A Method for Calculating the Association Degrees between Concepts of Concept Networks ()

Shi-Jay Chen^{}

Department of Information Management, National United University, Miaoli, China.

**DOI: **10.4236/jcc.2018.65005
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Department of Information Management, National United University, Miaoli, China.

Depicting the associating degrees between two concepts and their relationships are major works for constructing a multi-relationship fuzzy concept network. This paper indicates some drawbacks of the existing methods of calculating associating degrees between concepts, and proposes a new method for overcoming these drawbacks. We also use some examples to compare the proposed method with the existing methods for calculating the associating degrees between two concepts in a multi-relationship fuzzy concept networks.

Keywords

Document Retrieval, Fuzzy Query Processing, Geometric-Mean Averaging Operators, Fuzzy Concept Networks

Share and Cite:

Chen, S. (2018) A Method for Calculating the Association Degrees between Concepts of Concept Networks. *Journal of Computer and Communications*, **6**, 55-65. doi: 10.4236/jcc.2018.65005.

1. Introduction

Salton and Mcgill proposed information retrieval system based on the Boolean logic model [1] . Moreover, documents are retrieved only when they contain the index terms specified in the user’s queries. However, this method may be neglect some relevant documents that do not contain the index terms specified in user’s queries. Therefore, many researchers proposed intelligent information retrieval systems to retrieval documents intelligently by incorporating knowledge bases into the systems [2] - [13] . In [13] , Lucarella *et al.* presented the fuzzy concept networks for information retrieval based on fuzzy set theory [14] . The concept network can depict the relationships between concepts which are defined as index terms [15] or classes of documents [11] in a specific domain.

In [9] , Horng *et al.* proposed the method to automatically construct multi-relationship fuzzy concept networks for fuzzy information retrieval. In multi-relationship, there are four kinds of relationship to describe possible semantic relationships between concepts, such as fuzzy positive association relationship, fuzzy negative associating relationship, fuzzy generalization relationship and fuzzy specialization relationship [10] . The users of the fuzzy information retrieval system based on multi-relationship concept networks can submit a fuzzy query in which a search context is involved to provide the user’s perspective on the fuzzy relationships between concepts. Documents are retrieved if they contain concepts that have a specified fuzzy relationship with the concepts contained in the user’s query when concerning the search context. Thus, depicting the associating degrees between two concepts and their relationships are important for constructing a multi-relationship fuzzy concept network.

The rest of this study is organized as follows. Section 2 briefly reviews the concept of geometric mean, the fuzzy concept network [13] and the muti-relationship fuzzy concept network [10] . Section 3 reviews the existing methods of associating degrees between concepts for automatically constructing multi-relationship associating fuzzy concept networks, and indicates some drawbacks of existing methods for calculating associating degrees between concepts. Section 4 presents a new method for calculating associating degrees between concepts, and uses some examples to compare the proposed method with the existing methods. Conclusions are finally drawn in Section 5.

2. Preliminary

In [9] , the geometric mean of positive number ${a}_{1},{a}_{2},\cdots ,{a}_{n}$ is defined as

$\sqrt[n]{{\displaystyle {\prod}_{i=1}^{n}{a}_{i}}}=\sqrt[n]{{a}_{1}\times {a}_{2}\times \cdots \times {a}_{n}}$ , (1)

where $1\le i\le n$ . The geometric mean is well defined for sets of positive numbers, and is useful to deal with fuzzy aggregating problem and fuzzy decision-making problem.

2.1. Fuzzy Concept Networks

Lucarella *et al.* have proposed the fuzzy concept networks for fuzzy information retrieval [13] . A fuzzy concept network includes nodes and directed links. Each node represents a concept or document. Each directed link connects two concepts or directs from one concept *c _{i} *to one document

associated with a degree *μ*, where
$\mu \in \left[0,1\right]$ , indicating the degree of strength of the relationship between two concepts or the degree of strength that a document contains a concept. Figure 1 shows a fuzzy concept network, where
${c}_{1},{c}_{2},\cdots $ and c_{7} are concepts, and *d*_{1}, *d*_{2}, *d*_{3} and *d*_{4} are documents. A link in the fuzzy concept network is defined as:

$l=\left\{\langle c,r\rangle ,u\left(c,r\right)|c\in C\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}r\in C\right\}$ ,

where *C* represents the set of concepts, *u* is the membership function,
$u:C\times C\to \left[0,1\right]$ , which represents that the concept *c* and concept *r* are connected

Figure 1. A fuzzy concept network.

by the link *l*, and their relevant is *u*(*c*, *r*), where
$u\left(c,r\right)\in \left[\mathrm{0,1}\right]$ .

In the relevant value between concept* c* and concept *r* is *u*(*c*, *r*), and the relevant value between concept *r* and concept *s* is *u*(*r*, *s*). Then based on the transitivity of link relationship, we can obtain the relevant value between concept *c* and concept *s* by the following expression:

$u\left(c,s\right)=\mathrm{min}\left(u\left(c,r\right),u\left(c,s\right)\right)$ .

Similarly, if
$u\left({c}_{1},{c}_{2}\right),u\left({c}_{2},{c}_{3}\right),\cdots ,u\left({c}_{n-1},{c}_{n}\right)$ are known, then based on the transitivity of relationship, we can obtain the relevant value between concept *c*_{1} and concept *c _{n}* by the following expression:

$u\left({c}_{1},{c}_{n}\right)=\mathrm{min}\left(u\left({c}_{1},{c}_{2}\right),u\left({c}_{2},{c}_{3}\right),\cdots ,u\left({c}_{n-1},{c}_{n}\right)\right)$ .

2.2. Multi-Relationship Fuzzy Concept Networks

Kracker proposed the multi-relationship fuzzy concept network [10] . The concepts of multi-relationship fuzzy concepts are similar to the concepts of semantic networks [9] for expressing different types of relationship between keywords. Four types of relationship can be described the possible relationship between concepts in a multi-relationship fuzzy concept network as follows:

1) Positive association: It relates concepts with a fuzzy similar meaning (e.g. person—individual) in some contexts.

2) Negative association: It relates concepts with fuzzy complementary relationship (e.g. male—female), fuzzy incompatible relationship (e.g. unemployed—freelance) or fuzzy antonymous relationship (e.g. small—large) in some contexts.

3) Generalization: A concept regarded as a fuzzy generalization of another concept if it includes that concept in an analytic or partitive sense (e.g. person—student).

4) Specialization: It is the inverse of fuzzy generalization.

Let *C* be a set of concepts in a multi-relationship fuzzy concept network. The fuzzy relationships between concepts are defined as follows [10] .

1) Fuzzy positive associating *P* is a fuzzy relation,
$P:C\times C\to \left[0,1\right]$ , which is reflexive, symmetric, and max-*-transitive.

2) Fuzzy negative association *N* is a fuzzy relation,
$N:C\times C\to \left[0,1\right]$ , which is anti-reflexive, symmetric, and max-*-nontransitive.

3) Fuzzy generalization *G* is a fuzzy relation,
$G:C\times C\to \left[0,1\right]$ , which is anti-reflexive, anti-symmetric, and max-*-transitive.

4) Fuzzy specialization *S* is a fuzzy relation,
$S:C\times C\to \left[0,1\right]$ , which is anti-reflexive, anti-symmetric, and max-*-transitive.

A multi-relationship fuzzy concept network is denoted as *MRFCN *(*E*, *L*), where *E* is a set of nodes, and where represents a concept or a document as in Figure 2. *L* is a set of directed edges between nodes. If
$l\in L$ , then the directed edge *l* has following two formats:

1)
${c}_{i}\stackrel{\left(\langle {\mu}_{P},P\rangle ,\langle {\mu}_{N},N\rangle ,\langle {\mu}_{G},G\rangle ,\langle {\mu}_{S},S\rangle \right)}{\to}{c}_{j}$ , means that the directed edge *l* connect *c _{i}* to

2)
${c}_{i}\stackrel{\mu}{\to}{d}_{j}$ , means that document *d _{j}* has concept

Figure 2 shows a multi-relationship fuzzy concept network, where
${c}_{1},{c}_{2},\cdots ,{c}_{7}$ * *are concepts; *d*_{1}, *d*_{2}, *d*_{3} and *d*_{4} are documents.

Furthermore, Horng *et al.* proposed an algorithm with eight steps to construct multi-relationship fuzzy concept networks automatically [9] .

3. Analysis of the Existing Methods for Calculating the Relationships and the Associating Degrees between Concepts

In [9] , Horng *et al.* pointed out that calculating the relationships and the associating degrees between concepts is an important part of constructing a multi-relationship fuzzy concept network. They decided fuzzy relationship between two concepts by following six cases. Assume the concept *c _{i}* and the concept

Case 1: If concept *c _{i}* and concept

Case 2: If concept *c _{i}* and concept

Case 3: If concept *c _{i}* and concept

Case 4: If most words contained in concept *c _{j}* are also contained in concept

Figure 2. A multi-relationship fuzzy concept network.

Case 5: If most words contained in concept *c _{i}* are also contained in concept

Case 6: If concept *c _{i}* and concept

Young proposed a method for calculating the associating degree between concepts [16] . The proposed method uses a mapping function *M* to represent each concept by showing its corresponding fuzzy subset in the word set *WS*. The mapping function *M* shown as follows:

$M\left({c}_{i}\right)={w}_{i\text{1}}/{t}_{\text{1}}+{w}_{i2}/{t}_{2}+\cdots +{w}_{ih}/{t}_{h}$ , (2)

where
$M:C\to {\left[0,1\right]}^{W}$ , *w _{i}*

between concepts denoted *G*(*c _{i}*,

$G\left({c}_{i},{c}_{j}\right)=\{\begin{array}{l}\left(\frac{\left|M\left({c}_{i}\right)\cap M\left({c}_{i}\right)\right|}{\left|M\left({c}_{i}\right)\right|}\right)=\left(\frac{{\displaystyle {\sum}_{k=1}^{h}\mathrm{min}\left({w}_{ki},{w}_{kj}\right)}}{{\displaystyle {\sum}_{k=1}^{h}{w}_{ki}}}\right),\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}M\left({c}_{j}\right)\ne \varphi \hfill \\ 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}M\left({c}_{i}\right)\ne \varphi \hfill \end{array}$ (3)

where *w _{ki}* is the weight of word

$S\left({c}_{i},{c}_{j}\right)=G\left({c}_{i},{c}_{j}\right)$ . (4)

Moreover, based on Subsection 2.1, the degree of fuzzy positive association relationship between concept *c _{i}* and concept

$P\left({c}_{i},{c}_{j}\right)=\mathrm{min}\left(G\left({c}_{i},{c}_{j}\right),S\left({c}_{i},{c}_{j}\right)\right)$ . (5)

However, Horng *et al.* [9] founded that Young’s method cannot effectively reveal the generality of concept *c _{j}* over concept

Example 3.1: Assume that there are five words *t*_{1}, *t*_{2}, …, and *t*_{5} in the word set *W**S *and assume that the corresponding fuzzy subset *M*(*c _{i}*) and

$M\left({c}_{i}\right)=0.3/{t}_{1}+0.3/{t}_{2}+0.4/{t}_{3}+0.4/{t}_{4}+0.3/{t}_{5}$ ,

$M\left({c}_{j}\right)=0.8/{t}_{2}+0.9/{t}_{3}$ ._{ }

According to Case 4 of the above six cases for deciding fuzzy relationship between concepts, concept *c _{i}* should be more general than the concept

$G\left({c}_{i},{c}_{j}\right)=\frac{0.3+0.4}{0.3+0.3+0.4+0.4+0.3}=0.41$ ,

$G\left({c}_{j},{c}_{i}\right)=\frac{0.3+0.4}{3.8+0.9}=0.41$ .

According to the above results, we cann’t know which concept is more general than the other one.

Therefore, Horng *et al.* [9] proposed the formula (6) to overcome this drawback.

$G\left({c}_{i},{c}_{j}\right)=\{\begin{array}{l}{\left(\frac{{\displaystyle \underset{k=1}{\overset{h}{\sum}}\mathrm{min}\left({w}_{ki},{w}_{kj}\right)}}{{\displaystyle \underset{k=1}{\overset{h}{\sum}}{w}_{ki}}}\right)}^{\frac{WC\left({c}_{i}\right)}{\mathrm{max}\left(WC\left({c}_{i}\right),WC\left({c}_{j}\right)\right)}},\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}M\left({c}_{j}\right)\ne \varphi \hfill \\ 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}M\left({c}_{i}\right)\ne \varphi \hfill \end{array}$ (6)

where *w _{ki}* is the weight of word

However, we also found the formula (6) proposed by Horng *et al.* still has some drawbacks for dealing with associating degrees between concepts (*i.e.*, the result is not fitting for one of the above six cases). In the following, we use some examples to illustrate these drawbacks.

Example 3.2: Assume there are four words *t*_{1}, *t*_{2}, *t*_{3} and *t*_{4} in the word set *WS*, and assume that the corresponding fuzzy subset *M*(*c _{i}*) and

$M\left({c}_{i}\right)=0.3/{t}_{1}+0.1/{t}_{2}+0.2/{t}_{3}+0.8/{t}_{4}$ ,

$M\left({c}_{j}\right)=0.4/{t}_{1}+0.3/{t}_{2}+0.8/{t}_{3}$ ._{ }

According to Case 2 of the above six cases for deciding fuzzy relationship between concepts, concept *c _{j}* should be more general than concept

$G\left({c}_{i},{c}_{j}\right)={\left(\frac{0.3+0.1+0.2}{0.3+0.1+0.2+0.8}\right)}^{\frac{4}{4}}=0.42$ ,

$G\left({c}_{j},{c}_{i}\right)={\left(\frac{0.3+0.1+0.2}{0.4+0.3+0.8}\right)}^{\frac{3}{4}}=0.5$ .

Since *G*(*c _{j}*,

Example 3.3: Assume that there are six words
${t}_{1},{t}_{2},\cdots ,{t}_{6}$ in the word set *WS*, and assume that the corresponding fuzzy subset *M*(*c _{i}*) and

$M\left({c}_{i}\right)=0.6/{t}_{1}+0.3/{t}_{2}+0.4/{t}_{3}+0.7/{t}_{4}+0.6/{t}_{5}+1/{t}_{6}$ ,_{ }

$M\left({c}_{j}\right)=0.8/{t}_{1}+0.5/{t}_{2}+0.7/{t}_{3}+1/{t}_{4}+0.8/{t}_{5}$ .

According to Case 3 of the above six cases for deciding fuzzy relationship between concepts, concept *c _{j}* is general than

$G\left({c}_{i},{c}_{j}\right)={\left(\frac{0.6+0.3+0.4+0.7+0.6}{0.6+0.3+0.4+0.7+0.6+1}\right)}^{\frac{6}{6}}=0.6767$ ,

$G\left({c}_{j},{c}_{i}\right)={\left(\frac{0.6+0.3+0.4+0.7+0.6}{0.8+0.5+0.7+1+0.8}\right)}^{\frac{5}{6}}=0.6842$ .

Since *G*(*c _{j}*,

Example 3.4: Assume that there are seven words
${t}_{1},{t}_{2},\cdots ,{t}_{7}$ in the word set *WS*, and assume that the corresponding fuzzy subset *M*(*c _{i}*) and

$M\left({c}_{i}\right)=1/{t}_{1}+0.8/{t}_{2}+0.9/{t}_{3}$ ,_{ }

$M\left({c}_{j}\right)=0.2/{t}_{1}+0.1/{t}_{2}+0.2/{t}_{3}+0.1/{t}_{4}+0.2/{t}_{5}+0.1/{t}_{6}+0.1/{t}_{7}$ .

According to Case 4 of the above six cases for deciding fuzzy relationship between concepts, concept *c _{j}* should be more general than the concept

$G\left({c}_{i},{c}_{j}\right)={\left(\frac{0.2+0.1+0.2}{1+0.8+0.9}\right)}^{\frac{3}{7}}=0.485$ ,

$G\left({c}_{j},{c}_{i}\right)={\left(\frac{0.2+0.1+0.2}{0.2+0.1+0.2+0.1+0.2+0.1+0.1}\right)}^{\frac{4}{4}}=0.5$ .

Since *G*(*c _{i}*,

According to the above discussion, we found that formula (5) proposed by Horng *et al*. has some drawbacks for calculating the degrees between concepts. In order to obtain more accurate associating degrees between concepts for automatically constructing multi-relationship fuzzy concept networks, to develop a new method for calculating associating degrees between concepts is necessary.

4. A New Method for Calculating Associating Degrees between Two Concepts

In this section, we present a new method for calculating associating degrees between concepts based on geometric mean operator. The new method for calculating associating degrees between concepts shown as follows:

$G\left({c}_{i},{c}_{j}\right)=\{\begin{array}{l}\frac{{\displaystyle {\sum}_{k=1}^{h}\sqrt{{w}_{ki}\times {w}_{kj}}}}{2\times {\displaystyle {\sum}_{k=1}^{k}{w}_{ki}}}+{\left(0.5\right)}^{1+2\times ROUND\left(\frac{WC\left({c}_{i}\right)}{max\left(WC\left({c}_{i}\right),WC\left({c}_{j}\right)\right)}\right)},\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}M\left({c}_{j}\right)\ne \varphi \hfill \\ 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}M\left({c}_{i}\right)\ne \varphi \hfill \end{array}$ (7)

where *w _{ki}* is the weight of word

In the following, we use the examples discussed in Section 3 to compare the proposed method with existing methods.

1) If we use formula (7) to deal with Example 3.1, we can calculate the two associating degrees *G*(*c _{i}*,

$G\left({c}_{i},{c}_{j}\right)=\frac{\sqrt{0.3\times 0.8}+\sqrt{0.4\times 0.9}}{2\times \left(0.3+0.3+0.4+0.4+0.3\right)}+{\left(0.5\right)}^{1+2\times \text{ROUND}\left(\frac{5}{5}\right)}=0.44556$

$G\left({c}_{j},{c}_{i}\right)=\frac{\sqrt{0.3\times 0.8}+\sqrt{0.4\times 0.9}}{2\times \left(0.8+0.9\right)}+{\left(0.5\right)}^{1+2\times \text{ROUND}\left(\frac{2}{5}\right)}=0.82056$ .

Since *G*(*c _{j}*,

2) If we use formula (7) to deal with Example 3.2, we can calculate the two associating degrees *G*(*c _{i}*,

$G\left({c}_{i},{c}_{j}\right)=\frac{\sqrt{0.3\times 0.4}+\sqrt{0.1\times 0.3}\times \sqrt{0.2\times 0.8}}{2\times \left(0.3+0.1+0.2+0.8\right)}+{\left(0.5\right)}^{1+2\times \text{ROUND}\left(\frac{4}{4}\right)}=0.45343$ ,

$G\left({c}_{j},{c}_{i}\right)=\frac{\sqrt{0.3\times 0.4}+\sqrt{0.1\times 0.3}\times \sqrt{0.2\times 0.8}}{2\times \left(0.4+0.3+0.8\right)}+{\left(0.5\right)}^{1+2\times \text{ROUND}\left(\frac{3}{4}\right)}=0.43154$ .

Since *G*(*c _{j}*,

3) If we use formula (7) to deal with Example 3.3, we can calculate the two associating degrees *G*(*c _{i}*,

$\begin{array}{c}G\left({c}_{i},{c}_{j}\right)=\frac{\sqrt{0.6\times 0.8}+\sqrt{0.3\times 0.5}+\sqrt{0.4\times 0.7}+\sqrt{0.7\times 0.1}+\sqrt{0.6\times 0.8}}{2\times \left(0.6+0.3+0.4+0.7+0.6+1\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left(0.5\right)}^{1+2\times \text{ROUND}\left(\frac{6}{6}\right)}\\ =0.5609\end{array}$ ,

$\begin{array}{c}G\left({c}_{j},{c}_{i}\right)=\frac{\sqrt{0.6\times 0.8}+\sqrt{0.3\times 0.5}+\sqrt{0.4\times 0.7}+\sqrt{0.7\times 1}+\sqrt{0.6\times 0.8}}{2\times \left(0.8+0.5+0.7+1+0.8\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left(0.5\right)}^{1+2\times \text{ROUND}\left(\frac{5}{6}\right)}\\ =0.5380\end{array}$ .

Since *G*(*c _{i}*,

4) If we use formula (7) to deal with Example 3.4, we can calculate the two associating degrees *G*(*c _{i}*,

$G\left({c}_{i},{c}_{j}\right)=\frac{\sqrt{0.1\times 0.2}+\sqrt{0.8\times 0.1}\times \sqrt{0.9\times 0.2}}{2\times \left(0.1+0.8+0.9\right)}+{\left(0.5\right)}^{1+2\times \text{ROUND}\left(\frac{3}{7}\right)}=0.71376$ ,

$\begin{array}{c}G\left({c}_{j},{c}_{i}\right)=\frac{\sqrt{0.3\times 0.4}+\sqrt{0.1\times 0.3}\times \sqrt{0.2\times 0.8}}{2\times \left(0.2+0.1+0.2+0.1+0.2+0.1+0.1\right)}+{\left(0.5\right)}^{1+2\times \text{ROUND}\left(\frac{7}{7}\right)}\\ =0.70216\end{array}$ .

Since *G*(*c _{i}*,

From the previous discussions, we can obtain the proposed method is useful than the two existing methods proposed by Young and Horng *et al.* respectively for calculating the associating degrees between two concepts for deciding their relationship in a multi-relationship fuzzy concept network.

5. Conclusion

In this paper, we firstly pointed out some drawbacks of the existing methods for calculating the associating degree between two concepts, and presented a method based on geometric mean operator for overcoming these drawbacks. We used some examples to compare the proposed method with the existing methods. The proposed method is more useful than the existing methods to calculate the associating degrees between two concepts for constructing their relationship in a multi-relationship fuzzy concept networks for document retrieval.

Acknowledgements

This work was supported in part by the Ministry of Science and Technology, under Grant 104-2410-H-239-007.

Chen, S.-J. (2018) A Method for Calculating the Association Degrees between Concepts of Concept Networks. *Journal of Computer and Communications*, 6, 55-65. https://doi.org/10.4236/jcc.2018.65005

References

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- 2. Bezdek, J.C., Biswas, G. and Huang, L.Y. (1986) Transitive Closures of Fuzzy Thesauri for Information-Retrieval System. International Journal of Man-Machine Studies, 25, 343-356. https://doi.org/10.1016/S0020-7373(86)80065-7
- 3. Bhatia, S.K. and Deogun, J.S. (1998) Conceptual Clustering on Information Retrieval. IEEE Transactions on Systems, Man, and Cybernetics—Part B, Cybernetics, 28, 427-435. https://doi.org/10.1109/3477.678640
- 4. Chang, C.S. and Chen, A.L.P. (1998) Supporting Conceptual and Neighborhood Quiries on the World Wide Web. IEEE Transactions on Systems, Man, and Cybernetics—Part B, Cybernetics, 28, 300-308. https://doi.org/10.1109/5326.669578
- 5. Chen, C.L.P. and Lu, Y. (1997) FUZZY: A Fuzzy-Based Concept Information System That Integrates Human Categorization and Numerical Clustering. IEEE Transactions on Systems, Man, and Cybernetics—Part B, Cybernetics, 27, 79-94. https://doi.org/10.1109/3477.552187
- 6. Chen, S.M. and Horng, Y.J. (1999) Fuzzy Query Processing for Document Retrieval Based on Extended Fuzzy Concept Networks. IEEE Transactions on Systems, Man, and Cybernetics—Part B, Cybernetics, 29, 96-104.
- 7. Chen, S.M., Horng, Y.J. and Lee, C.H. (2000) Fuzzy Information Retrieval Method Based on Multi-Relationship Fuzzy Concept Networks. Proceedings of the 2000 International Computer Symposium: Workshop on Artificial Intelligence, Chiayi, 6-8 December 2000, 79-86.
- 8. Chen, S.M., and Wang, J.Y. (1995) Document Retrieval Using Knowledge-Based Fuzzy Information Retrieval Techniques. IEEE Transactions on Systems, Man, and Cybernetics, 25, 793-803.
- 9. Horng, Y.J., Chen, S.M. and Lee, C.H. (2003) Automatically Constructing Multi-Relationship Fuzzy Concept Networks for Document Retrieval. Applied Artificial Intelligence, 17, 303-328. https://doi.org/10.1080/713827141
- 10. Kracker, M. (1992) A Fuzzy Concept Network Model and Its Applications. Proceedings of the First IEEE International Conference on Fuzzy Systems, San Diego, 8-12 March 1992, 761-768. https://doi.org/10.1109/FUZZY.1992.258752
- 11. Liang, T. and Chang, C.C., (1999) Chinese Textual Retrieval Based on Fuzzy Concept Networks. Proceedings of National Computer Symposium, Tamsui, 20-22 December1999, 61-67.
- 12. Lin, C.C., Tseng, S.Y. and Chen, P.M (1999) A Fuzzy Document Retrieval System Based on Concept Networks and Cluster Analysis. Soochow Journal of Economics and Business, 25, 39-60.
- 13. Lucarella, D. and Morara, R. (1991) FIRST: Fuzzy Information Retrieval System. Journal of Information Science, 17, 81-91. https://doi.org/10.1177/016555159101700202
- 14. Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
- 15. Kim, K.J. and Cho, S.B. (2001) A Personalized Web Search Engine Using Fuzzy Concept Network with Link Structure. Proceedings of the Joint 9th IFSA Congress and 20th NAFIPS International Conference, Vancouver, 25-28 July 2001, 81-86.
- 16. Young, V.R. (1996) Fuzzy Subsethood. Fuzzy Sets and Systems, 77, 371-384. https://doi.org/10.1016/0165-0114(95)00045-3

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Salton, G. and Mcgill, M.J. (1983) Introduction to Modern Information Retrieval. McGraw-Hill Education, New York. |

[2] | Bezdek, J.C., Biswas, G. and Huang, L.Y. (1986) Transitive Closures of Fuzzy Thesauri for Information-Retrieval System. International Journal of Man-Machine Studies, 25, 343-356. https://doi.org/10.1016/S0020-7373(86)80065-7 |

[3] | Bhatia, S.K. and Deogun, J.S. (1998) Conceptual Clustering on Information Retrieval. IEEE Transactions on Systems, Man, and Cybernetics—Part B, Cybernetics, 28, 427-435. https://doi.org/10.1109/3477.678640 |

[4] |
Chang, C.S. and Chen, A.L.P. (1998) Supporting Conceptual and Neighborhood Quiries on the World Wide Web. IEEE Transactions on Systems, Man, and Cybernetics—Part B, Cybernetics, 28, 300-308. https://doi.org/10.1109/5326.669578 |

[5] |
Chen, C.L.P. and Lu, Y. (1997) FUZZY: A Fuzzy-Based Concept Information System That Integrates Human Categorization and Numerical Clustering. IEEE Transactions on Systems, Man, and Cybernetics—Part B, Cybernetics, 27, 79-94. https://doi.org/10.1109/3477.552187 |

[6] | Chen, S.M. and Horng, Y.J. (1999) Fuzzy Query Processing for Document Retrieval Based on Extended Fuzzy Concept Networks. IEEE Transactions on Systems, Man, and Cybernetics—Part B, Cybernetics, 29, 96-104. |

[7] | Chen, S.M., Horng, Y.J. and Lee, C.H. (2000) Fuzzy Information Retrieval Method Based on Multi-Relationship Fuzzy Concept Networks. Proceedings of the 2000 International Computer Symposium: Workshop on Artificial Intelligence, Chiayi, 6-8 December 2000, 79-86. |

[8] | Chen, S.M., and Wang, J.Y. (1995) Document Retrieval Using Knowledge-Based Fuzzy Information Retrieval Techniques. IEEE Transactions on Systems, Man, and Cybernetics, 25, 793-803. |

[9] | Horng, Y.J., Chen, S.M. and Lee, C.H. (2003) Automatically Constructing Multi-Relationship Fuzzy Concept Networks for Document Retrieval. Applied Artificial Intelligence, 17, 303-328. https://doi.org/10.1080/713827141 |

[10] | Kracker, M. (1992) A Fuzzy Concept Network Model and Its Applications. Proceedings of the First IEEE International Conference on Fuzzy Systems, San Diego, 8-12 March 1992, 761-768. https://doi.org/10.1109/FUZZY.1992.258752 |

[11] | Liang, T. and Chang, C.C., (1999) Chinese Textual Retrieval Based on Fuzzy Concept Networks. Proceedings of National Computer Symposium, Tamsui, 20-22 December1999, 61-67. |

[12] | Lin, C.C., Tseng, S.Y. and Chen, P.M (1999) A Fuzzy Document Retrieval System Based on Concept Networks and Cluster Analysis. Soochow Journal of Economics and Business, 25, 39-60. |

[13] |
Lucarella, D. and Morara, R. (1991) FIRST: Fuzzy Information Retrieval System. Journal of Information Science, 17, 81-91. https://doi.org/10.1177/016555159101700202 |

[14] | Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X |

[15] | Kim, K.J. and Cho, S.B. (2001) A Personalized Web Search Engine Using Fuzzy Concept Network with Link Structure. Proceedings of the Joint 9th IFSA Congress and 20th NAFIPS International Conference, Vancouver, 25-28 July 2001, 81-86. |

[16] | Young, V.R. (1996) Fuzzy Subsethood. Fuzzy Sets and Systems, 77, 371-384. https://doi.org/10.1016/0165-0114(95)00045-3 |

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