Cpx1,Cpx2}, Path2 = {Cpx2,Cpx3}.
Finally,

PS3(V0) = PS
(
PS2(V0)

)
lists all systems (collections of pathways), for example Sysα = {Path1,Path2}.

Set
V (3) = PS3(V0), E = PS(V (3)) \ {∅}, Q3 = 2|V

(3)| − 1.

In SuHG(3)(V0, p), each potential system e ∈ E is active independently with probability p.
Hence for any chosen edge-set E ⊆ E,

Pr
(
SuHG(3)(V0, p) = (V (3), E)

)
= p|E|(1− p)Q3−|E|.

Each selected e models a functional biological system formed by randomly activated path-
ways of protein complexes.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 11 of 29

3.3. Properties and Theorems of Random SuperHyperGraphs

This subsection presents the fundamental properties and theoretical results related to
random SuperHyperGraphs.

Theorem 1 (Generalization of G(N, p) and Hk(N, p)). The random superhypergraph
SuHG(n)(V0, p) simultaneously extends:

(i) the Erdős–Rényi graph G(N, p), and

(ii) the uniform k-hypergraph Hk(N, p).

Proof. First note that when n = 1,

V (1) = PS1(V0) = PS(V0), Q1 = 2N − 1.

Thus SuHG(1)(V0, p) is a random hypergraph on the N -element vertex set V0 whose po-
tential edges are all nonempty subsets of V0.

(1) Recovery of G(N, p). If we restrict attention to those random edges of size exactly
2, i.e.

E2 = { e ∈ E : |e| = 2},

then
(
V0, E2

)
is exactly an instance of the Erdős–Rényi random graph G(N, p), since each

2-element subset of V0 appears independently with probability p.

(2) Recovery of Hk(N, p). More generally, restricting to edges of size k,

Ek = { e ∈ E : |e| = k},

yields the k-uniform random hypergraph Hk(N, p), because each k-subset of V0 is included
independently with probability p.

In both cases, the marginal distributions and independence assumptions coincide with
the classical models, so SuHG(n)(V0, p) indeed generalizes G(N, p) and Hk(N, p).

Theorem 2 (Expectation of Number of Superedges). Let V0 be a finite set, n ∈ N, and
write

Mn =
∣∣PSn(V0)

∣∣,
E = PS

(
PSn(V0)

)
\ {∅},

Qn = |E| = 2Mn − 1.

In the random model SuHG(n)(V0, p), the expected number of superedges is

E
[
|E|

]
= pQn.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 12 of 29

Proof. For each potential hyperedge e ∈ E , let

Xe = 1{e∈E}

be its indicator. Since each e is included independently with probability p, we have
E[Xe] = p. Hence

|E| =
∑
e∈E

Xe, E
[
|E|

]
=

∑
e∈E

E[Xe] = Qn p.

Theorem 3 (Variance of Number of Superedges). Under the same notation,

Var
[
|E|

]
= p(1− p)Qn.

Proof. Since the Xe are independent Bernoulli(p) variables,

Var
[
|E|

]
=

∑
e∈E

Var(Xe)

=
∑
e∈E

p(1− p) = Qn p(1− p).

Theorem 4 (Chernoff Concentration). For any ε ∈ (0, 1),

Pr
(∣∣|E| − pQn

∣∣ > εpQn

)
< 2 exp

(
−ε2 pQn

3

)
.

Proof. This follows from the multiplicative Chernoff bound applied to the sum
∑

e∈E Xe

of independent indicators, noting that E[|E|] = pQn.

Theorem 5 (Substructure Appearance Threshold). Let H = (VH , EH) be a fixed n-
superhypergraph with |VH | = s and |EH | = t. Write

Mn =
∣∣PSn(V0)

∣∣,
(Mn)s = Mn (Mn − 1) · · · (Mn − s + 1).

In SuHG(n)(V0, p), let XH be the number of labeled embeddings of H. Then

E[XH ] = (Mn)s p
t.

In particular, if p ≪ M
− s/t
n then E[XH ] → 0 and hence with high probability no copy of

H appears; while if p ≫ M
− s/t
n then E[XH ] → ∞ and with high probability at least one

copy appears.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 13 of 29

Proof. Each labeled embedding of H into PSn(V0) is a choice of an ordered s-tuple of
distinct supervertices, of which there are (Mn)s possibilities, together with the requirement
that each of the t specified hyperedges of H appears. Since each potential hyperedge is
included independently with probability p, the probability that a given labeled embedding
is present is pt. Linearity of expectation then yields E[XH ] = (Mn)s p

t. The “0–infinity”
dichotomy follows by comparing pt to (Mn)−1

s .

Theorem 6 (Generalization of G(N, p) and Hk(N, p)). The random superhypergraph
SuHG(n)(V0, p) recovers:

• The Erdős–Rényi graph G(N, p) by restricting to hyperedges of size 2 when n = 1.

• The k-uniform hypergraph Hk(N, p) by restricting to hyperedges of size k when n = 1.

Proof. When n = 1, we have V (1) = PS(V0) and each nonempty subset of V0 is a
potential hyperedge. Restricting to those of size r yields exactly the model in which each
r-subset of V0 appears independently with probability p, i.e. G(N, p) if r = 2 and Hk(N, p)
if r = k.

Theorem 7 (Central Limit Theorem for Number of Hyperedges). Let

Qn = 2Mn − 1, |E| ∼ Binomial(Qn, p).

As Qn →∞,
|E| − pQn√
p(1− p)Qn

d−→ N (0, 1).

Proof. Since |E| is the sum of Qn independent Bernoulli(p) indicators, the classical
central limit theorem applies directly.

Theorem 8 (Strong Law of Large Numbers). In the same notation,

|E|
Qn

a.s.−−→ p (Qn →∞).

Proof. By Kolmogorov’s strong law for independent, identically distributed Bernoulli
trials, the sample proportion converges almost surely to the common success probability
p.

Theorem 9 (Distribution of Fixed-Size Hyperedges). Let V (n) be the set of n-supervertices
with |V (n)| = Mn, and let

E = PS
(
V (n)

)
\ {∅}

be the set of all potential hyperedges. For any integer r with 1 ≤ r ≤Mn, set

Mn,r =

(
Mn

r

)
, Yr =

∣∣{ e ∈ E : |e| = r}
∣∣.

Then
Yr ∼ Binomial

(
Mn,r, p

)
,

with E[Yr] = pMn,r and Var(Yr) = p(1− p)Mn,r.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 14 of 29

Proof. There are
(
Mn

r

)
= Mn,r possible hyperedges of size r, each included indepen-

dently with probability p. Hence the count Yr follows a Binomial(Mn,r, p) distribution,
from which the stated expectation and variance follow by standard formulas.

Theorem 10 (Monotonicity in p). For any 0 ≤ p1 < p2 ≤ 1, one can couple SuHG(n)(V0, p1)
and SuHG(n)(V0, p2) so that, almost surely,

E(p1) ⊆ E(p2),

where E(pi) is the edge-set under parameter pi.

Proof. Assign each potential hyperedge e ∈ E an independent Ue ∼ Uniform(0, 1).
Then define

E(pi) = { e ∈ E | Ue ≤ pi}, i = 1, 2.

Since p1 < p2 implies {Ue ≤ p1} ⊆ {Ue ≤ p2} for each e, we have E(p1) ⊆ E(p2) almost
surely. Marginally, each E(pi) has the law of SuHG(n)(V0, pi).

Theorem 11 (Vertex Degree Distribution). Let SuHG(n)(V0, p) = (V (n), E) and write
Mn = |V (n)|. For each v ∈ V (n), define its degree

deg(v) =
∣∣{ e ∈ E : v ∈ e}

∣∣.
Then

deg(v) ∼ Binomial
(
2Mn−1, p

)
,

so
E[deg(v)] = p 2Mn−1,

Var[deg(v)] = p(1− p) 2Mn−1,

and, as Mn → ∞, the normalized degree
(
deg(v) − p2Mn−1

)
/
√

p(1− p) 2Mn−1 converges
in distribution to N (0, 1).

Proof. Each potential hyperedge e ⊆ V (n) includes v with probability 1/2 indepen-
dently of other vertices, so there are 2Mn−1 such hyperedges. Since each is present in
E with probability p, deg(v) is Binomial(2Mn−1, p). The stated moments and CLT then
follow by standard properties.

Definition 10 (2-Section Graph). Given SuHG(n)(V0, p) = (V (n), E), its 2-section G2 is
the simple graph on V (n) where {u, v} is an edge of G2 if and only if there exists e ∈ E
with {u, v} ⊆ e.

Example 12 (2-Section Graph of a 2-SuperHyperGraph). Recall the 2-SuperHyperGraph
from the company task-force example:

V (2) = { v1, v2, v3},



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 15 of 29

where
v1 = {TeamXXX}, v2 = {TeamYYY},

v3 = {TeamXXX,TeamYYY},
and the hyperedge set is

E =
{
{v1, v3}, {v2, v3}

}
.

Its 2-section graph G2 = (V (2), E2) has the same vertex set V (2) and an edge between any
two vertices that appear together in some hyperedge. Hence

E2 =
{
{v1, v3}, {v2, v3}

}
.

In other words, G2 is the simple path

v1 − v3 − v2,

with v1 and v2 both adjacent to v3 but not to each other.

Theorem 12 (Giant-Component Phase Transition). Let G2 be the 2-section of SuHG(n)(V0, p).
Define

p2 = 1−
(
1− p

)2Mn−2

≈ p 2Mn−2.

• If p2 = 1−ε
Mn

for some fixed ε > 0, then with high probability G2 has all connected
components of size O(logMn).

• If p2 = 1+ε
Mn

, then with high probability G2 contains a unique “giant” component of
size Θ(Mn), while all other components are O(logMn).

Proof. For each pair {u, v} ⊂ V (n), the probability that no hyperedge contains both is

(1− p)2
Mn−2

, so in G2 we have Pr({u, v} ∈ E(G2)) = p2. These events are asymptotically
independent. Thus G2 behaves like an Erdős–Rényi random graph G(Mn, p2), which
exhibits the stated phase transition at p2 = 1/Mn by classical results.

Theorem 13 (Connectivity Threshold). With notation as above, if

p2 =
logMn + c

Mn

for some constant c, then

Pr
(
G2 is connected

)
−→ e−e−c

(Mn →∞).

Equivalently, in the original model this occurs when

p =
logMn + c

Mn 2Mn−2
.

Proof. Again G2 ≈ G(Mn, p2). In the classical Erdős–Rényi model, connectivity
emerges at p2 = (logMn + c)/Mn, with limiting probability exp(−e−c). Translating p2
back to p gives the claimed threshold.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 16 of 29

Algorithm 1 Generate a Random n-SuperHyperGraph

Require: A finite base set V0, integer n ≥ 1, probability p ∈ [0, 1]
Ensure: A random n-SuperHyperGraph SuHG(n) = (V (n), E)
1: PS0(V0)← V0

2: for k = 1 to n do
3: PSk(V0)← PS

(
PSk−1(V0)

)
4: end for
5: V (n) ← PSn(V0)
6: E ← PS(V (n)) \ {∅}
7: E ← ∅
8: for each potential hyperedge e ∈ E do
9: draw u ∼ Uniform(0, 1)

10: if u ≤ p then E ← E ∪ {e}
11: end if
12: end for
13: return SuHG(n) = (V (n), E)

4. Result: Algorithm for Generating Random SuperHyperGraphs

In this section, we give an algorithm to generate a random n-SuperHyperGraph. See
Algorithm 1 for a summary.

Remark 1. (i) Compute powersets. Starting from V0, iteratively form PSk(V0) up
to k = n.

(ii) Define vertices. Set V (n) = PSn(V0) as the supervertex set.

(iii) Define potential edges. Let E = PS(V (n)) \ {∅}.

(iv) Random selection. Include each e ∈ E in E independently with probability p.

(v) Output. Return the hypergraph (V (n), E).

Example 13 (University Curriculum as a 3-SuperHyperGraph). Let

V0 = {Algebra, Calculus,

Physics, Chemistry}.

Then
PS1(V0)

= {{Algebra,Calculus},

{Physics,Chemistry}, . . . }

represents all courses. Next,

PS2(V0) = PS
(
PS1(V0)

)



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 17 of 29

lists all majors (each a set of courses), for example MajorA = {Course1,Course2}. Finally,

PS3(V0) = PS
(
PS2(V0)

)
lists all degree programs (each a collection of majors), e.g. ProgramX = {MajorA,MajorB}.

In the random model SuHG(3)(V0, p), set

V (3) = PS3(V0),

E = PS(V (3)) \ {∅}.

Each candidate program P ∈ E is included independently with probability p. Thus Al-
gorithm 1 produces a random collection of active degree programs each academic year,
capturing the hierarchy: topics → courses → majors → programs.

Theorem 14 (Correctness of GenerateRandomNSuperHyperGraph). Let V0 be a
finite set, n ∈ N, and p ∈ [0, 1]. Write

V (n) = PSn(V0),

Mn =
∣∣V (n)

∣∣.
Then Algorithm 1 returns each pair (V (n), E) with probability

Pr
(
E is chosen

)
= p|E| (1− p)Mn−|E|,

and the events {x ∈ E} for x ∈ V (n) are independent Bernoulli(p). Hence its output
distribution coincides exactly with that of the random model SuHG(n)(V0, p).

Proof. After computing V (n) = PSn(V0), the algorithm processes each x ∈ V (n) once,
drawing ux ∼ Uniform(0, 1) and including x in E precisely if ux ≤ p. Therefore

Pr(x ∈ E) = p, Pr(x /∈ E) = 1− p,

and these events are independent over x. For any fixed E ⊆ V (n) with |E| = m, exactly
the m elements of E must satisfy ux ≤ p and the other Mn − m must satisfy ux > p,
yielding

Pr(algorithm returns E) = pm (1− p)Mn−m.

This matches the defining law of SuHG(n)(V0, p).

Theorem 15 (Time Complexity). Define Mk = |PSk(V0)| for 0 ≤ k ≤ n, so M0 = |V0|
and Mk = 2Mk−1. Then Algorithm 1 runs in

T (n) = Θ
(
Mn logMn

)
.

Proof. Powerset computation: At each 1 ≤ k ≤ n, listing all Mk = 2Mk−1 subsets of
PSk−1(V0) (each of size up to Mk−1 = log2Mk) costs Θ(Mk Mk−1) = Θ(Mk logMk). Sum-
ming over k is dominated by k = n, giving Θ(Mn logMn). Random selection: Testing each
of the Mn candidates costs Θ(Mn), which is lower order. Hence T (n) = Θ(Mn logMn).



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 18 of 29

Theorem 16 (Space Complexity). With Mn = |PSn(V0)|, the algorithm uses

S(n) = Θ
(
Mn logMn

)
space.

Proof. At stage k, storing PSk(V0) of size Mk requires Θ(Mk Mk−1) bits (each subset
encoded in Θ(Mk−1) bits). The maximum occurs at k = n, giving Θ(Mn logMn). The
additional Θ(Mn) space for the edge set E is asymptotically smaller.

Next, as an example of a graph algorithm that mitigates the aforementioned complex-
ity, we present a size-sampling–based approach. The algorithm is detailed in Algorithm 2.

Algorithm 2 Generate a Random n-SuperHyperGraph by Size-Sampling

Require: A finite base set V0, integer n ≥ 1, probability p ∈ [0, 1]
Ensure: A random n-SuperHyperGraph SuHG(n) = (V (n), E)
1: Compute V (n) ← PSn(V0), let M = |V (n)|
2: Initialize E ← ∅
3: for s = 1 to M do
4: Ms ←

(
M
s

)
5: Draw Ks ∼ Binomial(Ms, p)
6: for i = 1 to Ks do
7: Sample uniformly an s-subset e ⊆ V (n) without replacement
8: if e /∈ E then E ← E ∪ {e}
9: end if

10: end for
11: end for
12: return SuHG(n) = (V (n), E)

Remark 2. This size-sampling approach avoids iterating over all 2M − 1 possible hyper-
edges. Instead, for each edge-size s, it first draws the number Ks of s-hyperedges to include
(a Binomial(

(
M
s

)
, p) variate), then samples exactly Ks distinct s-subsets uniformly at ran-

dom. When p is small or one restricts to moderate s, the running time and memory scale
with

∑
sKs · s rather than

∑
s

(
M
s

)
.

Example 14 (Size-Sampling in Action). Let the base set be

V0 = {Hiroko, Yutaka}.

Then the level-1 supervertex set is

V (1) = PS1(V0)

= {∅, {Hiroko}, {Yutaka}, {Hiroko,Yutaka}},

so M = |V (1)| = 4. We choose p = 0.3. Algorithm 2 proceeds as follows:



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 19 of 29

• s = 1:

M1 =

(
4

1

)
= 4,

K1 ∼ Binomial(4, 0.3), K1 = 1.

Sample one 1-subset from V (1), e.g.

e1 = {{Yutaka}}.

• s = 2:

M2 =

(
4

2

)
= 6,

K2 ∼ Binomial(6, 0.3), K2 = 2.

Sample two 2-subsets, for instance

e2 = {{Hiroko}, {Hiroko,Yutaka}},

e3 = {{Yutaka}, {Hiroko,Yutaka}}.

• s = 3:

M3 =

(
4

3

)
= 4,

K3 ∼ Binomial(4, 0.3), K3 = 1.

Sample one 3-subset, e.g.

e4 = {{Hiroko}, {Yutaka}, {Hiroko,Yutaka}}.

• s = 4:

M4 =

(
4

4

)
= 1,

K4 ∼ Binomial(1, 0.3), K4 = 0.

Hence the sampled hyperedge set is

E = { e1, e2, e3, e4},

and we obtain
SuHG(1) =

(
V (1), E

)
.

This concrete run demonstrates how size-sampling generates a random superhypergraph by
drawing only 1+2+1+0 = 4 hyperedges, rather than enumerating all 24−1 = 15 possible
subsets of V (1).

Theorem 17 (Correctness of Size-Sampling Algorithm). Algorithm 2 produces a random
n-SuperHyperGraph with the same law as SuHG(n)(V0, p). In particular, each potential
s-hyperedge e ⊆ V (n), |e| = s, is included independently with probability p.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 20 of 29

Proof. Fix s ∈ {1, . . . ,M} and write Ms =
(
M
s

)
. Let Es be the collection of all s-subsets

of V (n). The algorithm first draws

Ks ∼ Binomial(Ms, p),

so

Pr(Ks = k) =

(
Ms

k

)
pk(1− p)Ms−k.

Conditioned on Ks = k, it chooses uniformly k distinct subsets from Es. Therefore for any
fixed e ∈ Es,

Pr
(
e ∈ E | Ks = k

)
=

(
Ms−1
k−1

)(
Ms

k

) =
k

Ms
.

Hence marginally

Pr(e ∈ E) =

Ms∑
k=1

Pr(Ks = k)
k

Ms

=
1

Ms

Ms∑
k=1

k

(
Ms

k

)
pk(1− p)Ms−k

=
Ms p

Ms
= p.

Moreover, for any set S ⊆ Es of size m,

Pr
(
S ⊆ E, Es \ S ∩ E = ∅

)
=

(
Ms

m

)
pm(1− p)Ms−m 1(

Ms

m

) = pm(1− p)Ms−m,

which matches the product measure of independent Bernoulli(p) trials on Es. Repeating
across all sizes s shows that every potential hyperedge is included independently with
probability p. Thus the algorithm exactly recovers the distribution of SuHG(n)(V0, p).

Theorem 18 (Time and Space Complexity). Let M = |V (n)| and Ms =
(
M
s

)
. Assume

that sampling one s-subset uniformly at random (without full enumeration) takes O(s)
time. Then Algorithm 2 runs in

O
( M∑
s=1

(
Tbinom(Ms, p) + Ks · s

))
,

where Tbinom(Ms, p) is the cost of drawing a Binomial(Ms, p) variate and Ks is its realized
value. In expectation,

E
[
time

]



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 21 of 29

= O
( M∑
s=1

(
polylog(Ms) + pMs s

))
.

Space usage is

O
(
M +

M∑
s=1

Ks · s
)
,

i.e. O(M + |E|).

Proof. Time. Computing V (n) costs O(M logM) by Theorem 15. For each s,
drawing Ks ∼ Binomial(Ms, p) takes Tbinom(Ms, p) time (e.g. O(logMs) with accep-
tance–rejection). Sampling Ks distinct s-subsets costs O(Ks · s) if each unranked in O(s)
time. Summing yields the stated bound. In expectation, E[Ks] = pMs, giving the second
formula.

Space. Storing V (n) uses O(M logM) bits. The edge set E requires space proportional
to the total size of sampled hyperedges,

∑
sKs ·s. Ignoring logarithmic factors, total space

is O(M + |E|).

5. Conclusions

In this paper, we investigated the notion of a random SuperHyperGraph and provided
a concise mathematical exploration of its structure. We also included a brief discussion
of the associated algorithm. We list below several scientific benefits of random SuperHy-
perGraphs. These points underscore the potential impact of this research in areas such as
computer science and social systems.

• Unified Framework: Merges random graphs and hypergraphs into one probabilis-
tic model.

• Hierarchical Representation: Encodes multi-level interactions via nested pow-
erset layers.

• Probabilistic Insights: Enables study of phase transitions and concentration in
layered networks.

• Algorithmic Foundations: Supports development of algorithms for complex,
probabilistic hierarchies.

To capture richer uncertainty in Random SuperHyperGraphs, we plan to integrate
fuzzy-based frameworks. In particular, we will extend the current model to encom-
pass Fuzzy Sets [59], Intuitionistic Fuzzy Sets [60, 61], Vague Sets [62], Hesitant Fuzzy
Sets [63, 64], HyperFuzzy Sets [65, 66], Picture Fuzzy Sets [67, 68], Pythagorean fuzzy
set [69, 70], Neutrosophic Sets [71, 72], and Plithogenic Sets [27, 73, 74]. We will also
develop directed[75, 76], bidirected [77, 78], and multidirected [79, 80] variants, as well as



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 22 of 29

hypergraph and superhypergraph analogues of Quasi-Random Graphs [81, 82] and Semi-
Random Graphs [83, 84]. For example, regarding Random SuperHyperGraphs, we expect
future investigations to explore extensions that incorporate Fuzzy random variables[85, 86],
Neutrosophic random variables[87–89], Plithogenic random variables[90, 91], and related
constructs. Finally, we aim to validate and refine these extensions through comprehensive
computational experiments.

Acknowledgements

We wish to thank everyone whose guidance, ideas, and assistance contributed to this
research. Our appreciation goes to the readers for their interest and to the scholars
whose publications provided the groundwork for this study. We are also indebted to the
individuals and institutions that supplied the resources and infrastructure necessary for
completing and disseminating this paper. Lastly, we extend our gratitude to all who
offered their support in various capacities.

Data Availability

This paper is purely theoretical and does not involve any empirical data. We welcome
future empirical studies that build upon and test the concepts presented here.

Ethical Approval

As this work is entirely conceptual and involves no human or animal subjects, ethical
approval was not required.

Conflicts of Interest

The authors declare no conflicts of interest in connection with this study or its publi-
cation.

Funding

No external or organizational funding supported this work.

Author Contributions

Conceptualization, Takaaki Fujita and Florentin Smarandache; Investigation, Takaaki
Fujita; Methodology, Takaaki Fujita; Writing – original draft, Takaaki Fujita; Writing –
review & editing, Takaaki Fujita and Florentin Smarandache.



T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6835 23 of 29

Research Integrity

The authors affirm that, to the best of their knowledge, this manuscript represents
their original research. It has not been previously published in any journal, nor is it
currently being considered for publication elsewhere.

Disclaimer on Computational Tools

No computer-based tools—such as symbolic computation systems, automated theorem
provers, or proof assistants (e.g., Mathematica, SageMath, Coq)—were employed in the
development, analysis, or verification of the results contained in this paper. All derivations
and proofs were conducted manually through analytical methods by the authors.

Disclaimer on Scope and Accuracy

The theoretical models and concepts proposed in this manuscript have not yet under-
gone empirical testing or practical deployment. Future work may investigate their utility
in applied or experimental contexts. While the authors have taken care to maintain ac-
curacy and provide appropriate citations, inadvertent errors or omissions may remain.
Readers are encouraged to consult original references for confirmation and further study.

The authors assert that all mathematical results and justifications included in this
work have been carefully reviewed and are believed to be correct. Should any inaccuracies
or ambiguities be discovered, the authors welcome constructive feedback and will provide
clarification upon request.

The conclusions presented are valid only within the specific theoretical framework
and assumptions described in the text. Generalizing these results to other mathematical
contexts may require further investigation. All opinions and interpretations expressed
herein are solely those of the authors and do not necessarily reflect the views of their
respective institutions.

Use of Generative AI and AI-Assisted Tools

I use generative AI and AI-assisted tools for tasks such as English grammar checking,
and I do not employ them in any way that violates ethical standards.

Consent to Publish

All authors have given their consent for submission of this manuscript to the journal.

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