Analysis indices of human relationships based on fuzzy model

Abstract

The purpose of our study is to evaluate an analysis indices and fuzzy decision method for human relationships using fuzzy models. Many studies have been conducted on human relationship analysis in various fields which are sociology, psychology, information science and so on. Most popular analysis method in sociology is sociometry analysis. It was proposed by Moreno. Sociometry analysis is a general method for social network analysis and this method is used in various studies. Sumida has detailed results by studying a group of elementary school students from the perspective of pedagogy. Yamashita et al. proposed a fuzzy model for human relationships based on sociometry analysis. Yamashita’s method is generated a partition tree from survey question results and to analyze human relationships based on the partition tree. When sociometry analysis is carried out, there is the disadvantage that it depends on the analyst. As preconditions for analysis, the nodes belonging to the same similarity in the partition tree have already been analyzed. In previous studies, the relationships among the groups belonging to different similarity levels was not specified. Therefore, we have newly developed an analytical method to solve the problem based on Yamashita’s research. In order to evaluate these new analysis indices and fuzzy decision method, we used one example of data and its results. As a result of verification, we were able to obtain results equivalent to those done by experts and confirmed the usefulness of this new method.

Keywords

Analysis for human relationships sociometry fuzzy theory partition tree fuzzy decision

1. Introduction

Many studies have been conducted on human relationship analysis in various field which are sociology, psychology, information science and so on. Most popular analysis method in sociology is sociometry analysis and it was proposed by Moreno [8]. In sociometry analysis, the analyst conducts a questionnaire survey and analyzes based on the survey results from the subjects selected. The results are represented by a diagram called a sociogram. A sociogram is a graph representation of the obtained data. An analyst analyzes sociograms. The method used for analysis differs depending on how the analyst wants to grow the group. In sociometry analysis, one result (sociogram) is obtained for one question. Sociometry analysis is a common analytical method in the field of sociology and pedagogy. However, in sociometry analysis, analysts mostly subjectively analyze on their own, based on their own knowledge. Therefore, analysts are required to have a very high level of knowledge about the field to be analyzed and can-not easily analyze. Yamashita et al. applied fuzzy theory to sociometry analysis [2, 11, 5, 4, 6]. Yamashita’s method consolidated the results obtained from multiple questions into one fuzzy graph. There are differences between the two methods. Sociometry analysis analyzes one-way relationships between subjects. On the other hand, Yamashita’s analysis method analyzed mutual relationships between subjects.

As a result, sociometry analysis revealed superficial consciousness, whereas Yamashita’s method revealed the subconscious mind of the subject indirectly by consolidating multiple question results. The edge value is 0 (non-connection) or 1 (connection) in the sociogram, whereas it is between 0.0 and 1.0 in the fuzzy graph.

Yamashita’s method provided basic methods for automatically analyzing subgroups, opinion leaders, and isolated members. However, human relationships are complex, and analysts are not satisfied only with the results obtained from a quantitative analysis method. For this reason, the method used for analysis differs depending on the purpose of the analyst. Depending on the purpose of the analyst, a directed graph like the sociogram may be more effective or the undirected graph may be more desirable. Therefore, there are both merits and demerits to using either method. Therefore, we aim to provide a new foundation that can analyze both of the two analysis methods.

Also, Yamashita’s method analyzes human relations based on the partition tree. As a precondition for analysis, only nodes belonging to the same similarity level in the partition tree are targets for analysis. Also, the optimal degree of similarity is calculated for which similarity is to be analyzed, and only the optimal similarity is the target of analysis [6]. We also aim to eliminate preconditions and provide a method that can analyze nodes belonging to any degree of similarity. In addition, we will compare the results of analysis using sociometry analysis and the analysis results using fuzzy theory. We evaluate the effectiveness of the analysis method applying fuzzy theory.

This paper is organized as follows: in Section 2 we introduce an analysis method for human relationships, while Section 3 explains the fuzzy model for human relationships. In Section 4, we propose an advanced analysis method for group relationships in human relationships and describe various analysis indices for fuzzy graphs. Section 5 presents a case study and the description of some results obtained through the proposed method. The results of comparative evaluation with sociometry analysis are also shown. Finally, conclusions and plans for future work are presented in Section 6.

2. Analysis method for human relationships

Various research studies have been made applying human relationships [3, 12]. Therefore, it is very important to analyze human relationships. Various analysis methods for human relationships have been proposed. In this section, we introduce existing analysis methods and their problems.

2.1 Sociometry analysis

Sociometry analysis is a measurement method proposed by Moreno as a way of measuring and evaluating the group structure of human relations using tables and figures [8]. Human society consists of mutually related groups and the group consists of social, economic and cultural ties between individuals. Sociometry analysis focuses on structural analysis in smaller groups, for example, classrooms and workplaces, rather than society as a whole. For analysis, the analyst uses data obtained by asking people to be surveyed: “Who will you choose as a friend (or colleague, advisor, etc.)?”. A graph representation of the obtained data is called a sociogram. In the sociogram, individuals are represented as nodes and selective relationships are represented by an arc.

An example of a sociogram is shown Fig. 1 [17]. Figure 1 shows a relationship between the first and second selected persons, who selected their preferred partners for dinner. The solid line represents the first and the dotted line represents the second selected persons.

Figure 1.

An example of sociogram.

2.2 Observation method and interview method for human relationships

In analyzing human relationships, the most appropriate method for grasping the boundaries of peer group membership and analyzing internal structure is the observation and interview method [9]. In the observation method, the analyst observes the activity of the group for a certain period and confirms the existence of the group by direct external observation. Then, analysts interview each member and check the group boundaries.

From the results of the survey using the observation and interview method, it is possible to analyze the internal structure of the peer group as shown below.

•
The moment and timing of group formation
•
The purpose of the group
•
The moment of joining the group
•
Emotional relationship to other members

2.3 Problems of existing analysis methods

In sociometry analysis, it is possible to analyze and evaluate group structure using only data obtained from simple questions. Sociometry analysis is a method of measuring and analyzing human relations and group structure. Therefore, it is not desirable for grasping the group itself or for clarifying the boundaries of group membership.

Analysis by the observation and interview method allows more detailed analysis than sociometry analysis, such as analysis of inner group structure. On the other hand, the fact that it takes a long time to investigate is a demerit. In a research study on the structure of a peer group of children by Sumida [9], it required approximately 2 weeks to obtain the results of the observation method investigation, and about 5.5 months to obtain results from the survey using the interview method. It takes a long time just for the survey, and presumably, even more time to include analysis. Also, in analysis using the observation and interview method, it is difficult to analyze unless the analyst possesses sophisticated knowledge, such as a specialist in sociology might have. Other methods include extracting interpersonal relationships using motion picture processing [13, 10]. In this method, it is necessary to prepare places and equipment for recording as motion pictures, and it is necessary to consider that the psychological state that results from being recorded affects subjects’ behavior.

In contrast, for human relations analysis applying fuzzy graphs, it is possible to analyze human relationships in a short time using a short question survey.

3. Fuzzy model for human relationships

In the sociometry analysis, human relationships are evaluated using tables and figures. However, it is not easy to show their features clearly using this method. Also, with the observation and interview method, it is difficult to analyze easily due to the precondition that it requires much time and advanced expertise. By applying fuzzy graph theory to the sociogram, human relationships can be quantitatively analyzed [2, 11, 5, 4, 6, 18].

3.1 Fuzzy graph model

Human relationships within the group can be analyzed using data obtained from a simple questionnaire. It is possible to analyze human relationships rationally and more clearly within the group by constructing fuzzy models from results obtained from a simple questionnaire and applying fuzzy clustering and fuzzy similarity.

3.2 Response table and response matrix

If we ask on the $M$ questionnaire to $L$ students: “Write your friendly $N(1\leqslant N\leqslant L)$ members in the order of those with whom you would like to do activities $Ap(1\leqslant p\leqslant M)$ ”, then we have a response table. Analyzing this table, we have the response matrix $K^{\left(p\right)}=\left(k^{\left(p\right)}_{ij}\right)$ . $k^{\left(p\right)}_{ij}$ is the ranking value $k(0\leqslant k\leqslant N)$ in which a student $S_{i}$ has selected another student $S_{j}(i,j=1,\ldots,L)$ . However, $k=n-m(0\leqslant k\leqslant N)$ , $m=$ (number of order) $-$ 1 for each activity, and $n$ is defined as follows:

$\displaystyle n=\left[\frac{\sum n_{i}}{ML}+0.5\right]$ (1)

Where $n_{i}$ is the sum of numbers by which $S_{i}$ selects his friendly members for $M$ activities, [ ] is a gaussian symbol.

Examples of questions are as follows.

•

Please write your $N$ friends in the order of those with whom

\left(Q1\right)

you would like to study in group

\left(Q2\right)

you would like to take lunch together

\left(A3\right)

you would like to do club activities

\left(A4\right)

you would like to do volunteer activities

3.3 Evaluation matrix

Analyzing the matrix $K^{\left(p\right)}=\left(k^{\left(p\right)}_{ij}\right)$ , we could define the evaluation matrix $R=\left(r_{ij}\right)$ as follows.

$\displaystyle r_{ij}=\sum_{p}\left(n-k^{\left(p\right)}_{ij}+1\right)$ (2)

3.4 Fuzzy matrix and fuzzy graph

Analyzing the matrix $R=\left(r_{ij}\right)$ , we could define the fuzzy degree of “friendliness” and that of “preference” among the members within the group. Let $f_{ij}=r_{ij}/nM$ , and we consider $f_{ij}$ as the degree of fuzziness to which a member $S_{i}$ prefers the member $S_{j}$ . We then have the fuzzy cluster matrix $F=\left(f_{ij}\right)$ , $0\leqslant f_{ij}\leqslant 1$ , where $f_{ij}=1$ if $i=j$ [1].

On the other hand, a fuzzy graph is defined mathematically as $G=\left(V,F\right)$ , where $V=\{v_{i}\}$ is the set of nodes and $F=\left(f_{ij}\right)$ is the relation matrix; where $f_{ij}$ is the value of the arc from $v_{i}$ to $v_{j}$ . A graph is called a fuzzy graph if its value is between 0.0 and 1.0. The fuzzy matrix $F$ is equivalent to a fuzzy graph. As observed above, we could build a fuzzy model representing the connective structure among the members in a group.

3.5 Cluster analysis for fuzzy graph

Let $2/s_{ij}=1/f_{ij}+1/f_{ji}$ , and we consider $s_{ij}$ as the degree of fuzziness to which two members $S_{i}$ and $S_{j}$ are mutually friendly. Thus, we have the fuzzy matrix $S=\left(s_{ij}\right)$ , $0\leqslant s_{ij}\leqslant 1$ , and $s_{ij}=0$ if $f_{ij}f_{ji}=0$ (Because it can not be divided if the denominator is zero). Here, $s_{ij}\sim 1$ means that $S_{i}$ and $S_{j}$ are very friendly, and $s_{ij}\sim 0$ means that they are not very friendly. In order to analyze the similarity structure of nodes for a fuzzy graph, we use the symmetric relational matrix $S=\left(s_{ij}\right)$ and compute it for transitive closure $\widehat{S}=\left(\widehat{s_{ij}}\right)$ [5]. After calculating transitive closure of the symmetric relational matrix, we define the c-cut matrix as follows.

$\displaystyle S^{c}=\left(s^{c}_{ij}\right),s^{c}_{ij}=\left\{\begin{array}[]{% ll}1,&\text{for }\widehat{s_{ij}}\geqslant c\\ 0,&\text{for }\widehat{s_{ij}}<c\\ \end{array}\right.$ (3)

The relation $R_{c}$ of the c-cut matrix $S^{c}$ gives a similarity of relations among nodes. Thence, we have the partition tree (dendrogram) by changing level $c$ of the c-cut matrix, which represents the clustering situation of nodes in a fuzzy graph. A value of level $c$ is defined by an element of the transitive closure. We call it a similarity level $R_{c}$ or, hereafter, a level $c$ . The partition tree is produced by analyzing transitive closure as shown in Fig. 2. In this way, partition trees’ clustering situation and the branching process among members are produced by cluster analysis using $\widehat{S}$ and its c-cut matrices.

Figure 2.

Partition tree from the question survey results.

Figure 3.

Number of clusters $p(z)$ and size of clusters $q(z)$ .

3.6 Fuzzy decision method

There are multiple similarity levels in the partition tree. It is necessary to decide the similarity level to be used in the analysis. As a method of deciding the to similarity level, various methods were proposed. Shinkai proposed a method applying the fuzzy decision [6].

The number and size of clusters in similarity level $R_{z}$ could be defined as $x(z)$ and $y(z)$ concerning the partition tree. The value of $y(z)$ is defined by the size of the power mean cluster of similarity level $R_{z}$ . Power mean $\tau_{n}$ is defined as follows.

$\displaystyle y_{i}>0(i=1,2,\ldots,n)$

(4) $\displaystyle\tau_{n}=\left(\frac{1}{n}\sum_{i=1}^{n}y_{i}^{n}\right)^{\frac{1% }{n}}$

As similarity level $R_{z}$ approaches 1.00, $x(z)$ monotonically increases. On the other hand, $y(z)$ monotonically decreases. The evaluation function $p(z)$ of the number of clusters and the evaluation function $q(z)$ of the size of clusters could be defined as follows:

$\displaystyle p(z)=\frac{x(z)-x(0.00)}{x(1.00)-x(0.00)}\in[0,1]$ (5) $\displaystyle q(z)=\frac{y(z)-y(1.00)}{y(0.00)-y(1.00)}\in[0,1]$ (6)

There is a trade off relationship between $p(z)$ and $q(z)$ that is shown Fig. 3. Therefore, the fuzzy decision of the optimal Interval $z_{I}$ and the optimal similarity level $R_{z^{}}$ is defined as follows:

$\displaystyle z_{I}=\{z|p(z)\wedge q(z)=\max_{0\leqslant i\leqslant 1}{(p(i)% \wedge q(i))}\}$ (7) $\displaystyle z^{}=\max\{z|p(z)\wedge q(z)=\max_{0\leqslant i\leqslant 1}{(p(% i)\wedge q(i))}\}$ (8)
4. Advanced analysis method for group relationships in human relationships

There are various indices for analyzing graphs and networks. Besides general analytical indices, there are also indices specialized for the analysis of human relationships. The target fuzzy graph to be analyzed is defined mathematically as $G=(V,S)$ , where $V=\{v_{i}\}$ is a set of nodes and $S=\left(s_{ij}\right)$ is a symmetric relation matrix; where $s_{ij}$ is the value of the edge between $v_{i}$ and $v_{j}$ .

In this section, we introduce the analysis method in which we proposed an advanced analysis method for group relations based on the fuzzy graph and described various indices for analyzing the fuzzy graph [7, 16, 14].

4.1 Fuzzy graph density

This is an indicator of closeness and complexity in a relationship. The existing index is the number of links divided by the number of links when all nodes are connected. Generally, edge values of graphs and networks are represented by 0 (no connection) or 1 (connection). However, in fuzzy graphs, edge values range from 0.0 to 1.0. Therefore, the fuzzy density ( $F D$ ) considering the edge value is defined as follows:

$\displaystyle FD=\frac{\sum_{v_{i},v_{j}\in V}s_{ij}}{|V\times V|}$ (9)

4.2 Node reachability

This identifies that there is a reachable route from one node to another node. It is possible to identify an isolated member ( $I S$ ) not belonging to a group in a human relationship using the following formula:

$\displaystyle IS=\min_{1\leqslant i\leqslant n}\left\{\sum_{j}^{n}s_{ij}\right\}$ (10)

4.3 Degree Centrality

Measures centrality from the number of edges connected to the node and identifies the node with the widest relationship to other nodes. Degree Centrality ( $D C$ ) is defined as follows:

$\displaystyle DC=\max{(d(v))}$ (11)

4.4 Similarity betweenness centrality [16]

The similarity between centrality (SBC) [14] in the two clusters $CL_{A}$ and $CL_{B}$ is defined by the following formula:

$\displaystyle v_{i}\in CL_{A},|CL_{A}|=m$ $\displaystyle v_{j}\in CL_{B},|CL_{B}|=n$ (12) $\displaystyle\textit{SBC}=\max_{1\leqslant i\leqslant m}\left\{\sum_{j}^{n}s_{% ij}\right\}$

Thus, in the route from one cluster to another cluster, the degree of contribution to the route with the highest degree of association is specified.

4.5 Analysis method for group relationships

In the fuzzy graph, a group is replaced as one node (hereafter referred to as a cluster node).

At this time, the number of elements (cluster size) $|CL_{A}|\geqslant 2$ of the cluster $CL_{A}$ . Therefore, to define the relationship between a node and a cluster node, it is necessary to aggregate the relationships in two or more existing nodes into one. There are various methods for aggregating two or more relationships.

When aggregating is based on the average, setting arbitrary cluster nodes $CL_{A}$ and $CL_{B}$ , the set of node pairs belonging to each cluster node is $C$ . At this time, the relationship CLR between the two cluster nodes is defined as follows [14]:

$\displaystyle C=CL_{A}\times CL_{B}:\{(i,j)|i\in CL_{A},j\in CL_{B},CL_{A}\neq CL% _{B},s_{ab}>0.0\}$ $\displaystyle\textit{CLR}=\left\{\begin{array}[]{ll}\frac{\displaystyle\sum_{i% =1}^{|CL_{A}|}\sum_{j=1}^{|CL_{B}|}s_{ij}}{|C|},&\text{for }|C|>0\\ 0,&\text{for }|C|=0\\ \end{array}\right.$ (13)

4.6 Relationship analysis for groups belonging to different similarity levels

In human relationship analysis applying the fuzzy graph, a partition tree is used. In the partition tree, it is clustered for each similarity level. The existing analysis method is a relation analysis of nodes belonging to the same similarity level. When the similarity level of the partition tree changes, the cluster structure also changes according to the similarity level. When analyzing the relationship between clusters, one cluster size may be too large or too small. Therefore, by analyzing two clusters belonging to different similarity levels, it is possible to analyze them with the same cluster size. Let the clusters to be analyzed be $CL_{A}$ , $CL_{B}(A_{R}\cup B_{R^{\prime}}=\phi)$ , with cluster similarity levels $R$ , $R^{\prime}$ . At this time, the relationship between the two clusters can be calculated by the mathematical formula equivalent to the mathematical expression defined by Eq. (4.5).

$\displaystyle C=A_{R}\times B_{R^{\prime}}:\{(i,j)|i\in A_{R},j\in B_{R^{% \prime}},s_{ij}>0.0\}$ (14) $\displaystyle\textit{DCR}=\left\{\begin{array}[]{l}\frac{\displaystyle\sum_{i% \in A_{R},j\in B_{R^{\prime}},A_{R}\cap B_{R^{\prime}}=\phi}s_{ij}}{|C|}\\ \quad\text{for }s_{ij}>0.0\\ 0,\quad\text{for }s_{ij}=0.0\\ \end{array}\right.$

Figure 4.

Fuzzy graph (similarity level $=$ 0.68).

5. Case study

In this section, we analyze using a fuzzy graph model based on the collective question survey from 21 students in an elementary school class which was carried out by Sumida [9].

5.1 Question survey

M. Sumida conducted the question surveys for 21 elementary school students. Considering that it is a collective question survey in elementary school, he excluded negative selection/nomination criteria. The 8 questions used for question survey are shown below.

1.
Please write your favorite friend’s name in this class. There is no limit on the number of selections.
2.
Please write the name of a friend who always plays with you. There is no limit on the number of selections.
3.
Please write the name of a friend who played with you after you returned home yesterday. There is no limit on the number of selections.
4.
Who are the group members with whom you play together in this class? Please write the names of all group members.
5.
Who are your friendly group members in this class? Please write the names of all group members.
6.
Please write the name of a friend who always plays with you after you go home. There is no limit on the number of selections.
7.
Who are the group members with whom you play together after going home? Please write the names of all group members.
8.
Who friendly group members with whom you are friendly after going home? Please write the names of all group members.

Figure 5.
SBC from cluster 89 to cluster 90.

5.2 Results of analysis

In this section, we will show the result of analyzing the question survey with 21 students.

The partition tree created from the response matrix from the above question survey results is shown in Fig. 2.

Also, Fig. 4 that we show is a fuzzy graph showing similarity level $R=0.68$ based on Fig. 2. Squares represent cluster nodes that are subgroups, and circles represent nodes that are individuals. The right hand side in Fig. 4 shows all nodes that cluster nodes divided into individual nodes. Figure 4 shows that there are four groups and six members not belonging to groups in this class.

5.2.1 Analysis of SBC

The results of SBC analysis from cluster 89 to cluster 90 are shown in Fig. 5. From Fig. 5, it is possible to give the greatest influence by mediating node 1.

5.2.2 Analysis of CLR

The results of CLR on the two clusters 89 and 90 of Fig. 4 are shown in Fig. 6. According to the analysis result of CLR, the relationship between the two clusters was 0.35. From this result, it can be read intuitively that these two clusters have slightly friendly relations.

Figure 6.

CLR between cluster 89 and 90.

Figure 7.

DCR between cluster 2 and 90.

Figure 8.

DCR between node 1 and cluster 90.

Figure 9.

Fuzzy graph ( $R_{z^{*}}=0.56$ ).

5.2.3 Analysis of DCR

Two clusters (89, 90) do not indicate friendly relations. We will further analyze the relationship between these two clusters. The similarity level of cluster 89 changes from level 0.68 to 1.00. Cluster 89 is divided into the node 1, 9 and cluster 2. Figures 7 and 8 show the result of DCR on divided clusters or nodes and cluster 90. DCR between node 9 and cluster 90 is 0.00, and the analysis result screen is omitted. DCR between cluster 2 and cluster 90 is 0.25. DCR is lower than in the case of clusters of the same similarity level. Therefore, it turns out that these two clusters truly do not get along well. Also, DCR between nodes 9 and cluster 90 is 0.00, which shows that they do not get along as well as cluster 2. On the other hand, DCR between node 1 and cluster 90 is 0.56, which shows that they have a certain degree of friendly relations.

5.3 Consideration

In this subsection, we consider optimal similarity level and analysis results.

The value of the $p(z)$ and $q(z)$ are calculated by the fuzzy decision method shown Table 1.

Table 1
The value of evaluate function $p(z)$ and $q(z)$

$z$	$x(z)$	$y(z)$	$p(z)$	$q(z)$	$p(z)\wedge q(z)$
0.00	1	21.00	0.00	1.00	0.00
0.22	2	14.16	0.06	0.63	0.06
0.33	4	10.61	0.18	0.44	0.18
0.56	5	10.15	0.24	0.41	0.24
0.63	7	4.25	0.35	0.09	0.09
0.64	9	4.26	0.47	0.09	0.09
0.68	10	4.26	0.53	0.09	0.09
0.73	12	4.07	0.65	0.08	0.08
0.75	13	4.11	0.71	0.08	0.08
0.88	15	3.34	0.82	0.04	0.04
0.93	16	3.36	0.88	0.04	0.04
1.00	18	2.56	1.00	0.00	0.00

By the calculated result, $z_{I}$ and $z^{*}$ are obtained as follows:

$\displaystyle p(z)\wedge q(z),z_{I}=(0.33,0.56]\rightarrow z^{*}=0.56$

Figure 9 that we show is afuzzy graph showing optimal similarity level $R_{z^{*}}=0.56$ . The right hand side in Fig. 9 shows all nodes that cluster nodes divided into individual nodes.

The clustering result of $R_{z^{*}}$ and Sumida’s clustering result are significantly different. However, the clustering results of $R_{0.63}\sim R_{0.68}$ are similar to Sumida’s result. This shows that the same results as the clustering done by experts can be obtained by a quantitative method. Therefore, it is necessary to establish a better method for deciding the optimal similarity level from the partition tree.

On the other hand, we use the similarity level $R_{0.68}$ that is similar to the clustering result by Sumida. According to the analysis in Fig. 6, the relationship is 0.35. However, when the similarity level is changed from 0.68 to 1.00, the relationship with subgroup 90 is 0.56. Therefore, when analyzing with subgroup 89 and 90, there were both members with very good friends and members who were not such good friends. By changing the similarity level, it is possible to analyze clearly which subgroup has an intimate relationship in more detail. Also, analysis of the relationship between subgroups and individuals has been dependent on analysts to this point. This analysis result showed that it can analyze quantitatively through aggregating the relationship by average.

Furthermore, we compared our analysis result with Sumida’s result [15]. About the analysis method, Sumida performed clustering of friendly relationships using not only the question survey but also the observation method and the interview method. However, his clustering model is undefined. Therefore, the quality of analysis results depends on the analysts. Sumida analyzed class mates using the results of a question survey, the observation method and the interview method, and members 1, 3, 4, and 5 are analyzed as one group. Analyzing only the results of the question survey, member 9 was also selected from group members (1, 3, 4, 5). However, the observation method showed that member 9 did not participate in group play. Therefore, member 9 is not included in the group members. It was a false recognition of the members that member 9 was selected as a group member even though they were not playing together. Sumida confirmed this through the interview method. Therefore, it was judged an incorrect result that member 9 was a group member, and member 9 was excluded. However, although member 9 does not belong to the group, s/he dose have a mutual selection relationship with other group members. Based on this, Sumida concluded that they have an intimate relationship.

On the other hand, the number of group members is 5 (1,3,4,5,9) from the partition tree in our analysis system. Members belonging to the group are different from Sumida’s analysis result. Sumida judged this to be an incorrect result from the participation degree of actual play and excluded member 9 from the group. However, in the question survey, there was the response that s/he was a member of the group, and this was a mutual selection relationship. Therefore, it is considered that no great error occurred in clustering as a clustering of friendly relations.

For other groups, clustering results similar to Sumida’s clustering result could be obtained. From that, it can be considered that there is reliability equivalent to the clustering results of experts with advanced knowledge.

In addition, we developed a method to identify the degree of influence among groups and influence of people using SBC. Sumida does not do the same research and analysis as these. It is an analysis method that has not been done in other studies which use sociometry analysis. It is possible to decide the route for efficiently communicating information to other groups and people using SBC. Furthermore, by repeatedly analyzing the SBC for the group or person who received the information, it is considered possible to find a route that can efficiently transmit information in a secondary and tertiary manner. We focused on subgroups existing in the group and divided the group into subgroups to analyze more detailed human relationships.

6. Concluding remarks

In this paper, we evaluate analysis indices and a fuzzy decision method to analyze human relationships using the question survey result by Sumida [9]. The clustering results of $R_{z^{*}}$ and Sumida’s clustering results are significantly different. However, we could obtain results equivalent to those done by experts and confirmed the effectiveness of our proposed method by using similarity level $R_{0.68}$ . We defined relationships on nodes and clusters belonging to different similarities and verified their effectiveness. Furthermore, we showed various analytical indices, and some of them were expanded to indices adapted to fuzzy graphs. Furthermore, we compared and evaluated the analysis results based on the question survey by Sumida and confirmed that our analysis method is effective. On the other hand, our proposed method focuses on analysis, and does not consider the intersection of edges and labels in drawing. Therefore, visibility may be severely impaired, for example, when the number of analysis targets increases. In future studies, we will propose a drawing algorithm for placement of edges and labels and an application for mapping data of multi-layers applying the fuzzy graph.

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Analysis indices of human relationships based on fuzzy model

Abstract

Keywords

1. Introduction

2. Analysis method for human relationships

2.1 Sociometry analysis

• The moment and timing of group formation • The purpose of the group • The moment of joining the group • Emotional relationship to other members 2.3 Problems of existing analysis methods

3. Fuzzy model for human relationships

3.1 Fuzzy graph model

3.2 Response table and response matrix

3.5 Cluster analysis for fuzzy graph

4.1 Fuzzy graph density

5.1 Question survey

5.2.1 Analysis of SBC

5.2.2 Analysis of CLR

5.3 Consideration

Table 1 The value of evaluate function p ⁢ ( z ) and q ⁢ ( z )

References

•
The moment and timing of group formation
•
The purpose of the group
•
The moment of joining the group
•
Emotional relationship to other members

2.3 Problems of existing analysis methods

Table 1
The value of evaluate function $p(z)$ and $q(z)$