Uncertain Hypergraphs: A Conceptual Framework and Some Topological Characteristics Indexes

: In practical applications of hypergraph theory, we are usually surrounded by the state of indeterminacy. This paper employs uncertainty theory to address indeterministic information. We initially put forward the idea of an uncertain hypergraph by combining hypergraph theory with uncertainty theory in order to provide a useful tool to deal with a variety of uncertain complex systems and to create a new interdisciplinary research ﬁeld. The main focus of this paper is to propose a conceptual framework of uncertain hypergraphs and to study the operations of uncertain hypergraphs. Moreover, some topological indexes are proposed to describe the characteristics of the structures of uncertain hypergraph. Additionally, some further research directions are discussed.


Introduction
Graph theory has a history of more than 280 years, beginning when mathematician Euler first used a graph approach to solve the problem of the Seven Bridges of Königsberg in 1736. It commenced its formal development during the second half of the 19th century, and substantial growth has been witnessed during the last 100 years. Until now, graphs have been extensively used in domains including mathematics and computer science as well as various other fields, ranging from neuroscience to social network analysis.
However, graphs have a major limitation in that they cannot represent complex multi-ary relations among things but only support binary relations between pairs of things in a system. In real life, the multi-ary relationships extending unary and binary relationships exist commonly in a complex system. N-ary relations describe relations of any arity, including unary, binary, ternary, etc. As a generalization of traditional graph theory (see, for example, Bondy and Murty [1]), the hypergraph was introduced by Berge [2] to describe the complex systems of multi-ary relationships other than binary relationships. The co-occurrence relation can be represented by a hyperedge that connects two or more vertices in a hypergraph. Since then, many popular topics of hypergraphs have been studied, such as connectivity (Dankelmann and Meierling [3], Zhao and Meng [4]), degree (Frankl [5], Hàn et al. [6]), matching problems (Yu et al. [7], Lu et al. [8]) and so forth.
In practice, indeterminacy is inevitable due to the lack of observed data. Sometimes, whether an edge exists cannot be completely determined. Then, how does one address such indeterministic information? Some researchers thought that whether an edge existed could be described as a random variable. As a result, probability theory (Kolmogorov [9]) was introduced into graph theory, and the random graph was defined by Erdös and Rényi [10,11] and Gilbert [12] at nearly the same time. Subsequently, the random hypergraph has been studied by many scholars. For instance, Cooper [13] studied the problem of asymptotical descriptions of the adjacency eigenvalues of random and complete uniform hypergraphs. Semenov and Shabanov [14] addressed the weak chromatic In this section, we introduce basic notions about hypergraphs. We suppose that readers are familiar with the basic knowledge of graph theory. As hypergraphs are an important generalization of ordinary graphs, many of the definitions of graphs can be extended to hypergraphs in a nearly verbatim way.

Definition 1.
A hypergraph H is a pair H = (V, E), where V is a finite set of elements and E is a family of subsets of V. Customarily, V = {v 1 , v 2 , · · · , v n } is called the vertex set of H, denoted by V(H) sometimes for clarity, and E = {e i |e i = {v i 1 , v i 2 , · · · , v i n i }} is called the hyperedge set of H, denoted by E(H) sometimes for clearness.
We say that the cardinality |V| of V is the order of the hypergraph H = (V, E), and the cardinality |E| of E is the size of the hypergraph H = (V, E). It can be seen from the above definition that a hyperedge in a hypergraph may connect any number of vertices. The hyperedge set E is a subset of the power set P (V) of the vertex set V in a hypergraph.
A hyperedge e ∈ E such that |V(e)| = 1 is a loop. Now we consider the question: How can we visually represent the hypergraph in a reasonable way? There are two different kinds of representations; one is a geometrical method and the other is an algebraical method.
Then H has 8 vertices and 6 hyperedges. The hypergraph H is graphically illustrated in Figure 1. Example 2. Suppose eight members comprise a research team V = {v 1 , v 2 , · · · , v 8 }. Now consider the collaborative relationship between them for published papers. The set E = {e 1 , e 2 , e 3 , e 4 , e 5 } means a total of 5 articles have been published, and the authors of each article are 6 , v 7 } and e 5 = {v 8 }, respectively. Therefore, the research team's paper collaboration hypergraph in a knowledge network analysis is H = (V, E) with 8 vertices and 5 hyperedges, which is shown in Figure 2. In fact, the hypergraph H may stand for the kinds of coauthor relationships among 8 authors.  3. Nowadays, WeChat (or QQ) is one of the most popular social media platforms for Chinese social network users. This new form of media, instead of telephone and email, is powerfully changing the way of social communication. Consider somebody's contact objects and private groups of WeChat. Assume that V = {v 1 , v 2 , · · · , v 12 } and E = {e 1 , e 2 , e 3 , e 4 , e 5 10 , v 11 , v 12 }}. Then H has 12 vertices, which represent contact objects in his or her WeChat address list, and 5 hyperedges, which represent his or her private group in WeChat. The hypergraph H = (V, E) has the visualization depicted in Figure 3.

Definition 2. (Liu
The triplet (Γ, L, M) is said to be an uncertainty space. An uncertain variable (Liu [22]) is defined as a measurable function ξ from an uncertainty space to the set of real numbers. The product uncertain measure was defined by Liu [31] in 2009, thus producing the fourth axiom of uncertainty theory. Axiom 4. Let (Γ k , L k , M k ) be uncertainty spaces for k = 1, 2, · · · . The product uncertain measure M is an uncertain measure satisfying where Λ k are arbitrarily chosen events from L k for k = 1, 2, · · · , respectively.   [31]) Let ξ be an uncertain variable with uncertainty distribution Φ. Then its entropy is defined by As discussed in the introduction, many scholars have paid attention to research topics on uncertain graphs. We begin by recalling the most standard definitions of uncertain graphs (Zhang and Peng [25]). Note that an uncertain graph can be redefined by the method of set and mapping in the following way: Definition 5. An uncertain graph is defined as a triple system G = (V, E, ϕ) where V is the vertex set of G, E(⊆ V × V) is the edge set of G and ϕ : E → [0, 1] is a belief degree function in which ϕ(e) assigns the uncertain measure of the existence of the edge e ∈ E.
It follows from Definition 5 that in an uncertain graph, the vertices are all predetermined, while edges are uncertain. The existence possibility of an edge is described by uncertain measure. Generally, a Boolean uncertain variable ξ i can be used to describe the existence of edge e i . That is, ϕ(e i ) = M{ξ i = 1} indicates the existence possibility of e i , where ξ i = 1 means edge e i exists. Apparently, the larger the value of ϕ(e i ), the more the true it is that the edge e i exists. In particular, ϕ(e i ) = 1 means that the edge e i exists completely; on the contrary, ϕ(e i ) = 0 means that the edge e i does not exist. For more research on uncertainty theory and uncertain graphs, the interested readers may refer to Liu [32] and Zhang and Peng [25].
is an uncertain graph with the uncertain adjacency matrix

Uncertain Hypergraphs
Despite the fact that a hypergraph can only represent the determining multi-ary relations (including binary relations and unary relations) in a complex system, it cannot be used to deal with the uncertain multi-ary relations in a complex system under the presence of indeterminacy.
We have introduced the fundamental concepts of hypergraph theory and uncertainty theory as above-mentioned. We now aim to combine the two by exploring the notions of uncertain hypergraphs.
In this section, we introduce the basic notions about uncertain hypergraphs. From a theoretical point of view, on one hand, an uncertain hypergraph is the generation of a hypergraph stated in Section 1; on the other hand, an uncertain hypergraph can be considered the extension of an uncertain graph, which will be reviewed in the coming subsection.

Concept of Uncertain Hypergraphs
As the generalization of a hypergraph, an uncertain hypergraph is defined as follows: An uncertain hypergraph is defined as a triple system H = (V, E, ψ) where V is the vertex set, E is a family of subsets of V and ψ : E → [0, 1] is a set function in which ψ(e) assigns the uncertain measure of the existence of the hyperedge e ∈ E. Correspondingly, H = (V, E) is called the underlying hypergraph of uncertain hypergraph H and ψ is known as the hyperedge assignment function of uncertain hypergraph H.
A simple hypergraph is a hypergraph having no loops and multiple hyperedges among (or between) a set of vertices. Every hypergraph in this paper is a simple uncertain hypergraph.

Remark 1.
When the hyperedge set E(∈ P (V)) of H degrades into the classical edge set E(⊆ V × V), an uncertain hypergraph degenerates to an uncertain graph. Thus, an uncertain hypergraph is an extension of an uncertain graph.

Remark 2.
When the hyperedge assignment function ψ ≡ 1 of H, an uncertain hypergraph can be regarded as an ascertained hypergraph. Thus, an uncertain hypergraph is also an extension of a determined hypergraph.

Remark 3.
When each hyperedge in E is an ordered set with a known head and tail, an uncertain hypergraph will become an uncertain directed hypergraph, or uncertain dihypergraph. Thus, an uncertain dihypergraph is both an extension of an uncertain digraph and an extension of a determined dihypergraph.

Representation of Uncertain Hypergraphs
Assume H = (V, E, ψ) is an uncertain hypergraph of order n with V = {v 1 , v 2 , · · · , v n }, E = (e 1 , e 2 , · · · , e m ) and ψ(E) = (ψ(e 1 ), ψ(e 2 ), · · · , ψ(e m )). How can we visually represent the uncertain hypergraph in a reasonable way? There are two different kinds of representations aside from the set method as stated in the definition; one is a geometrical method and the other is an algebraical method.
In a geometrical way, we can put in the uncertain measure on each hyperedge based on the underlying hypergraph H = (V, E) (see, for example, Figure 4). In an algebraical way, we can use two types of uncertain hyper-adjacency matrices. In other words, each uncertain hypergraph can be represented by the minimum hyper-adjacency matrix or the maximum hyper-adjacency matrix.

Definition 7.
The minimum hyper-adjacency matrix of an uncertain hypergraph H = (V, E, ψ) of order n is defined as where α ij represent that the hyperedges containing vertices v i and v j exist with uncertain measures α ij = min{ψ(e)|v i , v j ∈ e, e ∈ E}, i, j = 1, 2, · · · , n, respectively.

Definition 8.
The maximum hyper-adjacency matrix of an uncertain hypergraph H = (V, E, ψ) of order n is defined as where α ij represent that the hyperedges contain vertices v i and v j exist with uncertain measures α ij = max{ψ(e)|v i , v j ∈ e, e ∈ E}, i, j = 1, 2, · · · , n, respectively.

Variations of Uncertain Hypergraphs
The above-mentioned uncertain hypergraph is considered an adjacency uncertain hypergraph. We can now consider the other type of uncertain hypergraph, that is, the incidence uncertain hypergraph. Definition 9. An incidence uncertain hypergraph is defined as a triple system H = (V, E, ψ ) where V is the vertex set, E is the hyperedge set of H and ψ : V × E → [0, 1] is a set function in which ψ ((v, e)) assigns the uncertain measure of the incidence existence between vertex v ∈ V and hyperedge e ∈ E. Correspondingly, H = (V, E) is still called the underlying hypergraph of uncertain hypergraph H and ψ is known as the incidence assignment function of uncertain hypergraph H.
Assume H = (V, E) is a hypergraph of order n and size m with V = {v 1 , v 2 , · · · , v n } and E = {e 1 , e 2 , · · · , e m }. Each hypergraph can be represented by the incidence matrix as stated in Definition 9.

Definition 10.
A hypergraph H = (V, E) of order n and size m is said to be of uncertain incidence if it has uncertain incidence matrix B = (β ij ) n×m as follows: where β ij ∈ [0, 1] represents that the hyperedge e j containing vertex v i exists with uncertain measure β ij , i = 1, 2, · · · , n; j = 1, 2, · · · , m, respectively.
As a matter of fact, if β ij ∈ {0, 1}, then the above-mentioned uncertain incidence matrix will be degraded to the traditional incidence matrix.

Example 7.
Let H = (V, E) be the hypergraph in Example 1. Then the incidence matrix is If the incidence matrix is replaced with then the uncertain incidence matrix B implies a new type of uncertain hypergraph with uncertain vertex-hyperedge relationships.
Generally, another type of uncertainty may actually exist in hypergraphs in the case that both the adjacency relationships among the vertices and the incidence relationships between the vertices and the hyperedges are uncertain.

Definition 11.
A double uncertain hypergraph is defined as a quaternary system H = (V, E, ψ, ψ ) where V is the vertex set, E is the hyperedge set of H, ψ : E → [0, 1] is a set function in which ψ(e) assigns the uncertain measure of the existence of the hyperedge e ∈ E and ψ : V × E → [0, 1] is a set function in which ψ ((v, e)) assigns the uncertain measure of the incidence existence between vertex v ∈ V and hyperedge e ∈ E.
In this article, we restrictively focus on the first type of uncertain hypergraphs which have uncertain topology adjacency structures.

Definition 12. Given two uncertain hypergraphs H
Alternatively, we also call H 2 a supergraph of H 1 , denoted by H 2 ⊇ H 1 .
The isomorphism relationship of uncertain hypergraphs is an important relationship between uncertain hypergraphs.
We now introduce the complement, union and joint operations on uncertain hypergraphs as follows: The proof is completed.
Definition 15. Given two uncertain hypergraphs H 1 = (V 1 , E 1 , ψ 1 ) and H 2 = (V 2 , E 2 , ψ 2 ), the union H 1 ∪ H 2 of the uncertain hypergraphs H 1 and H 2 is an uncertain hypergraph H = (V, E, ψ) Theorem 3. Let H 1 = (V 1 , E 1 , ψ 1 ) and H 2 = (V 2 , E 2 , ψ 2 ) be two simple uncertain hypergraphs. Then the following properties hold: Note that In the following, we give a definition of the α-cut to describe the relationship between a hypergraph and an uncertain hypergraph: Definition 17. The α-cut H α of an uncertain hypergraph H = (V, E, ψ) is defined as a certain hypergraph H α = (V α , E α ) where V α = V is the vertex set, E α is the hyperedge set of H α such that e ∈ E α iff e ∈ E and ψ(e) ≥ α.
It is not so difficult to prove that the α-cuts of the union and joint operations of two uncertain hypergraphs H 1 and H 2 have the following properties:

Some Topological Indexes of Uncertain Hypergraphs
By topological indices, we mean the numerical parameters of an uncertain hypergraph which characterize its topology structure. We are very interested in some questions of uncertain hypergraphs similar to the concepts and solutions in graphs or hypergraphs.

Degrees in Uncertain Hypergraphs
Definition 18. Let H = (V, E, ψ) be an uncertain hypergraph with vertices V = {v 1 , v 2 , · · · , v n } and hyperedges E = {e 1 , e 2 , · · · , e m }. For a vertex v i ∈ V, the star H(v i ) centered in v i is the set of hyperedges containing v i . That is to say, H(v i ) = {e|v i ∈ e} = {e 1 , e 2 , · · · , e n i }.

Distances in Uncertain Hypergraphs
It is natural to consider the distance between two vertices in an uncertain hypergraph. Since the hyperedges exist with uncertain measure, the concept of path in an uncertain hypergraph is also related to uncertain measure.
The integer s is the length of the uncertain path P. Denote α = min e i ∈P {ψ(e i )} and β = 1 − max e i ∈P {ψ(e i )}. Then the uncertain path P is called the α-maximum belief degree path from x to y, or antisymmetrically, it is also called the β-minimum risk path from x to y. The uncertain path P is called an uncertain cycle if Notice that if there is an uncertain path with α-maximum belief degree from x to y, then there is also an uncertain path with α-maximum belief degree from y to x. In this case, we say that the uncertain path P connects x and y with α-maximum belief degree. An uncertain hypergraph is α-connected if for any pair of vertices, there is an uncertain path with α-maximum belief degree which connects these vertices; otherwise, it is not αconnected, which is also called α-disconnected. It is obvious that if an uncertain hypergraph H is α-connected and α ≥ γ, then H is γ-connected.
We define the relation xRy with α-maximum belief degree if and only if either there is an uncertain path with α-maximum belief degree from x to y or x = y. Then relation R is an equivalence relation, which can classify the equivalence classes of this relation into the α-connected components of the uncertain hypergraph.
In a hypergraph, the distance between two vertices is the minimal number of hyperedges that connect the two vertices. By contrast, the α-belief degree distance between two vertices in an uncertain hypergraph is the minimal number of hyperedges that connect the two vertices at α-belief degree level.
What we concern ourselves about here is the topological existence of a belief degree path between x and y other than the physical distance between the two vertices. There may exist a not unique path with α-maximum belief degree from x to y in an uncertain hypergraph.

Definition 20.
Let H = (V, E, ψ) be an uncertain hypergraph. The distance d α (x, y) at α-belief degree level between two vertices x and y is the minimum length of a path with α-maximum belief degree which connects x and y. The diameter diam α (H) at α-belief degree level of H is defined by diam α (H) = max{d α (x, y)|x, y ∈ V}.
If there is a pair of vertices x, y with no path from x to y at α-belief degree level (or from y to x), we define d α (x, y) = ∞. An α-connected component is a maximal set of vertices X ⊆ V such that d α (x, y) = ∞ for all x, y ∈ X.

Reliability Indexes of Uncertain Hypergraphs
The reliability index of the topological structure of an uncertain hypergraph is a topic of our interest and concern. It is also the starting point of reliability analysis for uncertain hypergraphs based on uncertainty theory. Note that uncertain reliability analysis and uncertain risk analysis have duality in mathematics.
Dually, the maximum adjacency belief degree is called the supremum adjacency risk index, and the minimum adjacency belief degree R (ψ(e 1 ), ψ(e 2 ), · · · , ψ(e m )) is called the infimum adjacency risk index.

Entropy of Uncertain Hypergraphs
Entropy is an important concept which provides a quantitative measurement for the degree of uncertainty of an uncertain hypergraph. We introduce below the notion of entropy associated with an uncertain hypergraph.

Definition 22.
Let H = (V, E, ψ) with V = {v 1 , v 2 , · · · , v n } be an uncertain hypergraph. Then we can define the entropy I(H) of H by is the degree of the vertex v i for i = 1, 2, · · · , n.

Conclusions and Future Work
Graphs, uncertain graphs, hypergraphs and uncertain hypergraphs are important and useful concepts or tools in describing systems in our real lives. In our view, on one side, a graph is used to describe a system with relative simplicity, while a hypergraph is used to describe a system with relative complexity. On the other side, a graph is used to describe a system with determinacy, while an uncertain graph is used to describe a system with uncertainty. In a more profound view, an uncertain graph is used to describe a system with indeterminacy and relative simplicity, while an uncertain hypergraph is used to describe a system with uncertainty and relative complexity.
The main contributions of this paper can be summarized as follows: Firstly, we constructed the foundation for a new interdisciplinary theory for uncertainty theory and hypergraph theory termed uncertain hypergraph theory. The concept of an uncertain hypergraph was proposed and the representations of an uncertain hypergraph were discussed. Subsequently, the operations of uncertain hypergraphs were studied, and some properties were verified. Finally, some topological indexes were introduced. The importance and originality of this study are that we hope this work can serve as an opportunity of reflection for experienced complexity scientists and experienced uncertainty scientists as well as an introductory resource for new researchers.
Some directions for future work are described here: As part of our future work, we would first like to introduce the uncertain hypernetwork, which is different from the hypergraph in some aspects. Second, we would like to study ways in which we could improve the computational efficiency of the proposed methods. Third, we would like to think about hypergraph theory under more a complex environment, such as an uncertain random environment. Chance theory was introduced by Liu [33] to model complex systems related to uncertainty and randomness. Hopefully, the related topics, such as connectivity, regularity and operations on hypergraphs, can be further studied in the future within the framework of chance theory.
Author Contributions: Conceptualization, J.P.; methodology, J.P. and B.Z.; investigation, J.P. and B.Z.; writing-original draft preparation, J.P.; writing-review and editing, J.P. and K.A.S. All authors have read and agreed to the published version of the manuscript.