Modeling and Numerical Methods of Supply Chain Trust Network with the Complex Network

Abstract

Finding reliable partners is the key to supply chain management. However, the symmetrical evaluation of enterprise trust is complex, so the decision-makers must understand its quantitative and qualitative characteristics in order to realize a reasonable evaluation. Based on the analysis of the causes and influencing factors of supply chain trust, this paper constructed four primary indexes and 16 secondary indexes to define enterprise trust, and used analytic network process (ANP) to evaluate and rank the indicators. Then, the paper constructed a supply chain directed weighted trust evolution network model based on complex network theory, integrated trust into the network with edge weights, and put forward the merit index of comprehensive node degree, weight, and efficiency to study the supply chain network evolution. The simulation results show that the node degree distribution in the trust evolution network conforms to the power-law distribution rule, and the trust evolution model of the complex network has obvious scale-free characteristics, which effectively avoid the situation that the node influence is too high due to the excessive strength of a single index. At the same time, it can quickly evaluate the node influence of the directed weighted complex network, and provide certain practical value for the node trust prediction of the supply chain network.

1. Introduction

In recent years, the development of science and technology has further deepened the degree of information globalization and economic globalization, and the importance of supply chain management has also increased. The supply chain is a huge value chain with high complexity and high dynamics. Because of the large number of members in the chain and frequent conflicts and cooperation, the operation process is full of risks and uncertainties, which increase the difficulty of selecting good partners. Inadvertently, “partners” may also cause huge losses to enterprises [1]. Having a good cooperative relationship is not only a major boost for enterprises to stabilize their operations and increase their profits but also an important driving force for enterprises to reduce costs, increase efficiency, innovate and develop [2]. Therefore, the study of enterprise trust is particularly important for the supply chain. Therefore, the study of enterprise trust is particularly important for the supply chain. Research of the supply chain trust relationship is of great significance to enterprise operation and supply chain management. In supply chain management, decisions are usually made on the basis of trust. Evaluating the trust degree of supply chain partners can play a controlling role, which is conducive to the rational choice of enterprises and provides a better choice for cooperative enterprises.

The supply chain network is an open and dynamic complex network. Supply chain complexity is an important part of supply chain management, which is mainly represented by structural complexity, operational complexity, and decision-making complexity. Structural complexity describes the physical relationships among supply chain members [3]. With the expansion of the supply chain scale and the increasing complexity of the network, the overall structure of the supply chain is getting more and more attention. The supply chain is an upstream and downstream enterprise model, which usually consists of three or more levels. A multi-level supply chain has become one of the research hotspots. Different types of enterprises constitute different supply chain structures, and also play a role in connecting upstream and downstream. Each level is not only the information provider, but also the information demand. Defining the role of node enterprises contributes to the decision-makers of the supply chain to understand and identify the way and tightness of the connection between node enterprises in the whole supply chain, identify the position of each node enterprise in the supply chain, and make reasonable adjustments and effective allocation of resources in the supply chain. The multi-layer network model can accurately reflect the multi-attribute characteristics of node connection [4], and the topological connection among levels also reflects the position of node enterprises in the supply chain to a certain extent. However, the supply chain is a very complex system, and taking the whole supply chain as the research object is facing tremendous difficulties. In order to simplify this problem, this paper selected the local supply chain as the research object. Although the simplified model can enlighten people to a certain extent, the two-level supply has the defects of oversimplification, unable to fully reflect the cohesion of levels, and it is difficult to accurately describe the supply chain structure. In contrast, the three-tier structure simplifies the model while avoiding the above-mentioned shortcomings. At the same time, the three-tier structure is also considered to be suitable for the study of multi-tier supply chain problems [5,6]. Thus, this paper chooses the three-tier supply chain model as the research subject.

Complex network theory is recognized as an effective method to study complex supply chains, which is beneficial to develop a deep understanding on the structure of supply chain networks from the perspective of network structure. At present, most studies simply use BA, BBV, or other networks to simulate the influence of parameter changes on the overall network evolution. Trust is an essential factor in the research of supply chain operation and decision-making complexity. At present, the trust evaluation of supply chain node enterprises is still a challenging problem in the supply chain operation process. There is no consistent standard for trust evaluation scholars, and there is a relative lack of research on trust network evolution. In addition, many scholars at home and abroad have analyzed the competition and cooperation between enterprises in the supply chain network from the perspective of complex network and its development and evolution law, but most of the research is at the initial stage, and the specific application method of trust network in supply chain management has not been deeply discussed. In view of this research gap, the following research questions are put forward: (1) How to construct a supply chain trust evaluation index system in a complex supply chain? (2) Research on the evolution model of the preferred trust supply chain based on the combination of partner trust assessment and complex network theory. (3) There are certain dependencies among interconnected nodes in the supply chain. In addition to considering the influence of the enterprise’s own characteristics, trust is also influenced by the network connection structure. Therefore, the dynamic reliability evaluation of joints should also refer to structural characteristics.

Many researchers apply the ANP method and complex network to supply chain management respectively, but few scholars apply them to partner selection. Based on the complex network theory and ANP method, this paper constructed a supply chain network evolution model to study the influence of enterprise characteristics and network structure characteristics on supply chain trust evolution. Firstly, the ANP method is used to build a trust evaluation index which is used to evaluate the trust between enterprises. Secondly, in the complex network model, supply chain enterprises are regarded as nodes in the network, and trust between enterprises is defined as the weight of connecting edges between nodes. Finally, a preferred evolution mechanism combining node degree, weight, and network efficiency are proposed, and this mechanism is applied to supply chain network simulation. The main contributions this study makes are as follows:

(1)

The causes and influencing factors of supply chain trust are analyzed in detail, and the trust evaluation model is constructed by network analysis;

(2)

There is little literature on the evolution mechanism of trust networks with the development of network scale from the perspective of complex network structure. This paper fills the blank of research on the evolution direction of trust in complex networks;

(3)

With the expansion of complex network scale, the network topology also changes. A single metric has certain defects when judging the importance of nodes. Given this, a merit-based mechanism considering both structure and node influence is proposed, which provides an effective method for node trust measurement;

(4)

The directional characteristics of trust and the directional weighted network are combined to further distinguish the influence of side and power directional. Research shows that under the optimal connection mechanism, the trusted network has a scale-free attribute and is more discriminatory in the judgment of trust value than the powerless network.

The remainder of the paper is organized as follows, the second part is the literature review, the third part introduces the trust evaluation model and trust evolution model in detail, the fourth part is data simulation, and the fifth part is the summary and prospect.

2. Literature Review

2.1. Research on Supply Chain Trust

The effect of trust in supply chain has received a great deal of attention from scholars. Hou et al. [7] studied the influence of priority trust rules, priority price rules, and priority stochastic rules on supply chain networks by using dynamic multi-agent and multi-stage models, and found that priority trust rules are more beneficial to the supply chain to obtain higher profits. Ojiha et al. [8] developed and tested a proposed supply chain organization design model, which proved that trust plays an important role in the development and innovation of supply chain enterprises. Shen et al. [9] studied the importance of contract execution and goodwill trust in supply chain enterprises at different stages of the relationship life cycle. Hou et al. [10] constructed a dynamic multi-agent supply chain simulation model, which determined suppliers according to trust, selling price, and random selection, respectively, and analyzed the influence of enterprise trust on the supply chain structure and its elasticity, proving the trust between enterprises played an important role in the long-term cooperation.

All the above documents affirm the role of trust as the cornerstone of cooperation and selection in the supply chain. Apart from the advantages of complementary advantages, cost reduction, and efficiency enhancement, the trust also provides a certain flexible space for cooperation between the two parties, making the cooperation relationship more stable. Researchers at home and abroad agree that supply chain trust measure is an important part of the supply chain network. Many experts and scholars have studied the selection and evaluation of supply chain partners, among which the quantification and prediction of supply chain trust indicators have become the focus of scholars’ attention. The analytic network process and fuzzy evaluation methods [11] are applied to the construction of the supply chain trust evaluation index system. The research of Akkermans [12] shows that the characteristics of cooperative enterprises have a certain influence on the choice of trust behavior in cooperation; Hail et al. [13] used quantitative and qualitative methods to study the influence of partner types on trust in the supply chain, which showed that public customers paid more attention to the frequency and effectiveness of communication, while private customers paid more attention to ability. Literature [14] is based on trust and reputation model, and proposes a multi-criteria decision-making method based on variable weight and satisfaction principle. Simulation results show that the proposed trust and reputation model can effectively filter unfair ratings from those customers who did lie, and the proposed multi-criteria decision-making method can help customers make the right decisions. The research of Nold [15] shows that the elements of organizational culture, especially trust, enable enterprises to turn knowledge and learning plans into tangible performance recognized by financial markets. Nowicka’s research shows that the type and scope of shared information affect the trust level [16]. The evaluation of corporate trust involves a variety of methods and influencing factors, and a good partner is a prerequisite for a successful supply chain. Therefore, a reliable partner evaluation method is crucial to supply chain management.

Without a foundation of trust, collaborative alliances can neither be built nor sustained [17]. Reasonable analysis and selection methods can help enterprises to deal with diversification problems and improve supply chain performance [18]. There are fuzzy evaluation methods, AHP, ANP, and other methods for the evaluation method of supply chain trust. Compared with AHP, which does not take into account the interaction between different decision-making levels or the same level, the ANP method which allows interdependent feedback between indicators has more advantages and is widely used in supply chain evaluation. Patil proposed a fuzzy analysis network process (ANP) method, which was used to select the best KM strategy to build elastic SC [19]. Chung integrates the features of ANP and IPA to establish a green supplier selection and guidance mechanism [20]. Mubarik applied GDANP method to supplier selection, and made an empirical study on the automobile industry in emerging markets [21]. The analytic network process (ANP) is utilized as a popular multi-criteria decision-making technique to provide effective decision-support models [22,23].

2.2. Complex Network Theory

Complex network theory has become an acknowledged effective tool for supply chain research. With the help of complex network theory, scholars have focused on risk propagation [24] and cascading failure [25,26], and other aspects have made great achievements. From the perspective of social networks, Galaskiewicz [27] proved that the supply chain network with small-world characteristics is beneficial to enhance the trust of small groups, which coincides with the conclusions of Capaldo and Giannoccaro’s research [28,29]. Studies have shown that the topology of trust network is very important for optimizing its trust perception [30].

At present, most of the research results of supply chain network are the directed weighted supply chain network [31] and layered supply chain network [32]. Li [32] proposed a layered supply chain weighted complex network model, which effectively described the topological characteristics and formation and evolution mechanism of the real network of agricultural product supply chain network; Wei [33] evaluates the supplier evaluation system from two indicators—risk and greenness—and uses the evaluation values as weights to build a directed weighted network to study the partner selection problem. Jiang et al. [34] proposed a global optimization algorithm based on a three-level supply chain network and solved the optimal solution problem by using the simulated annealing and gradient descent method. Tang [35] established a three-level supply chain model composed of manufacturers, collaborative design enterprises, and customers to study the scheduling problem among members. Wang-Mlynek [6] studied the multi-tiered supply chain risk management problem based on the case study of the automotive and civil aircraft industry.

The influence of nodes in network structure is the foundation and hotspot of complex network research and it is also one of the indicators of credibility evaluation. Most of the methods based on network topology structure do not consider the attribute information between nodes, while network structure has certain limitations in expressing node information. Traditional algorithms for the evaluation of node importance mainly compare the changes in network performance before and after node removal. As a matter of fact, the removal of nodes may lead to network fragmentation and inaccurate prediction [36]. Aiming at the shortcomings of the node deletion method, contraction method, and intermediate method, Liu et al. [37] considered the global structure of the network and constructed an evaluation index based on the degree of nodes and nearest neighbor information. Zhao et al. [38] proposed a novel method that takes into account not only the importance of itself but also the influence of all nodes in the graph into consideration, and verified the effectiveness of the method on six kinds of networks. Liu et al. [39] proposed a generalized mechanical model that combines global information with local information to evaluate networks. All the methods mentioned above prove that the evaluation results conducted by an integrated method for determining the local and global importance of nodes are more accurate. Meng et al. [40] studied the effect of complex network structure changes on node importance based on URT networks and proposed a multi-attribute decision method, revealing that the development of node importance is affected by the changes in topology and passenger flow. Therefore, it is necessary to make specific analysis in combination with the actual situation, and the literature summary is shown

Table 1. Literature summary.

	Researcher	Major Findings
Trust perspective	Hou et al. [7]	Using dynamic multi-agent and multi-stage model to study the influence of trust mechanism on supply chain network
	Ojha et al. [8]	A supply chain organization design model was established to study the role of trust and learning in developing entrepreneurship and innovation supply chains
	Shen et al. [9]	Research on contract–trust relationship by quasi-longitudinal analysis
	Hou et al. [10]	Use an agent-based method to characterize the supply chain network as a complex adaptive system
	Hail et al. [12], Niu et al. [13], Nold [14], Nowicka [16]	The characteristics of enterprises and other factors have a certain relationship with the trust behavior of enterprises in cooperation
	Patil et al. [19], Chung et al. [20], Mubarik et al. [21], Moons et al. [22], Yücenur et al. [23]	Analytic network process (ANP) is utilized as a popular technique to provide effective decision support for the supply chain
Multi-tier perspective	Wang-Mlynek [6]	Studied the multi-tiered supply chain risk management problem
	Galaskiewicz et al. [27]	Trust is central in most theories of social network effectiveness
	Capaldo et al. [28]	The interdependence pattern has a significant moderating effect on the relationship between trust and supply chain performance
	Capaldo et al. [29]	Influence of supply chain interdependence structure on network-level trust
	Yuan et al. [30]	Optimize the rating prediction mechanism of the conventional trust-aware recommender system
	Silva et al. [31]	A directional weighted supply chain network is constructed based on Brazilian company data
	Li et al. [32]	Layered agri-food supply chains weighted complex network model
	Wei et al. [33]	Studies the partner selection problem from the perspective of supplier network global optimization
	Jiang et al. [34]	Study the balance between cost and delivery time in the three-level supply chain of complex networks
	Tang et al. [35]	Studies the scheduling problem among three-level supply chain members
Node importance perspective	Liu et al. [37]	Improved an important node identification algorithm based on the structural hole and K-shell decomposition algorithm
	Zhao et al. [38]	Identifying influential nodes in complex networks from global perspective
	Liu et al. [39]	A generalized mechanical model is proposed that uses global information and local information
	Meng et al. [40]	A multi-attribute decision-making method based on URT network is proposed

.

Based on the above literature, this paper constructs a three-level supply chain model, and studies the evolution of trust network based on the characteristics of trust and supply chain network structure.

3. Supply Chain Trust Network Evolution Model

The supply chain network is a typical complex network consisting of suppliers, manufacturers, distributors, retailers, consumers, and other entities. Its complexity is reflected in the structure, mutual influence, and interaction. As the supply chain network develops and expands, the network environment is becoming increasingly complex. To simplify the analysis of the dynamic trust evolution process of a three-tier supply chain consisting of only suppliers, manufacturers, and retailers, the sets of suppliers, manufacturers, and retailers are respectively regarded as $S$, $M$, and $R$, and the satisfying conditions are as follows:

$$\begin{gathered} ①S\cap M\cap R=\varnothing \\ ②S\cup M\cup R=V \\ ③S\leftrightarrow M\leftrightarrow R \end{gathered}$$

The supply chain is a model of upstream and downstream enterprises. Considering the fact that enterprises at the same level are generally competitive in reality, this paper assumes that the cooperative relationship in the supply chain only occurs between enterprises at adjacent levels and is connected in the order of supplier–manufacturer–retailer, i.e., the trust relationship between node enterprises in the network only exists between enterprises at adjacent levels, and there is no cross-level cooperation. In addition, the supply chain network is a two-way weighted network, and there must be two different directional edges between two nodes with edges, which reflects the directional characteristics of trust. The edge weights represent the trust between enterprises.

3.1. The Analytic Network Process

Trust has different meanings in different supply chain environments. Therefore, quantifying trust and determining the appropriate factors to evaluate supply chain trust is a very tedious task. The trust factors tracked are not exactly the same from different perspectives, but there is a certain degree of homogeneity [41]. This paper reviews the relevant trust research literature, summarizes and constructs trust-related indicators, and determines the index weight through network analysis. The analytic network process (ANP) is the most comprehensive framework. The analytic network process allows interdependence and feedback among various standards. Such feedback best captures the complex effects of interplay in human society, especially when risk and uncertainty are involved [42]. Therefore, this paper uses the ANP method to study the trust problem of a complex supply chain.

Trust is defined as an expectation that the other party can meet the transaction requirements in the future cooperation between the subject enterprise and the object enterprise in supply chain cooperation. In this paper, the trust between supply chain nodes is integrated into the network model with edge weights. In the process of constructing the evaluation index model, we refer to the relevant literature at home and abroad, and strive to seek a reliable basis for the trust evaluation of supply chain partners. By analyzing and summarizing relevant literature and questionnaire survey, it is concluded that there are 4 primary evaluation indexes and 16 secondary evaluation indexes. The framework of supply chain trust evaluation indicators is shown in

Table 2. Trust evaluation index system.

Goal	Primary Index	Secondary Index
Trust evaluation $G$	Enterprise characteristics $R_1$	Capability
		Scale
		Region
		Nature
	Characteristics of cooperative enterprises $R_2$	Capability
		Interpersonal relationship
		Importance of products
		Reputation
	Cooperative comprehensive evaluation $R_3$	Profitability
		Cooperative frequency
		Communication initiative
		Information sharing level
		Interest relevance
	Trust characteristics $R_4$	Entrepreneurial characteristics
		Existing trust level
		Third-party trust

.

The main indicators in Table 1 are as follows:

• Enterprise characteristics $R_1$

Enterprises must fully understand their situation, and make clear their capabilities and advantages, so as to make appropriate decisions. Therefore, the enterprise feature module selects enterprise capability $r_{11}$, scale $r_{12}$, region $r_{13}$, and nature $r_{14}$ as secondary indicators. Capability $r_{11}$: An enterprise’s capabilities include technical capabilities, market development capabilities, innovative capabilities, and other capabilities, which all affect the competitiveness of an enterprise. Enterprises are more inclined to choose partners with a strong ability to cooperate [17]. Scale $r_{12}$: The scale of an enterprise affects its competitiveness and risk tolerance. Region $r_{13}$: Regional development, local culture, overall environment, and policies affect the trust of enterprises, and geographical location also affects the development of trust [43]. Nature $r_{14}$: Enterprises of different natures, such as state-owned enterprises, private enterprises, and listed companies have different advantages, and the nature of enterprises affects the trust of enterprises.

• Characteristics of cooperative enterprises $R_2$

Cooperative characteristics refer in particular to the characteristics of candidate enterprises. The cooperative enterprise’s capability and interpersonal relationship have a significant impact on enterprise decision-making. Capability $r_{21}$, interpersonal relationship $r_{22}$, importance of products $r_{23}$, and reputation $r_{24}$ are selected as secondary indicators. Capability $r_{21}$: Most supply chains are centered around core enterprises, and enterprises in the supply chain are distinguished by their capabilities. Generally, the enterprises that master the core technologies have more power to speak and can even affect the income distribution of both parties. Therefore, enterprises in the supply chain pay attention to the balance between fairness and development when choosing cooperative enterprises. Interpersonal relationship $r_{22}$: China is a typical human society. Interpersonal relationship among enterprises has an impact on long-term cooperation and trust between organizations. At the same time, a good interpersonal relationship has a certain moderating effect on the supply chain vulnerability [44]. Importance of products $r_{23}$: The importance of products are evaluated based on three aspects: product status, substitutability, and the impact on enhancing the competitiveness of enterprises. Reputation $r_{24}$: Reputation is inversely proportional to the probability of taking opportunistic actions. Compared with enterprises with low reputations and high reputations, enterprises with high reputations are more trustworthy. In addition, factors such as enterprise ability, scale, and risk tolerance affect the reputation of enterprises [45].

• Cooperative comprehensive evaluation $R_3$

Apart from the above inherent characteristics of enterprises, the performance in past cooperation also affects the important factors of enterprise trust. The profitability $r_{31}$, cooperation frequency $r_{32}$, communication initiative $r_{33}$, information sharing degree $r_{34}$, and interest relevance $r_{35}$ are selected as the secondary indicators. Profitability $r_{31}$: The stronger the profitability, the greater the possibility of subsequent cooperation and mutual trust. Cooperation frequency $r_{32}$: The more contacts or cooperation between the two parties in multiple markets [46], the higher the cooperation frequency, indicating the higher the trust between enterprises. The communication initiative $r_{33}$ and the degree of information sharing $r_{34}$ reflect the degree of communication and information sharing between the two parties [47]. Interest correlation $r_{35}$: In the information age, there is a general interest correlation between enterprises [48]. The higher the correlation, the more willing they are to spend more financial and material resources to maintain the relationship between the two parties, and the higher the degree of trust they have.

• Trust characteristics $R_4$

Entrepreneur characteristics $r_{41}$: Entrepreneur characteristics are crucial to the survival and success of an enterprise. Entrepreneurs who have a sense of self-efficacy, confidence in solving problems, trust in their ability to accept challenges, and a high level of self-confidence have a high probability of success. Similarly, the credibility of an enterprise is higher [49]. Existing trust level $r_{42}$: Existing trust refers to trust derived from past cooperation and interpersonal relationship, and can be regarded as computational trust and relational trust. The higher the trust, the higher the possibility of future cooperation. Third-party trust $r_{43}$: It integrates the trust values of enterprises to be evaluated from other enterprises, and the objective data of the third party can provide a certain reference for enterprise decision-making.

In order to increase the scientific of the index weight ratio, the expert evaluation method is used to determine the relevance and relative importance of indexes. Subsequently, according to the dependency relationship between indicators, the enterprise trust evaluation system was constructed using Super Decisions software. As shown in Figure 1, the mutual influence exists not only among the secondary indicators within the same group but also among the secondary indicators across the element group level.

3.2. Supply Chain Trust Network Evolution Model

Table 3. Key indicators table.

Key Indicators	Explanation
KI	The value of the in-degree of node I
KO	The value of the out-degree of node I
UI	The set of nodes pointing to node $i$
UO	The set of nodes pointing from $i$
SI	The sum of the edge weights that this node points to other nodes
SO	The sum of the edge weights of other nodes pointing to this node
wij	The weight of the directed edge i→j
W	Trust matrix
Ii	The efficiency value of the node i

is the key index table of the supply chain trust network evolution model, and the indicators are described in detail below.

3.2.1. Key Indicators

Node Degree

Node degree refers to the number of neighbors with this node. In a complex network supply chain, node degree is regarded as one of the indexes to evaluate the influence and importance of nodes. Node degree in a directed network is divided into in-degree and out-degree. Taking the manufacturer node as an example, the in-degree value reflects its purchasing ability, and the out-degree value reflects its sales channel and customer resources. In a complex supply chain network, the degree represents the sum of the number of enterprises that establish a relationship with the enterprise and reflects the influence of the node. Generally speaking, users with a large number of neighbors have greater social influence.

The value of adjacency matrix e represents the trust relationship, the value of $e_{ij}$ equals 1 represents the existence of trust relationship between nodes $i$ and $j$, and the value of $e_{ij}$ equals 0 represents the absence of trust relationship between nodes $i$ and $j$. $UI\left(i\right)$ represents a set of nodes in the network whose edges point to node $i$, $UO\left(i\right)$ represents the set of other nodes pointed by node $i$. $KO$ represents the out-degree of node $i$, and $KI$ represents the in-degree of node $i$.

$$KI(i)={\displaystyle \sum _{j\in UI(i)}{e}_{ji}}$$

(1)

$$KO(i)={\displaystyle \sum _{j\in UO(i)}{e}_{ij}}$$

(2)

Edge Weight

The weights of connected edges represent the relationship and reliability of enterprises to a certain extent. In the directed network, the edge weight represents the trust between node enterprises, and $w_{ij}$ represents the weight of the directed edge $\langle i\to j\rangle$. The greater the edge weight, the higher the trust, and the more stable the cooperation. Among the existing directed weighted complex networks, the bidirectional weighted network can better reflect the directionality and asymmetry of trust, and usually $w_{ij}\ne w_{ji}$. The matrix $W$ represents the directed trust between nodes in the overall network and the trust decays with time. The time decay factor $\theta \in \left[0,1\right]$. R represents the evaluation index, $g$ is the evaluation score of the index, and $g\in \left[0,1\right]$.

$$w_{ij}={\displaystyle \sum _{x=11,\dots 43}^{}{r}_x\cdot g_x}$$

(3)

$$W\left(T\right)={\theta }^{T-1}\left(\begin{array}{ccc}{w}_{11}& \cdots & w_{1n}\\ ⋮& \ddots & ⋮\\ w_{m1}& \cdots & w_{mn}\end{array}\right)+\left(\begin{array}{ccc}\Delta w_{11}^{}& \cdots & \Delta w_{1n}\\ ⋮& \ddots & ⋮\\ \Delta w_{m1}& \cdots & \Delta w_{mn}\end{array}\right)$$

(4)

Node Strength

The stronger the node, the closer the relationship between the node and other nodes in the network, and the higher the importance of the node in the network [50]. At the same time, the strength of nodes represents the reliability of enterprise relationship to a certain extent. In the directed supply chain network, the increase or decrease of the connected edges will cause the change of the output and input intensity of its neighbor nodes The weight of connected edges can reflect the closeness of connections between nodes, and the greater the weight, the higher the comprehensive trust of the node. $w_{ij}$ represents the trust of node $i$ to $j$, $UI$ represents the set of nodes pointing to node $i$, and $UO$ represents the set of nodes pointing from $i$.

In a directed weighted network, the node strength consists of the input strength $SI$ and the output strength $SO$, $SI$ represents the sum of the edge weights of adjacent nodes pointing to this node, and $SO$ represents the sum of the weights of the edge points from this node to adjacent nodes.

$$SI(i)={\displaystyle \sum _{j\in UI}{w}_{ji}}$$

(5)

$$SO(i)={\displaystyle \sum _{j\in UO}{w}_{ij}}$$

(6)

Node Distance

The internal cohesion of the nodes and the actual connection distance between the nodes are also indicators for evaluating the importance of the nodes [33]. In the trusted network, the distance between nodes is expressed in the form of contribution degree. In the supply chain trust network, the higher the degree of intimacy between nodes, the more important it is, the higher the reliability, the smaller the resistance in cooperation, and the higher the transaction response speed and transaction efficiency. $d_{ij}$ in this paper is the distance between two nodes is defined as the inverse of the contribution of the weight between two nodes to the overall weight of two nodes. The greater the contribution, the closer the distance between the two points.

$$d_{ij}=1/\frac{w_{ij}+w_{ji}}{\sqrt{SO\left(i\right)\cdot SI\left(j\right)}\cdot \sqrt{SO\left(j\right)\cdot SI\left(i\right)}}$$

(7)

Node Efficiency

The efficiency value $I_i$ is used to describe the degree of centralization of the node in the network, indicating the difficulty of the node reaching other nodes in the network. The definition form is:

$$I_i=\frac{1}{n}{\displaystyle \sum _{j=1}^n\frac{1}{d_{ij}}}$$

(8)

The greater the efficiency value of a node, the higher the degree that the node is in the center of the network. The greater the role that the node plays in the operation process of the supply chain, the higher the corresponding importance.

3.3. Evolutionary Mechanism

A supply chain directed trust network $G\left(V,E,W\right)$ was constructed: The node set is represented as $V=\left(v_1,v_2,\cdots ,v_n\right)$, the directed edge set $E_i=\left\{v_x\to v_y\right\}$, the adjacency matrix $W=\left\{w_{ij}\right\}$ represents the network edge weight set, w_ij represents the weight of the edge in the direction $i$ to $j$, and $w_{ij}\ne w_{ji}$. Considering that enterprises prefer to select better enterprises to cooperate for the reliability and sustainability of cooperation, it is assumed that the new connected edges follow the preferential connection principle, and the disconnected edges follow the reverse preferential principle. The specific evolution process is as follows.

3.3.1. Construction of the Initial Network

The complexity of supply chain structure and node entities is one of the manifestations of supply chain complexity. The supply chain network is divided into three levels, with a total of $n_1$ suppliers, $n_2$ manufacturers, and $n_3$ retailers. The initial connection network is randomly generated, and the existence of connection edges indicates the existence of a trust relationship between the two nodes. In addition, the network has a good growth mechanism.

3.3.2. Preferential Connection Mechanism

In a completely competitive market environment, enterprises can freely enter and exit the network and freely select cooperative enterprises. However, in the process of establishing the supply chain network, the newly-added connection edge is preferentially connected to the node with better connectivity, so the random connection selection preferential connection mechanism is abandoned. According to the preferential connection evolution of the network, $m_0$ new nodes are added to each level, and $L_1$ incoming connection edges and $L_2$ outgoing connection edges connected with the new nodes are added to the network. The weight $w$ of the new connection edges obeys uniform distribution in the interval $\left[w_{\min },w_{\max }\right]$.

Network topology and relative distance between nodes also have a great influence on complex network nodes. Therefore, the merit-based probability needs to take into account the degree, weight, and relative efficiency of nodes. $v$ represents a set of nodes of the same type as $i$, and $P_1$ and $P_2$ represent the attraction of node $I$ as an ingress node and an egress node, respectively:

$$P_1(i)={\left[I\cdot \left(KI(i)/{\displaystyle \sum _{j\in v}KI(j)}\right)\cdot \left(SI(i)/{\displaystyle \sum _{j\in v}SI(j)}\right)\right]}^{1/3}$$

(9)

$$P_2(i)={\left[I\cdot \left(KO(i)/{\displaystyle \sum _{j\in v}KO(j)}\right)\cdot \left(SO(i)/{\displaystyle \sum _{j\in v}SO(j)}\right)\right]}^{1/3}$$

(10)

The connection probability of each node was calculated according to the preferred connection to obtain a normalized state transition probability matrix, as shown in Formula (11):

$$P_{+}=\{\begin{array}{l}{\left(p_1(i)/{\displaystyle \sum _{j=1}^n{p}_1(i)}\right)}_{m\times n}\\ {\left(p_2(i)/{\displaystyle \sum _{j=1}^n{p}_2(i)}\right)}_{m\times n}\end{array}$$

(11)

3.3.3. Add or Delete Edges

The increase or decrease of connected edges is based on the preferential connection mechanism, which results in the change of the weights and degrees of nodes. If the added edge is duplicate of the existing edge, the weight $w^{+}$ is superimposed on the original weight. Considering that the trust between nodes decays during the evolution, the decay factor $\theta \in \left[0,1\right]$ is set. In the case of deleting the connection edges, the weight loss $w^{-}$ is subtracted from the current edge weight until the connection is disconnected when the weight value is negative. Scholars generally assume that the connection between a node and other nodes in the network is randomly disconnected when they study the node exit behavior. However, this behavior does not conform to the fact that the node gradually disconnects from other nodes when it exits. In real networks, the greater the degree and weight of nodes, the smaller the possibility of losing connection. The disconnection node is selected according to the principle of reverse merit, and the probability transfer matrix of reverse merit is shown in Formula (12):

$$P_{-}=\{\begin{array}{l}{\left(\left(1-p_1\right)/{\displaystyle \sum _{j=1}^n\left(1-p_1\right)}\right)}_{m\times n}\\ {\left(\left(1-p_2\right)/{\displaystyle \sum _{j=1}^n\left(1-p_2\right)}\right)}_{m\times n}\end{array}$$

(12)

4. Simulation Results and Analysis

4.1. ANP Model Calculation

Since there are many influencing factors involved in the ANP model and the calculation process is extremely complicated, the calculation processes of unweighted hypermatrix, weighted hypermatrix, and limit hypermatrix are completed by Super Decisions software. Due to the limitation of space,

Table 4. Judgment matrix for R₁.

	R1	R2	R3	R4	Weight
R1	1	2	1	3	0.35639
R2	1/2	1	1	3	0.25078
R3	1	1/3	1	3	0.29464
R4	1/3	1/3	1/3	1	0.09821

,

Table 5. Unweighted hypermatrix (R₁, R₂ parts).

		R1				R2
		r11	r12	r13	r14	r21	r22	r23	r24
R1	r11	0	0	0	0.6	0	0	0	0
	r12	0.333	0	0.5	0.2	0.5	0	0	0
	r13	0.333	0	0	0.2	0.5	0	0	0
	r14	0.333	1	0.5	0	0	0	0	0
R2	r21	0	0	0	0	0	0.333	0.190	0
	r22	0.25	0	0	0	0	0	0.263	0
	r23	0.75	0	0	0	0.5	0.333	0	1
	r24	0	0	0	0	0.5	0.333	0.547	0

,

Table 6. Weighted hypermatrix (R₁, R₂ parts).

		R1				R2
		r11	r12	r13	r14	r21	r22	r23	r24
R1	r11	0	0	0	0.328	0	0	0	0
	r12	0.119	0	0.238	0.109	0	0	0	0
	r13	0.119	0	0	0.109	0	0	0	0
	r14	0.119	0.476	0.238	0	0	0	0	0
R2	r21	0	0	0	0	0	0.060	0.038	0
	r22	0.063	0	0	0	0	0	0.053	0
	r23	0.188	0	0	0	0.089	0.060	0	0.179
	r24	0	0	0	0	0.089	0.060	0.110	0

and

Table 7. Limit hypermatrix (R₁, R₂ parts).

		R1				R2
		r11	r12	r13	r14	r21	r22	r23	r24
R1	r11	0.018	0.018	0.018	0.018	0.018	0.018	0.018	0.018
	r12	0.036	0.036	0.036	0.036	0.036	0.036	0.036	0.036
	r13	0.053	0.053	0.053	0.053	0.053	0.053	0.053	0.053
	r14	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054
R2	r21	0.065	0.065	0.065	0.065	0.065	0.065	0.065	0.065
	r22	0.123	0.123	0.123	0.123	0.123	0.123	0.123	0.123
	r23	0.034	0.034	0.034	0.034	0.034	0.034	0.034	0.034
	r24	0.043	0.043	0.043	0.043	0.043	0.043	0.043	0.043

only show a part of the matrix content.

The weights of each index are calculated from the limit super matrix data generated by Super Decisions, and the normalized values are shown in

Table 8. Index weight.

Primary Index	R1		R2		R3		R4
	0.161		0.265		0.474		0.101
Secondary indicators	r11	0.018	r21	0.065	r31	0.04	r41	0.002
	r12	0.036	r22	0.123	r32	0.065	r42	0.093
	r13	0.053	r23	0.034	r33	0.095	r43	0.006
	r14	0.054	r24	0.043	r34	0.154
					r35	0.119

.

4.2. Network Evolution Simulation

In order to verify the effectiveness of the above evolution mechanism, the degree distribution, weight distribution and other network characteristics of the supply chain trust network are tested by multiple numerical simulations. MATLAB was used to simulate the evolution of the supply chain network. In the initial network, the ratio of supplier, manufacturer, and retailer is set to 3:1:1, and the number of initial nodes is $n_1=15$, $n_2=5$, $n_3=5$. The nodes of adjacent levels are randomly connected. For the convenience of calculation, a node ($m_0=1$) was added to each level in each cycle, and two preferred edges ($L_1=1$, $L_2=1$) are provided for the new node. The weight of the new connected edge was w ∈ [0, 1]. The trust attenuation factor was $\theta =1$, the gain values $w^{+}\in \left[0,1\right]$, and attenuation values $w^{-}\in \left[0,0.5\right]$. During the evolution process, the nodes in the network randomly increase or decrease the connection edges according to the priority probability in each cycle. When the scale of manufacturers increases to 500 nodes, the model stops developing. Two sets of simulations were performed. The number of manufacturers in the network was recorded as N. The evolution stopped when the scale of manufacturers in the network expanded to 500 or 1000 (N = 500, N = 1000) nodes. The simulation result of node degree distribution in the supply chain trust network is shown in Figure 2.

Figure 2a,b shows simulation results of node degree distribution in the supply chain trust network when N = 500 and 1000, respectively. The degree value is the sum of node access degrees (K = KI + KO). According to the simulation results, it can be seen that the degree of most nodes is very low, while the degree of a few nodes is very high, and the degree distribution of nodes has obvious power-law characteristics. Generally speaking, the trusted network of the supply chain has obvious scale-free characteristics.

In the real supply chain network, some enterprises gain a dominant position in the supply chain network by virtue of their first mover advantage. This kind of enterprise usually has a high influence in the industry, and new joining nodes tend to establish connections with this kind of nodes, so that the connections of old nodes are generally high, while the connections of new nodes are few. However, there are still a few latecomers who can get far more connections than those of the same/adjacent nodes through reasonable competition, which is in line with the fact of supply chain operation.

Figure 3a,b shows logarithmic coordinate systems of node degree distribution when N = 500 and 1000, respectively, where k represents the degree of the node (K = KI + KO) and p(k) refers to the ratio of the node with the degree value k to the total number of network nodes. On the whole, most of the nodes in the supply chain trust network have low degrees, and a few nodes have high degrees. In the evolution of supply chain networks, there are usually a few core nodes that establish close ties with other nodes by virtue of their own resource advantages, and gradually accumulate more partners with the evolution, which promotes the development of trust networks.

In summary, the network model established in this paper conforms to reality and reflects the evolutionary trend of supply chain cooperation in reality. In addition, the numerical distribution of the node partners in the supply chain network obeys the power-law distribution, and the scale-free property of the supply chain network is well reflected and in line with existing research conclusions.

Figure 2 and Figure 3 show the distribution of node degrees from the whole angle, and do not reflect the influence of trust directionality. Figure 4 shows the degree-weight distribution diagram of supply chain trust network nodes. In which, subgraphs (a) and (c) are the input degree-input strengths relations when N = 500 and 1000 respectively, and subgraphs (b) and (d) are the output degree-output strengths relations when N = 500 and 1000 respectively. Most nodes with low degrees usually have low weight values. Similarly, nodes with high degrees usually have high weights. There is a linear correlation between the out-degree and out-weight, and in-degree and in-weight of most nodes. Combined with the supply chain trust network model, there is a positive correlation between node access weight and degree value. Enterprises with high trust in the supply chain networks often have more partners. Enterprises with more partners are willing to spend more energy to maintain and develop trust relationship relationships. There is a positive correlation between cooperation opportunities and the credibility of nodes. It also shows that the consolidation and development of existing trust relationships are critical for enterprises. The research also shows that the output and input intensity distributions of the nodes also have power-law characteristics, and the trusted network of the supply chain has obvious scale-free characteristics.

In the previous studies, the complex networks of supply chains mostly exist in the form of undirected graphs, and the degree-centrality (DC) index was the most direct measurement index in network analysis, but the DC index did not consider the influence of directionality. In the trust network model proposed in this paper, the nodes have both weight and direction differences.

In order to verify the effectiveness of the directed trust evolution mechanism in this paper, 10 nodes are randomly selected for verification, and P₁ and P₂ represent the influence of nodes as inbound nodes and outbound nodes, respectively. From the data in

Table 9. Node importance assessment (N = 500).

Node	Degree	SI	SO	DC	Node Importance
					P1	P2
v1	20	17.881	17.187	0.0083	0.0049	0.0049
v2	21	11.121	13.010	0.0087	0.0027	0.0028
v3	17	9.848	9.304	0.0071	0.0033	0.0032
v4	8	6.666	6.107	0.0033	0.0021	0.0021
v5	9	6.598	5.493	0.0037	0.0023	0.0022
v6	15	6.749	8.109	0.0062	0.0040	0.0042
v7	14	6.680	6.428	0.0058	0.0033	0.0034
v8	11	6.649	5.348	0.0046	0.0030	0.0028
v9	19	10.880	8.317	0.008	0.0048	0.0045
v10	14	6.148	7.302	0.0058	0.0036	0.0038

, we can clearly see the difference between the DC algorithm and the node influence algorithm of this paper. As the DC indicator only focuses on the number of neighbor nodes, nodes 7 and 10 are mistakenly considered to be of equal importance. In fact, the access weights of nodes 7 and 10 are significantly different. From the perspective of DC, the top three nodes of influence are 2, 1, and 3 in turn, while the top three nodes in this method are 1, 9, and 6 in turn. Taking nodes 1 and 2 as examples, the degree-centricity method considers that node 2 is more important than node 1, but from the perspective of weight, node 1 has a higher degree of directed trust and has a significantly better quality of trust relationship than node 2.

In comparison, the measurement index considering the direction proposed in this paper was more differentiated, which effectively avoids the deficiency of the DC method, and can well distinguish the influence between each node and the difference between the input influence and the output influence of the same node. Therefore, the form of the directed network was more suitable for distinguishing the influence of supply chain nodes under a complex network environment.

Based on Figure 2, Figure 3 and Figure 4 and Table 9, the simulation results of the supply chain trust evolution model showed obvious scale-free characteristics. With the continuous evolution of the supply chain, the topology structure and node importance of the supply chain trust network is changing all the time. Therefore, the practical evolution mechanism is of great significance for the research of supply chains. In addition, compared with the single degree-centered index, the method comprehensively considering the structure index and the node index in this paper can obtain more accurate results. This paper clarifies the differences of trust values between the node enterprises in the position of cooperative initiator and responder, and fills the blank in research of complex trust network evolution in the supply chain.

5. Conclusions and Recommendations

On the basis of the three-level supply chain, an ANP trust evaluation model based on the influencing factors of supply chain trust was proposed, and combined the relevant theories of complex networks to model and simulate the evolution process of supply chain network trust in reality. In terms of the evolution characteristics of supply chain trust, a merit-based evolution mechanism, which consists of node degree, weight, and efficiency, was proposed based on the two-way weighted network model. The characteristics of network evolution are deeply analyzed by describing the formation and evolution of supply chain network trust. The conclusions are as follows: (1) The distribution of node degree and strength in the supply chain trust network have power-law characteristics, and the supply chain trust network presents obvious scale-free characteristics; (2) compared with the simple degree-centered index, the preferred connection method proposed in this paper further clarifies the influence of degree, and weight directionality of network nodes, and it is more distinctive in evaluating the influence of direction weighted complex network nodes. (3) The merit-based index integrating the node’s own and structural characteristics further clarifies the difference of access degree and weight between nodes and its influence in the actual evolution process, which not only reflects the local connection characteristics of nodes, but also reflects the influence of the overall connection relationship on the importance of nodes in the directional weighted complex network from the global perspective, which is convenient for in-depth understanding of the evolution trend of the supply chain trust network.

The trust evolution model between supply chain levels established in this paper provides a new way of thinking for the research of supply chain networks. However, there is no intra-level or inter-level connection in this paper. With the cooperation becoming more and more diversified, there are cooperation possibilities, strategic cooperation, alliance, and other forms even among enterprises of the same type. In addition, a wide range of complex relationships such as alliances, horizontal and vertical cooperation, forward and backward integration, and other interconnected systems in the supply chain also bring new challenges to the trust of the supply chain. The purpose of this paper is to build a widely applicable framework of a supply chain trust evaluation system. However, the evaluation indicators and weights of supply chains will be different in different environments. Subsequent research can be adjusted and improved through large-scale surveys or secondary data. Besides, supply chain trust evaluation only uses the ANP method, but with the development of evaluation methods, fuzzy evaluation -ANP, DEMATEL-ANP, and other combined evaluation methods can provide new ideas for supply chain evaluation. The greatest characteristic of a multi-level supply chain network lies in the interaction and dependence among network levels. How to change the trust evolution model of the supply chain from theory to the practical problem of improving supply chain structure by using multi-level network theory will become the focus of future research.