Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 524328, 12 pages

http://dx.doi.org/10.1155/2015/524328

## Research on Conflict Behavior of Top Management Team in Family Enterprises: A Complex Network Perspective

^{1}School of Finance & Economics, Jiangsu University, Zhenjiang 212013, China^{2}School of Management, Jiangsu University, Zhenjiang 212013, China

Received 12 March 2015; Accepted 2 July 2015

Academic Editor: Xavier Delorme

Copyright © 2015 Mengyun Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

At present time, methods of research about top management team’s conflict become more and more prosperous with the help of complex system theory and evolutionary game. Taking family enterprise as an example, this paper makes an attempt on exploring complex network modeling to study data processing method and abstraction method of complex network of TMT conflict. And the paper will consider the attribute and relational mapping of top management team as nodes and edges in complex network to discuss the direct correspondence between complex network structure and management team characteristics. Besides that, according to the multiple attribute decision making, the method to dig into core members of the top management team will be created on the basis of the degree, closeness, cluster coefficient, and betweenness. And then the article will devote to studying the impact of attributes to the inner mechanism of TMT conflict and team cohesion through the network characteristic analysis.

#### 1. Introduction

In the era of knowledge-based economy society, as one of the decisive factor to get sustained viability and greater market development of family enterprises, conflict of top management team attracts more and more attention. Social investigation and mathematical statistics are usually adopted as basic tools in traditional research to discuss TMT (top management team) conflict and analyze the relationship between team performance and the motivation. In recent years, there is little progress in the description of nonlinear properties, microscopic mechanism, and interaction process of the TMT conflict. Different from the overall top-down research method, this paper tries to make use of grounded theory research method to describe attributes, microscopic behavior, and macrobehavior of TMT conflict with tools of complex network and multiagent complex system model. And finally the author will discuss the process and mechanism of team conflict and its influence on the cohesion and team performance from different perspectives.

#### 2. Analysis on Static Network Characteristics of TMT Conflict in Family Enterprises

##### 2.1. Network Relational Mapping of TMT in Family Enterprises

According to self-organization characteristics of family enterprises, TMT members’ constitution has the characteristics of knowledge complementarities, special affiliation subordination, and specific properties of demography and task other than normal organization. As a result, the paper considers individual members in family enterprises as different network nodes. Furthermore, the relationship between member and member will be treated as the connection of nodes. On the basis of it, we can describe the relationship among team members with the help of network diagram. This paper will reveal the intrinsic link between team members’ personal and relational information effectively and find out the effective mechanism of how the link influences the conflict through further analysis.

TMT members in family enterprise play important roles of conflict body in decision making and program delivering. The network analysis and abstraction of conflict should be based on the attribute extraction of team members [1]. The attributes of key members that affect the conflict will be shown by a collection of [2, 3]; we can define . And various properties of team members are listed by using enumeration. For example, can be used to describe the relationship attributes of TMT members, and 0 means relationship is not so good, 1 means just ordinary, 2 means good, and 3 means relatives; similarly, when is used to refer to education, it shows that 0 refers to primary education, 1 refers to high school education, 2 refers to college education, and 3 refers to graduate and doctoral education.

What a team should focus on are task and relationship. Thus, the internal conflict relationship among TMT members under the real conditions contains task conflict () and relationship conflict () [1, 4]. These two different relationship types can be shown by the finite set , and their corresponding color set is which is denoted as . In addition, by using a quad , the paper will describe the relationship network of TMT members in detail.

suggests network node set. The elements of shown by ordinal values represent TMT members. And the value of set determines the size of the team. In other words, more nodes means more complicated team relationships.

refers to coloring schemes which are corresponding to relationships. The color of red means task conflict () and green represents relationship conflict ().

is a set which refers to members relationship. The manifestation of task conflict () in this set is color edge, that is, from member to member , which means and are connected. Meanwhile, the manifestation of relational conflict () is bidirectional color edge between and , which represents interpersonal relationship of member and member .

is the attribute value matrix for TMT members and its element means attribute values of TMT member .

Therefore, this network can be defined in the form of the adjacency matrix as formula (1) below:

In formula (1), indicates the number of color edges whose length is 1 and it means nodes and are connected. And colors and are coloring vectors. It is worth noting that and .

Due to the basic concept of graph theory [5, 6], the finite sequence is called the th path from to . Therefore, we can count as a path whose color is from to in the TMT relationship network. If each side and each node of path are all different, this path will be called Hamilton [7]. And the existence of in the network means that that nodes and have relationship with color. The path with shortest length which connects and is noted as the shortest path .

##### 2.2. The Analysis of Network Attributes of TMT in Family Enterprises

If we can promptly determine the key personnel who definitely influence cohesion of TMT and focus on how to deal with them it will be possible to reduce the potential conflicts and improve TMT efficiency [8]. Based on the accomplishment of complex network mapping of TMT, the paper chooses four properties indexes, which consist of degree, closeness, cluster coefficient, and betweenness, and tries to set up correlation between characteristics of TMT and specialty of complex networks. Finally, by analyzing these four properties, we can efficiently obtain nonlinear information and situation of TMT network and digest core information about TMT conflict effectively [9].

###### 2.2.1. Network Attributes of Top Management Team

*(1) Degree*. Degrees can be defined as the numbers of edges connected to the node , which are the basic characteristics of the network nodes. And, in this paper, the degree of TMT relational network can be seen as vector sum of different color edges associated with . ConsiderIn formula (2), we consider the numbers of different color edges which are connected with as color , and they will be calculated with the following formula: With the help of adjacency matrix properties of order simple undirected graph , the main diagonal elements of matrix can be expressed asWe can know easily that is the degree of corresponding node .

The importance of the node can be shown through the value of nodes degrees. In other words, the node with high degrees in the network means the member has a complex interpersonal relationship with others in the task and human communication. So these members should be skilled in the management experience, communication skills and technical level; otherwise lacking of all these skills might result in the task or interpersonal conflict easily [10]. So the node can be considered as the core reference characteristics of potential conflict in top management team.

*(2) Closeness*. We use closeness to measure the centric degree of nodes according to shortest path. And closeness refers to the reciprocal of the sum of all the shortest distances from node to others. If are assumed to be the numbers of sides contained in the paths which started from point and ended in the terminal point , the closeness of nodes can be described as

Based on formula (5), the closeness of TMT network can be known as a vector which consisted of two kinds of colors. And the closeness of color of node can be shown by following formula:

The central location of every node in the network depends on the differences of closeness. The larger closeness means more important position. Communication will be more smooth, task transfer will be more convenient, and the conflict will be reduced greatly when network distance becomes small in the task and interpersonal interaction among the TMT members. Therefore, if the closeness of a TMT member is large, he/she will be in the center of the network. His/her strong cohesive force can fully and effectively mobilize and utilize human resources of the team; thus it is beneficial to improve the efficiency of TMT in family enterprise [11, 12].

*(3) Cluster Coefficient*. The relevance characteristics of social capital network are obvious in some sense. For example, your two different acquaintances may also by coincidence be familiar with each other. This kind of network structure which is known as integration can be described as cluster coefficient quantitatively, and cluster coefficient means the connection probability two nodes which are all have connections with another node in the network. Correspondingly, the member with high cluster coefficient will have more complex business relationships and closer human relationships in TMT network. If is assumed to be a collection of nodes which are connected with node in TMT network, we have the ability to calculate the cluster coefficient of TMT network according to the definition of cluster coefficient:

*(4) Betweenness*. Freeman (1977) [13, 14] proposed the concept of betweenness can be considered as global network characteristics. Usually, we use betweenness to identify and measure the effect and influence of nodes or edges in the network. Generally, the high value of betweenness of a member’s node indicates that his/her interpersonal interaction channels are also relatively close. As a result, he/she can influence the team performance greatly. The betweenness of a Node can be defined as the number of the shortest paths which go through node between all nonadjacent nodes and . Based on this definition, nodes’ betweenness of color network of TMT can be expressed as

It is important to note that shows the number of the shortest paths between and , and represents the number of the shortest paths with color which go through node .

###### 2.2.2. Data Processing of Characteristic Attributes

Degree, closeness, cluster coefficient, and betweenness, four characteristic properties which are commonly used to describe TMT network, are usually observed and measured by value of different colors. And due to their vector characteristics, the author has to make further processing before analyzing the core nodes of network. In TMT network, we use and to show the weight of task conflict (conflict arising from task) and interpersonal conflict (conflict arising from personal relationship), respectively. The weight vector can be denoted as . will hold without thinking about other factors in the team work. When the family enterprise mainly pays attention to business strategic orientation and task management, the value of will be relatively higher; while the family enterprise focuses more on the interpersonal interactions and relationships, the value of usually will be relatively higher.

If is assumed as a particular attribute value of one company, we can obtain that . On the basis of all the above, we can construct a matrix , which consists of degree, closeness, cluster coefficient, and betweenness. And standardization is essential with the help of following formula because of different dimension of these attributes.

After calculation with formula (9), we are able to obtain matrix which is standardized from matrix . In addition, the canonical matrix also needs normalization processing to simplify calculation. Consider

Finally we can obtain the normalized matrix .

##### 2.3. The Method to Mine Core Nodes of TMT Relation Network in the Family Enterprise

The team leadership which is a key factor to influence the team cohesion determines that the author should look for the core nodes of TMT network. System science [15] believed that the importance of network nodes is equivalent to its destructiveness, so we should observe how the node removal influences network connectivity based on system function. However, social network research [16] thought that the importance of the node is equivalent to its significance. So we should collect the effective information in the network under the condition of network connectivity to analyze the differences among the nodes so as to count and calculate the degree and betweenness of nodes and finally confirm quantitatively the importance of nodes in the network.

###### 2.3.1. The Sequence of Node Weight

According to the definition and analysis of the degree, we could know that if you want to estimate the importance of a node, it is best to measure it from the perspective of global network. Besides, the value of the degree of adjacent nodes is equally important. It is assumed that refers to a collection of nodes which are adjacent to node , and can be seen as the weight between node and its adjacent node . It is easy to infer from these definitions that in a simple undirected network. In this formula is defined as the degree of node , means the degree of node , and is a control parameter. Based on it, the node weight of node can be expressed as

If it is difficult to estimate the importance of the relationship between two nodes; we can assume the value of is 0. According to the degree of nodes it can be easy to calculate vertex weight. It is known to be -node. The node weight of TMT network includes two dimensions which are work-related network and human network. Therefore, we are able to find out position features and importance of each node in different relational network.

###### 2.3.2. The Sequence of Multiple Attribute Decision Making

Four properties which are degree, closeness, cluster coefficient, and betweenness can be used to elaborate the characteristics of the network nodes from different sides. So we can establish characteristic attribute value matrix , standardization matrix , and normalization matrix to evaluate the importance of TMT network node under complicated environment based on Multiple Attribute Group Decision Making Theory. Ordered weighted geometric (OWG) operator described as will be used for collecting attributes. In this formula, means maximum element of a column’s data in the matrix , and refers to the index weighting vector which connects with . In addition, what we need to note is the fact that and the value of is 1. So we can get :

On the basis of all the above, four values of weight , and which will respect the degree, closeness, cluster coefficient, and betweenness, respectively, can be obtained. And we have ability to get an order in matrix according to the following formula:

If we can obtain the order relationship with the help of formula (13), it will be easy to sort these network nodes and find the position attributes of a node consisting of degree, closeness, cluster coefficient, and betweenness. Then, it is possible for us to find out key members of TMT, analyze the position of each member in the network, and explore the underlying cause of the conflict.

The members of Family Business’s TMT have individual differences and their behavior is dynamic, while the individual members have the abilities of self-learning and self-adaption and are able to adjust their behavioral decision making based on their own experience and the interaction with the other members in the later work; thus, the conflict between team members is complex management problems; it has a nonlinear characteristic. This study pays attention to making a comparison between the different values of sequence calculated through node weight sorting and multiple attribute decision making sorting to identify and measure the position and role of the members in the team.

#### 3. The Calculation Model and Experimental Analysis

##### 3.1. The Model for Calculating

According to the definition of the attributes and operator algorithm, we can divide the calculation model into several steps. Firstly, draw TMT network graph and calculate characteristic attribute of two color vectors. Secondly, construct attribute value matrix by the weights of two colors (red and green); these two colors represent task conflict and relationship conflict, and also construct attribute value matrix by the weights of two colors. Thirdly, through normative approach and normalization processing, establish the standard matrix and the normalized matrix. Finally, count the value of weight and sort the attribute value, and precisely describe the key nodes [17, 18]. The calculation model to judge and look for the key node of top management team network is shown in Figure 1.