*Fuzzy Neighbourhood Hypergraphs*

The concepts of fuzzy open neighbourhood and fuzzy closed neighbourhood are given in Definition 13.

**Definition 13.** *[16] The fuzzy open neighbourhood of a vertex y in a fuzzy graph G* = (*μ*, *λ*) *is a fuzzy set* N (*y*)=(*Xy*, *<sup>μ</sup>y*), *where Xy* = {*w*|*λ*(*yw*) > 0} *and μy* : *Xy* → [0, 1] *a membership function defined by <sup>μ</sup>y*(*w*) = *<sup>λ</sup>*(*yw*)*.*

**Definition 14.** *[16] The fuzzy closed neighbourhood* N [*y*] *of a vertex y in a fuzzy graph G* = (*μ*, *λ*) *is defined as* N [*y*] = N (*y*) ∪ {(*y*, *μ*(*y*))}*.*

**Definition 15.** *The fuzzy open neighbourhood hypergraph of a fuzzy graph G* = (*μ*, *λ*) *is a fuzzy hypergraph* N (*G*)=(*μ*, *λ* ) *whose fuzzy vertex set is the same as G and there is a hyperedge E* = {*<sup>x</sup>*1, *x*2, ... , *xr*} *in* N (*G*) *if* N (*<sup>x</sup>*1) ∩ N (*<sup>x</sup>*2) ∩ ... ∩ N (*xr*) = ∅*. The membership function λ* : *X* × *X* → [0, 1] *is defined as*

> *λ* (*E*) = -*μ*(*<sup>x</sup>*1) ∧ *μ*(*<sup>x</sup>*2) ∧ ... ∧ *μ*(*xr*). × *h*-N (*<sup>x</sup>*1) ∩ N (*<sup>x</sup>*2) ∩ ... ∩ N (*xr*)..

The fuzzy closed neighbourhood hypergraph is defined on the same lines in the following definition.

**Definition 16.** *The fuzzy closed neighbourhood hypergraph of G* = (*μ*, *λ*) *is a fuzzy hypergraph* N [*G*] = (*μ*, *λ*∗) *whose fuzzy set of vertices is same as G and there is a hyperedge E* = {*<sup>x</sup>*1, *x*2, ... , *xr*} *in* N [*G*] *if* N [*<sup>x</sup>*1] ∩ N [*<sup>x</sup>*2] ∩ *X* ... ∩ N [*xr*] = ∅*. The membership function λ*∗ : *X* × *X* → [0, 1] *is defined as*

$$\lambda^\*(E) = \left(\mu(\mathbf{x}\_1) \land \mu(\mathbf{x}\_2) \land \dots \land \mu(\mathbf{x}\_{\mathcal{I}})\right) \times h(\mathcal{N}[\mathbf{x}\_1] \cap \mathcal{N}[\mathbf{x}\_2] \cap \dots \cap \mathcal{N}[\mathbf{x}\_{\mathcal{I}}]).$$

**Example 6.** *Consider the fuzzy graph G* = (*μ*, *λ*) *on set Y* = {*y*1, *y*2, *y*3, *y*4} *as shown in Figure 9. The fuzzy open neighbourhoods are given in Table 5.*

*Define a relation f* : *X* → *X by f*(*yi*) = *yj if yj* ∈ *supp*(N (*yi*)) *as shown in Figure 10. If, for yi* ∈ *X,* | *f* <sup>−</sup><sup>1</sup>(*yi*)| > 1, *then f* <sup>−</sup><sup>1</sup>(*yi*) *is a hyperedge of* N [*G*]*. Since, from Figure 10, f* <sup>−</sup><sup>1</sup>(*y*1) = {*y*2, *y*3, *y*4} = *E*1*, f* <sup>−</sup><sup>1</sup>(*y*2) = {*y*1, *y*4} = *E*2 *and f* <sup>−</sup><sup>1</sup>(*y*4) = {*y*1, *y*2}<sup>3</sup>*, therefore, E*1, *E*2, *E*3 *are hyperedges of* N (*G*)*. The degree of membership of each hyperedge can be computed using Definition 15 as follows.*

*For f* <sup>−</sup><sup>1</sup>(*y*1) = *E*1 = {*y*2, *y*3, *y*4}*, λ* (*<sup>E</sup>*1) = -*μ*(*y*2) <sup>∧</sup>*μ*(*y*3) <sup>∧</sup>*μ*(*y*4).<sup>×</sup> *h*-N (*y*2) ∩N (*y*3)<sup>∩</sup> N (*y*4). = 0.4 × 0.4 = 0.16*. Similarly, λ* ({*y*1, *y*4}) = 0.4 × 0.3 = 0.12 *and λ* ({*y*1, *y*2}) = 0.5 × 0.3 = 0.15*. The fuzzy open neighbourhood hypergraph constructed using Definition 13 from* #» *G is given in Figure 10.*

**Table 5.** Fuzzy open neighbourhood of vertices.


The fuzzy closed neighbourhoods of all the vertices in *G* are given in Table 6. Since N [*y*1] ∩ N [*y*2] ∩ N [*y*3] ∩ N [*y*4] = {(*y*1, 0.4)}, therefore, *E* = {*y*1, *y*2, *y*3, *y*4} is a hyperedge of N [*G*] and *λ*∗(*E*) = 0.4 × 0.4 = 0.16. The fuzzy closed neighbourhood hypergraph is given in Figure 11.

**Figure 9.** Fuzzy graph *G*.

**Figure 10.** Fuzzy open neighbourhood hypergraph of *G*.

**Figure 11.** Fuzzy closed neighbourhood hypergraph.

**Table 6.** Fuzzy closed neighbourhood of vertices.


Using different types of fuzzy neighbourhood of the vertices, some other types of fuzzy hypergraphs are defined here.

**Definition 17.** *Let k be a non-negative real number; then, the fuzzy* (*k*)*-competition hypergraph of a fuzzy graph G* = (*μ*, *λ*) *is a fuzzy hypergraph* N*k*(*G*)=(*μ*, *λ kc*) *having the same fuzzy set of vertices as G and there is a hyperedge E* = {*<sup>x</sup>*1, *x*2, ... , *xr*} *in* N*k*(*G*) *if* |N (*<sup>x</sup>*1) ∩ N (*<sup>x</sup>*2) ∩ ... ∩ N (*xr*)| > *k. The membership value of E is defined as*

$$\boldsymbol{\lambda}\_{kr}^{'}(E) = \frac{l-k}{l} (\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_l)) \times h(\mathcal{N}(\mathbf{x}\_1) \cap \mathcal{N}(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}(\mathbf{x}\_l)),$$

*where* |N (*<sup>x</sup>*1) ∩ N (*<sup>x</sup>*2) ∩ ... ∩ N (*xr*)| = *l.*

**Definition 18.** *The fuzzy* [*k*]*-competition hypegraph of G is denoted by* N*k*[*G*]=(*μ*, *<sup>λ</sup>*<sup>∗</sup>*kc*) *and there is a hyperedge E in* N*k*[*G*] *if* |N [*<sup>x</sup>*1] ∩ N [*<sup>x</sup>*2] ∩ ... ∩ N [*xr*]| > *k. The membership value of E is defined as*

$$
\lambda\_{kx}^\*(E) = \frac{p-k}{p} (\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_r)) \times h(\mathcal{N}[\mathbf{x}\_1] \cap \mathcal{N}[\mathbf{x}\_2] \cap \dots \cap \mathcal{N}[\mathbf{x}\_r]),
$$

*where* |N [*<sup>x</sup>*1] ∩ N [*<sup>x</sup>*2] ∩ ... ∩ N [*xr*]| = *p.*

**Definition 19.** *[16] Let* #» *G* = (*μ*, #» *λ*) *be a fuzzy digraph. The underlying fuzzy graph of* #»*G is a fuzzy graph* <sup>U</sup>(#»*G*)=(*μ*, *λ*) *such that*

$$
\lambda(xw) = \begin{cases}
\overrightarrow{\lambda}(xw), & \text{if } \overrightarrow{w}\overrightarrow{\mathbf{x}} \notin \overrightarrow{E}, \\
\overrightarrow{\lambda}(w\mathbf{x}), & \text{if } \overrightarrow{x}\overrightarrow{w} \notin \overrightarrow{E}, \\
\overrightarrow{\lambda}(xw) \wedge \overrightarrow{\lambda}(w\mathbf{x}), & \text{if } \overrightarrow{w}\overrightarrow{\mathbf{x}}, \overrightarrow{xw} \in \overrightarrow{E}.
\end{cases}
$$

where #» *E* = *supp*( #» *<sup>λ</sup>*). The relations between fuzzy neighbourhood hypergraphs and fuzzy competition hypergraphs are given in the following theorems.

**Theorem 3.** *Let* #» *G* = (*μ*, #» *λ*) *be a symmetric fuzzy digraph without any loops; then,* <sup>C</sup>*k*(#»*G*) = <sup>N</sup>*k*(U(#»*G*)), *where* U(#»*G*) *is the underlying fuzzy graph of* #»*G.*

**Proof.** Let <sup>U</sup>(#»*G*)=(*μ*, *λ*) correspond to the fuzzy graph #»*G* = (*μ*, #»*<sup>λ</sup>*). In addition, let <sup>N</sup>*k*(U(#»*G*)) = (*μ*, *λ kc*) and <sup>C</sup>*k*(#»*G*)=(*μ*, *<sup>λ</sup>kc*). Clearly, the fuzzy *k*-competition hypergraph <sup>C</sup>*k*(#»*G*) and the underlying fuzzy graph have the same fuzzy set of vertices as #» *G*. Hence, <sup>N</sup>*k*(U(#»*G*)) has the same vertex set as #»*G*. It remains only to show that *<sup>λ</sup>kc*(*xw*) = *λ kc*(*xw*) for every *x*, *w* ∈ *X*. Thus, there are two cases.

Case 1: If, for each *x*1, *x*2, ... , *xr* ∈ *X*, *<sup>λ</sup>kc*({*<sup>x</sup>*1, *x*2, ... , *xr*}) = 0 in <sup>C</sup>*k*(#»*G*), then |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... N +(*xr*)| ≤ *k*. Since #»*G* is symmetric, |N (*<sup>x</sup>*1) ∩ N (*<sup>x</sup>*2) ∩ ... N (*xr*)| ≤ *k* in <sup>U</sup>(#»*G*). Thus, *λ kc*({*<sup>x</sup>*1, *x*2,..., *xr*}) = 0 and *<sup>λ</sup>kc*(*E*) = *λ kc*(*E*) for all *x*1, *x*2,..., *xr* ∈ *X*.

Case 2: If, for some *x*1, *x*2, ... , *xr* ∈ *X*, *<sup>λ</sup>kc*(*E*) > 0 in <sup>C</sup>*k*(#»*G*), then |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... N +(*xr*)| > *k*. Thus,

$$\lambda \,\lambda\_{\text{kr}}(E) = \frac{l-k}{l} [\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_r)] h(\mathcal{N}^+(\mathbf{x}\_1) \cap \mathcal{N}^+(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}^+(\mathbf{x}\_r)),$$

where *l* = |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)|. Since #»*G* is a symmetric fuzzy digraph, |N (*<sup>x</sup>*1) ∩ N (*<sup>x</sup>*2) ∩ ... N (*xr*)| > *k*. Hence, *<sup>λ</sup>kc*(*E*) = *λ kc*(*E*). Since *x*1, *x*2, ... , *xr* were taken to be arbitrary, the result holds for all hyperedges *E* of <sup>C</sup>*k*(#»*G*).

**Theorem 4.** *Let* #» *G* = (*<sup>C</sup>*, #»*D*) *be a symmetric fuzzy digraph having loops at every vertex; then,* <sup>C</sup>*k*(#»*G*) = <sup>N</sup>*k*[U(#»*G*)], *where* U(#»*G*) *is the underlying fuzzy graph of* #»*G.*

**Proof.** Let <sup>U</sup>(#»*G*)=(*μ*, *λ*) be an underlying fuzzy graph corresponding to fuzzy digraph #»*G* = (*μ*, #»*<sup>λ</sup>*). Let <sup>N</sup>*k*[U(#»*G*)] = (*μ*, *λ kc*) and <sup>C</sup>*k*(#»*G*)=(*μ*, *<sup>λ</sup>kc*). The fuzzy *k*-competition graph <sup>C</sup>*k*(#»*G*) as well as the underlying fuzzy graph have the same vertex set as #» *G*. It follows that <sup>N</sup>*k*[U(#»*G*)] has the same fuzzy vertex set as #» *G*. It remains only to show that *<sup>λ</sup>kc*({*<sup>x</sup>*1, *x*2, ... , *xr*}) = *λ kc*({*<sup>x</sup>*1, *x*2, ... , *xr*}) for every *x*1, *x*2, ... , *xr* ∈ *X*. As the fuzzy digraph has a loop at every vertex, the fuzzy out neighbourhood contains the vertex itself. There are two cases.

Case 1: If, for all *x*1, *x*2, ... , *xr* ∈ *X*, *<sup>λ</sup>kc*(*E*) = 0 in <sup>C</sup>*k*(#»*G*), then, |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... N +(*xr*)| ≤ *k*. As #» *G* is symmetric therefore, |N ([*<sup>x</sup>*1] ∩ N [*<sup>x</sup>*2] ∩ ... N [*xr*]| ≤ *k* in <sup>U</sup>(#»*G*). Hence, *λ kc*(*E*) = 0 and so *<sup>λ</sup>kc*(*E*) = *λ kc*(*E*) for all *x*1, *x*2,..., *xr* ∈ *X*.

Case 2: If for some *x*1, *x*2, ... , *xr* ∈ *X*, *<sup>λ</sup>kc*(*E*) > 0 in <sup>C</sup>*k*(#»*G*), then |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... N +(*xr*)| > *k*. As #»*G* is symmetric fuzzy digraph and has loops at every vertex; therefore, |N ([*<sup>x</sup>*1] ∩ N [*<sup>x</sup>*2] ∩ ... N [*xr*]| > *k*. Hence, *<sup>λ</sup>kc*(*xy*) = *λ kc*(*xy*). As *x*1, *x*2, ... , *xr* were taken to be arbitrary, the result holds for all hyperedges *E* = {*<sup>x</sup>*1, *x*2,..., *xr*} of <sup>C</sup>*k*(#»*G*).

#### **3. Applications of Fuzzy Competition Hypergraphs**

In this section, we present several applications of fuzzy competition hypergraphs in food webs, business marketing and social networks.

#### *3.1. Identifying Predator–Prey Relations in Ecosystems*

We now present application of fuzzy competition hypergraphs in order to describe the interconnection of food chains between species, flow of energy and predator–prey relationship in ecosystems. The strength of competition between species represents the competition for food and common preys of species. We will discuss a method to give a description of species relationship, danger to the population growth rate of certain species, powerful animals in ecological niches and lack of food for weak animals.

Competition graphs arose in connection with an application in food webs. However, in some cases, competition hypergraphs provide a detailed description of predator–prey relations than competition graphs. In a competition hypergraph, it is assumed that vertices are defined clearly but in real-world problems, vertices are not defined precisely. As an example, species may be of different type like vegetarian, non-vegetarian, weak or strong.

Fuzzy food webs can be used to describe the combination of food chains that are interconnected by a fuzzy network of food relationship. There are many interesting variations of the notion of fuzzy competition hypergraph in ecological interpretation. For instance, two species may have a common prey *(fuzzy competition hypergraph)*, a common enemy (*fuzzy common enemy hypergraph*), both common prey and common enemy (*fuzzy competition common enemy hypergraph*), and either a common prey or a common enemy (*fuzzy niche hypergraph*). We now discuss a type of fuzzy competition hypergraph in which species have common enemies known as *fuzzy common enemy hypergraph*.

Let #» *G* = (*μ*, #» *λ*) be a fuzzy food web. The fuzzy common enemy hypergraph CH(#»*G*)=(*μ*, *<sup>λ</sup>c*) has the same vertex set as #» *G* and there is a hyperedge consisting of vertices *x*1, *x*2, ... , *xs* if N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xs*) = ∅. The degree of membership of hyperedge *E* = {*<sup>x</sup>*1, *x*2, ... , *xs*} is defined as

$$
\lambda\_{\mathfrak{c}}(E) = \left[ \mu(\mathbf{x}\_1) \land \mu(\mathbf{x}\_2) \land \dots \land \mu(\mathbf{x}\_{\mathfrak{s}}) \right] \times h(\mathcal{N}^+(\mathbf{x}\_1) \cap \mathcal{N}^+(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}^+(\mathbf{x}\_{\mathfrak{s}})).
$$

The strength of common enemies between species can be calculated using Algorithm A3. Consider the example of a fuzzy food web of 13 species giraffe, lion, vulture, rhinoceros, African skunk, fiscal shrike, grasshopper, baboon, leopard, snake, caracal, mouse and impala. The degree of membership of each species represents the species' ability of resource defence. The degree of membership of each directed edge represents the strength to which the prey is harmful for the predator. The fuzzy food web is shown in Figure 12. The directed edge between the giraffe and the lion shows that the giraffe is eaten by the lion and similarly.

The degree of membership of the lion is 0.9, which shows that the lion has 90% ability of resource defence, i.e., it can defend itself against other animals as well as survive many days if the lion does not find any food. The directed edge between giraffe and lion has degree of membership 0.25, which represents that the giraffe is 25% harmful for the lion because a giraffe can kill a lion with its long legs. This is an acyclic fuzzy digraph. The fuzzy out neighbourhoods are given in Table 7.

The fuzzy common-enemy hypergraph is shown in Figure 13. The hyperedges in Figure 13 show that there are common enemies between giraffe and rhinoceros, rhinoceros, African skunk and leopard, grasshopper and snake, mouse and impala, and baboon and impala. The membership value of each hyperedge represents the degree of common enemies among the species.

The hyperedge {impala, baboon} has a maximum degree of membership, which shows that the impala and the baboon have the largest number of common enemies, whereas the mouse and the impala have the least number of common enemies.

**Figure 12.** Fuzzy food web.

**Table 7.** Fuzzy out neighbourhoods of vertices.


**Figure 13.** Fuzzy common enemy hypergraph.

#### *3.2. Identifying Competitors in the Business Market*

Fuzzy competition hypergraphs are a key approach to studying the competition, profit and loss, market power and rivalry among buyers and sellers using fuzziness in hypergraphical structures. We now discuss a method to study the business competition for power and profit, success and business failure, and demanding products in market.

In the business market, there are competitive rivalries among companies that are endeavoring to increase the demand and profit of their product. More than one company in the market sells identical products. Since various companies regularly market identical products, every company wants to attract a consumer's attention to its product. There is always a competitive situation in the business market. Hypergraph theory is a key approach to studying the competitive behavior of buyers and sellers using structures of hypergraphs. In some cases, these structures do not study the level of competition, profit and loss between the companies. As an example, companies may have different reputations in the market according to market power and rivalry. These are fuzzy concepts and motivates the necessity of fuzzy competition hypergraphs. The competition among companies can be studied using a fuzzy competition hypergraph known as *fuzzy enmity hypergraph*.

We present a method for calculating the strength of competition of companies in the following Algorithm 1.

**Algorithm 1:** Business competition hypegraph.



Consider the example of a marketing competition between seven companies DEL, CB, HW, AK, LR, RP, SONY, RA, LR, three retailers, one retailer outlet and one multinational brand as shown in Figure 14.

**Figure 14.** Fuzzy marketing digraph.

The vertices represent companies, retailers, outlets and brands. The degree of membership of each vertex represents the strength of rivalry (aggression) of each company in the market. The degree of membership of each directed edge # »*xy* represents the degree of rejectability of company's *x* product by company *y*. The strength of competition of each company can be discussed using fuzzy competition hypergraph known as *fuzzy enmity hypergraph*. The fuzzy out neighbouhoods are calculated in Table 8.


**Table 8.** Fuzzy out neighbourhoods of companies.

The fuzzy enmity hypergraph of Figure 14 is shown in Figure 15. The degree of membership of each hyperedge shows the strength of rivalry between the companies.

**Figure 15.** Fuzzy competition hypergraph.

The strength of rivalry of each company is calculated in Table 9, which shows its enmity value within the business market. Table 9 shows that SONY is the biggest rival company among other companies.

**Table 9.** Strength of rivalry between companies.


#### *3.3. Finding Influential Communities in a Social Network*

Fuzzy competition hypergraphs have a wide range of applications in decision-making problems and decision support systems based on social networking. To elaborate on the necessity of the idea discussed in this paper, we apply the notion of fuzzy competition hypergraphs to study the influence, centrality, socialism and proactiveness of human beings in any social network.

Social competition is a widespread mechanism to figure out a best-suited group economically, politically or educationally. Social competition occurs when individual's opinions, decisions and behaviors are influenced by others. Graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network. Fuzzy hypergraphs can be used to study the influence and competition between objects more precisely. The social influence and conflict between different communities can be studied using a fuzzy competition hypergraph known as *fuzzy influence hypergraph*.

The fuzzy influence hypergraph *G* = (*μ*, *<sup>λ</sup>c*) has the same set of vertices as *G* and there is a hyperedge consisting of vertices *x*1, *x*2, ... , *xr* if N <sup>−</sup>(*<sup>x</sup>*1) ∩ N <sup>−</sup>(*<sup>x</sup>*2) ∩ ... ∩ N <sup>−</sup>(*xr*) = ∅. The degree of membership of hyperedge *E* = {*<sup>x</sup>*1, *x*2,..., *xr*} is defined as

#»

$$
\lambda\_{\mathsf{c}}(E) = [\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_{\mathsf{f}})] \times h(\mathcal{N}^+(\mathbf{x}\_1) \cap \mathcal{N}^+(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}^+(\mathbf{x}\_{\mathsf{f}})).
$$

The strength of influence between different objects in a fuzzy influence hypergraph can be calculated by the method presented in Algorithm 2. The complexity of algorithm is *<sup>O</sup>*(*n*<sup>2</sup>).

#### **Algorithm 2:** Fuzzy influence hypergraph.


$$\deg(\mathbf{x}\_i) = \sum\_{\mathbf{x}\_i \in E} \lambda\_\mathbf{c}(E) \text{ and } A\_i = \sum\_{\mathbf{x}\_i \in E} (|E| - 1).$$


$$S(x\_i) = \frac{\deg(x\_i)}{A\_i}.$$

## 8. **end do**

Consider a fuzzy social digraph of Florientine trading families Peruzzi, Lambertes, Bischeri, Strozzi, Guadagni, Tornabuon, Castellan, Ridolfi, Albizzi, Barbadori, Medici, Acciaiuol, Salviati, Ginori and Pazzi. The vertices in a fuzzy network represent the name of trading families. The degree of membership of each family represents the strength of centrality in that network. The directed edge # » *xy* indicates that the family *x* is influenced by *y*. The degree of membership of each directed edge indicates to what extent the opinions and suggestions of one family influence the other. The degree of membership of Medici is 0.9, which shows that Medici has a 90% central position in a trading network. The degree of membership between Redolfi and Medici is 0.6, which indicates that Redolfi follows 60% of the suggestions of Medici. The fuzzy social digraph is shown in Figure 16.

**Figure 16.** Fuzzy social digraph.

To find the most influential family in this fuzzy network, we construct its fuzzy influence hypergraph. The fuzzy in neighbourhoods are given in Table 10.


**Table 10.** Fuzzy in neighbourhoods of all vertices in social networks.

The fuzzy influence hypergraph is shown in Figure 17. The degree of membership of each hyperedge shows the strength of social competition between families to influence the other trading families. The strength of competition of vertices using Algorithm 2 is calculated in Table 11, where *<sup>S</sup>*(*x*) represents the strength to which each trading family influences the other families. Table 11 shows that Acciaiuol and Medici are most influential families in the network.

**Figure 17.** Fuzzy influence hypergraph.

**Table 11.** Degree of influence of vertices.


A View of Fuzzy Competition Hypergraphs in Comparison with Fuzzy Competition Graphs

The concept of fuzzy competition graphs presented in [16,17] can be utilized successfully in different domains of applications. In the existing methods, we usually consider fuzziness in pairwise competition and conflicts between objects. However, in these representations, we miss some information about whether there is a conflict or a relation among three or more objects. For example, Figure 15 shows the strong competition for profit among SONY, LR and AK. However, if we draw the

fuzzy competition graph of Figure 14, we cannot discuss the group-wise conflict among companies. Sometimes, we are not only interested in pair-wise relations but also in group-wise conflicts, influence and relations. The novel notion of fuzzy competition hypergraphs are a mathematical tool to overcome this difficulty. We have presented different methods for solving decision-making problems. These methods not only generalize the existing ones but also give better results regarding uncertainty.
