A Multi-Step Interaction Opinion Dynamics Group Decision-Making Method and Its Application in Art Evaluation

Due to the development of intelligent decision-making, social network group decision making (SNGDM) has become increasingly valued. Self-persistence is a significant topic in SNGDM problems, while it is ignored in most existing research. Besides, existing opinion evolution models often ignore high-order interactions because they assume individuals only communicate with their friends or neighbors. With these issues in mind, the authors propose a multi-step interaction opinion dynamics model to manage the consensus in SNGDM problem with self-persistence evolution. Given the decision makers’ social network information, the centrality degree, interaction strength, and high-order interactions are combined to construct a social influence network. Inspired by the social influence model, the authors develop an opinion dynamic consensus model which also describes the evolution of self-persistence in a group of decision makers. Finally, an example and detailed simulation experiment are presented to demonstrate the efficiency of the proposed consensus model.


INTRODUCTION
Group decision making (GDM) can be regarded as a powerful tool to select the most desirable alternatives in the situation where a group of decision makers (DMs) participate to achieve a common solution (Bezdek, 1978) and has been applied in various fields (García-Zamora, 2022;Li, Liu & Li, 2021, 2022;Yu & Li 2022;Yu, Fei & Li 2019).The development of the economy and web technologies pushes GDM problem to more complex situations.Consequently, social network group decision making (SNGDM) problem is becoming an important topic and gaining popularity in the decision making field (Gong, Wang, Guo, Gong & Wei, 2020;Wang, Liang & Li, 2022;Liao, & Liu, 2017;Li, Rodríguez & Wei, 2021).But this situation also brings new challenges, such as effective social network information supervision and difficulties of reaching consensus, etc. Reaching consensus is very important in GDM problems, because it can increase the efficiency in the implementation of the obtained decision (Palomares, Estrella, Martínez & Herrera, 2014;Quesada, Palomares & Martínez, 2015;Rodríguez, Labella, Tré & Martínez, 2018;Yu, Li & Fei, 2018;Zuo, Li & Yu, 2020).It is usually very difficult to achieve a common solution accepted by all DMs.
Studies on consensus model of SNGDM are still in the inceptive stage with only a small amount of researches (Ding, Wang, Shang & Herrera, 2019;Liu, Zhou, Ding & Palomares, 2019;Liang, Guo & Liu, 2022, Zuo, Li & Yu 2020).Notably, there are mainly two types of model based on social network (Dong, Zhan, Kou, Fujita, Chiclana & Herrera-Viedma, 2018), the one is based on trust relationship (Wu, Chiclana, Fujita, & Herrera-Viedma, 2017;Zhang, Wang, Dong, Chiclana & Herrera-Viedma, 2022;Wu, Zhang, Liu & Cao, 2019) and the other one is based on opinion evolution (Capuano, Chiclana, Fujta, Herrera-Viedma & Loia, 2018;Dong, Ding, Martínez & Herrera, 2017).As for the former one, for instance, Wu, Chang, Cao, & Liang (2019) and Wu, Chiclana, Fujita & Herrera-Viedma (2017) studied the trust propagation approaches and developed some consensus models based on the trust relationship between DMs.Ding, Wang, Shang & Herrera (2019) developed a social network analysis-based conflict relationship investigation process and a conflict degree-based consensus reaching process (CRP) for GDM problems.Wu, Zhang, Liu & Cao (2019) utilized a network partition algorithm based on trust relationship to reduce the complexity of GDM problem and then the weights of independent sub-networks and their individual members are computed by their trust values.Liu, Zhou, Ding & Palomares (2019) proposed a model considering both the preference inconformity and the relationship disharmony to detect and eliminate the conflict for GDM in social network context.Zhang, Iván, Dong & Wang (2019) defined some principles to detect the DMs' non-cooperative behaviors, and then social network analysis was used for the non-cooperative behavior management.
Opinion evolution is capable of playing a key role in SNGDM.Several scholars have recently focused on GDM based on opinion evolution.For instance, Pérez, Mata, Chiclana, Kou & Herrera-Viedma (2016) applied the FJ model to simulate the DMs' discussion process and study the CRP.Dong, Ding, Martínez & Herrera (2017) studied the DeGroot model based on leadership to support the individuals' CRP.Li & Wei (2020) proposed a two stage dynamic influence model for handling the consensus reaching process in GDM with incomplete information, in which the social network is assumed to be static during the decision process.Ureña, Chiclana, Melançon & Herrera-Viedma (2019) proposed a consensus model based on bounded confidence opinion dynamic mechanism.Based on DeGroot model, Zhou, Wu, Altalhi & Herrera (2020) proposed a two-step communication opinion dynamics model, and provided three opinion control strategies for controlling the opinion formation process.Yang, Wang, Ding, Xu & Li (2022) proposed an opinion management-based consensus model to investigate the evolution process of opinions of DMs during the CRP.
The above research on the opinion dynamic consensus model have enriched the theory and application of SNGDM.The self-persistence is first introduced by Friedkin and Johnsen to represent the degree of an agent's adherence to their initial opinion (Friedkin & Johnsen, 1999).Analyzing self-persistence to assist the CRPs in SNGDM, however, has rarely been previously considered.The real SNGDM cases involve not only the mathematical formulation of the social network analysis but also the DMs' self-persistence psychological behaviors.Although there have been significant developments for consensus model based on opinion evolution, they still suffer from some limitations.
(1) Most existing opinion evolution models ignore high-order interactions because they assume that individuals only communicate with their friends or neighbors and do not consider the opinions of others.Nowadays, this may no longer be true, because online social networks, such as Facebook and WeChat, facilitate the discussion and communication among agents.Therefore, it is necessary to develop an opinion dynamics model that considers the high-order interactions between DMs.
(2) The self-confidence/persistence is regarded as a prominent indicator of influence and plays a key role in opinion evolution.In general, the confidence degrees of DMs are given in advance and are assumed to be unchanged during the CRP (Dong, Ding, Martínez & Herrera 2017;Gupta, 2018;Liu, Dong, Chiclana, Cabrerizo & E. Herrera-Viedma, 2016;Zhou, Wu, Altalhi & Herrera 2020).However, in real decision making cases this may no longer be true.Taking the psychological concept of reflected appraisal into account (Cooley, 1902), where actors' selfappraisals are influenced by the appraisals provided by other actors.And during the discussion process, the DMs' self-persistence degrees should be modified along the discussion process.
Based on the above analysis, we will try to solve the open problem: how to assist the DMs to achieve a consensus by considering the high-order interactions and self-persistence evolution in social network.This paper introduces an opinion dynamics model in which the DMs interact with other DMs using a multi-step interaction.In the proposed model, the self-persistence and opinion evolve simultaneously.The main novelties of this paper are enumerated as follows: (1) In the proposed framework we proposed two measures for each DM.The first one is the global reputation based on all the DMs' the centrality degrees and interaction strength.The second one is the pairwise social influence which is a measure that indicates the level of influence or trust between two DMs by considering high-order interactions.Based on these two measures, we construct a social influence network and corresponding influence matrix.
(2) The self-confidence degrees of DMs play a prominent role in opinion evolution.Due to the fact that the self-confidence is always difficult to measure in our lives, we allow DMs to voice their mutual evaluations.Then we apply social influence network theory to analyze the changing process of selfpersistence and analyze the influence of self-persistence on opinion dynamic in SNGDM problem.
In what follows, some basic knowledge, such as graph and the social influence model are introduced in Section 2. Section 3 introduces a multi-step interaction opinion dynamics model to manage the consensus in SNGDM problem with self-persistence evolution in detail.An application example and simulation experiment are utilized to illustrate the validity of our model in Section 4. Finally, some conclusions are pointed out in Section 5.

PRELIMINARIES
In this section, we present some preliminary knowledge which is helpful to understand our methodology, regarding graph and social influence model.

Graph
In GDM problems, DMs are not independent individuals, generally, they often have some relationships with each other.We introduce some basic concepts about graph which is helpful to study the relationships among DMs.
Definition 2.1.(Bondy & Murty, 1976 , , , is a set of nodes, E represents the set of ordered pairs, and the elements of E are called edges.We assume that the sets V and E are finite and V is nonempty.
There are two types of graphs: the directed graph and the undirected graph.In the undirected Definition 2.2.(Bondy & Murty, 1976) In the directed graph G V E , ( ) , a sequence of edges → .The number of the edges in the directed path is called the length of the path and is denoted aslen v v i j
Definition 2.3.(Bondy & Murty, 1976) is defined as the zeroone matrix, where s ij = 1 indicates that there is an edge from v i to v j ; s ij = 0 indicates that there is no edge from v i to v j .
Definition 2.4.(Wasserman & Faust, 1994) , , , be a set of nodes and be the adjacent matrix corresponding to a directed graphG V E , ( ) , then (1) deg ( ) is called the out-degree centrality of the node v r .
(2) deg ( ) is called the in-degree centrality of the node v r .
An adjacent matrix is used to describe whether there is a connection between nodes and it is a binary relation.However, in some situations, we not only want to know whether the connection between nodes exists or not, but also the connection strength between them.So a weighted adjacent matrix is proposed for this purpose.
Definition 2.5.(Wasserman & Faust, 1994) , , , , , where s ij means the social connection strength from v i to v j .

Social Influence Model
Actors' emotions and thoughts are liable to change resulting from actors' interaction.In real GDM situations, actors' opinions will evolve due to the social influence.where ω ij represents the direct influence of actor e j to e i and satisfies ω ij ≥ 0 and be the vector of actors' opinions of time t .Different social influence models (Degroot, 1974;Krause, 2000;Dittmer, 2001;Jia, Friedkin, & Bullo, 2015;Pérez, Mata, Chiclana, Kou, & Herrera-Viedma 2016;Basu & Sly, 2017) were proposed to model the influence among the actors.The DeGroot model is considered as a seminal model (Degroot, 1974) and can be described as , , ) .
Definition 2.6.(Dong, Ding, Martínez, & Herrera, 2017) All actors can reach a consensus if In the DeGroot model, the consensus condition has been proposed as shown in Lemmas 2.1.Lemma 2.1.(Dong, Ding, Martínez, & Herrera, 2017) In DeGroot model, all actors can reach a consensus if and only if there exists at least one globally reachable node.
The FJ model (Friedkin & Johnsen, 1999) is a variation of the classical DeGroot model.The opinion evolution of actor d i can be described by the following expressions: From Eq. ( 2), the opinion evolution process can be compactly written as Where A is an m m × diagonal matrix, a ii ii = − 1 ω and I is the m m × identity matrix.
Assume the influence process can reach an equilibrium, then lim y y where V is called control matrix and can be calculated by 1 (5)

MODELING OPINION DyNAMICS IN A SOCIAL INFLUENCE NETwORK
In this paper, we focus on a GDM problem in social network context, where there are a lot of interactions among individuals.The social network can be described by a directed graphG D E , By consideration of the self-persistence and opinion evolution, we aim to obtain the consensus opinion.The main notations used in this paper are as follows.
Y ( ) is the final consensus opinions for all the alternatives, where a ij c is the consensus opinion corresponding to the alternative x j with respect to attribute c j .

Influence Network Considering Multi-Step Communication
Multi-step communication or interaction indicates that the agents trust not only their own neighbors but also the neighbors of their neighbors.In other words, the agents consider the opinions of those they can reach in several steps, which is referred to as multi-step communication.
be DM d i 's communication set.In the opinion dynamic model, DMs modify their opinions by referring to the opinions of their communication sets at each step.How should d i assign weights to the DMs inU d k i ( ) ?A novel method based on node centrality and interaction strength will be developed to address this problem.The higher the local centrality, the more central the node is in the social network.Generally, there are two kinds of centrality measurements for a DMd i , namely in-centrality & Herrera, 2020).The comprehensive centrality can be represented by The comprehensive centrality measures the DM's fame value, which provides a better reference for the weight allocation.The interaction strength between DMse h and e l in social network can be measured by their interaction frequency f hl .The normalized interaction strength between e l and e h can be represented by (Li &Lai, 2014): where f h min and f h max are the minimum and maximum interaction frequency from e h to other DMs, respectively.Note that IS hl does not have to be equal to IS lh .We combine the centrality degree and interaction strength to characterize the global reputation of DMd i , be the k -step interaction matrix which is a row-stochastic matrix with c ii = 0 for i m = 1 2 , , , .Parameter ρ ρ ( ) 0 1 < ≤ represents a direct (one-step) communication path has the force of ρ probability of effectiveness.A k -step communication path has probability ρ k of being effective.The interaction matrix C is calculated as follows, Where con GR GR GR and is referred to as the weight adjacency matrix of graph G .The influence matrix can be written as where I is m m × identity matrix, A is a diagonal matrix with a ii ii = − 1 ω where ω ii is the selfpersistence degree of d i to his/her initial opinion.
For example, we consider the social network of six agents with the information of normalized interaction strength between agents as shown in Fig. 1.

Modeling Self-Persistence and Opinion Dynamic in Social Influence Network
In GDM process, DMs usually negotiate and interact with each other for achieving an agreement.In this section our aim is to model how the DMs' opinions evolve during the interaction process.DMs may differ in some of the opinions and it is expected that they talk to each other to clarify, defend and modify their opinions.Taking the psychological concept of reflected appraisal into account (Cooley, 1902), the DMs' self-persistence will be modified along the discussion process.During each discussion, the DMs' mutual evaluation matrix will be updated as follows whereY 0 and y t ij ( ) is the updated evaluation value of d i to d j at time t and y t ii ( ) is the self-evaluation of DM d i of time t .Group members elevate or dampen their self-persistence degrees by averaging their own and others' appraisals.A time-dependent self-persistence can be written as The proposed model assumes that the interaction matrix C is constant.If the influence process can reach an equilibrium, i.e., lim t Y t Y →∞ ( ) = ∞ ( ) , then the original equation becomes where V can be calculated by and Z t diag z t ( ) = ( ) ( ) , then a time-dependent influence matrix can be written as During the discussion process, DMs' opinions and self-persistence evolve simultaneously.Then model can be represented as Theorem 3.1.If the influence network has at least one globally reachable node, then the DMs can reach a consensus by the model as defined in Eq. ( 16).
Proof.The mutual evaluation matrix will reach reach an equilibrium, i.e., Y VY ∞ ( ) = ( ) 0 , as we explained in Section 2.2.Thus we know z t ( ) converge to z * for any initial mutual evaluation matrixY 0 ( ) ,i.e., lim z t z t →∞ ( ) = * .In other words, the inf luence matrixW t ( ) converge to ) .We suppose that W t ( ) reach an equilibrium at time t * , then after time t * , the model becomes the DeGroot model.For the reason that there is at least one globally reachable node in the influence network, according to Lemma 2.1, the proof of theorem3.1 is completed.

The Main Steps of the Proposed Model
To summarize, the main steps in the consensus process are described in Algorithm 1 as follows: Algorithm 1.

Output:
The final consensus opinions X * , and the number of discussion rounds t .
Step 1.According to Eqs. ( 6), ( 7), ( 8) and ( 9), we can obtain the comprehensive centralityCEN i , the normalized interaction strength IS hl , the global reputationGR i , and the interaction matrix C respectively.
Step 2. Calculate the influence matrix W by the Eq. ( 10).
Step 3. Update the DMs' mutual evaluation matrix Y t ( ) according to Eq. (11).Then, a time-dependent self-persistence z t ii ( ) can be calculated by Eq. ( 12).
= + 1 and go back to step 3. End

ILLUSTRATIVE EXAMPLE AND SIMULATION EXPERIMENT
This section provides an example to show the effectiveness of our proposed model, and design a simulation experiment to explore the effects of the proposed model on the final consensus solution.

Illustrative Example
We consider a group composed of six experts to reaching a consensus in art evaluation.The experts are with the network described in Section 3.1 that is given in Fig. 1.Set k = 3 and ρ = 0 5 . .To avoid over raising or belittling, let the mutual evaluation value be y ij 0 0 2 0 9 According to Eq. ( 10) and the obtained interaction matrix C in Section 3.1, we can obtain the initial influence matrix Based on Eq. ( 11), the evolution of DMs' self-persistence degrees is shown in Fig. 2.
. We assume that the consensus has been reached if con ≤ 0 001 .
. By the model ( 16), we obtain the final consensus decision matrix

Comparison and Simulation Experiments
To show the effectiveness of the proposed model, we use the consensus model based on DeGroot model to solve the decision making problem above.Here we take the opinion evolution of the opinions on alternative x 1 for example, shown in Fig. 3.The comparison and analysis will be given together after the simulation experiment below.
To better illustrate the influence of the DMs' self-persistence evolution on the opinion formation, in this section, we use the ER random network (Ersős and Rényi, 1960) for this opinion dynamics simulation experiment.In the simulation experiment, the initial DMs' mutual evaluation matrix and their initial opinions are randomly generated from 0 3 0 9 ., . .The parameter eps is the consensus threshold value representing the consensus has been reached if con eps ≤ .In the simulation experiment, we set different input parameters m , eps and p , and run the simulation method 500 times by using the proposed model and the DeGroot model, respectively, to obtain the average values of t , which are described as Fig. 4.
Based on this, the proposed consensus model and the consensus based on DeGroot model will be compared.As result of the illustrative example and simulation experiment the following points must be highlighted: (1) The proposed model performs effectively by adapting the self-persistence in the discussion process as can be seen in Figs. 2, 3 and 4. As future research work, we will study how the social network analysis can be used in the study of non-cooperative behavior to assist the decision makers for achieving a consensus and how to manage experts' non-cooperative behavior that can make difficult to reach a consensus.

ACKNOwLEDGMENT
The research described in this paper is supported by National Natural Science Foundation of China under Grants 71971190 and 71702087, Shanxi Province Philosophy and Social Science Planning Project under Grants 2021YJ145 and Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2021L406).
, , ) be a vector representing the actors' initial opinions on an issue.A matrix W depict the influence relationship among the actors, of m DMs and E represents the set of edges, which represents the relationship between them.Let x decision matrix provided by d k , where a ij k represents the attribute values for x i with respect to attributec j .The DMs provide their initial mutual evaluation matrix Y 0 ( ) and initial opinions A k 0 ( ) .
The social influence network can be represented by a directed graph G D E , ( ) where D is a set of DMs connected by a set of directed edges E that interconnect the DMs in pairs with a set of weights attached to it, that is matrix C c ij m m = × ( )

Figure 1 .
Figure 1.The experts' social network . The experts' initial mutual evaluation matrix and their initial opinions are randomly generated form 0 respectively.The influence network is randomly generated ER random graph where each pair of nodes connect with the probability of p the weight of each edge is randomly generated from 0

Figure
Figure 2. The DMs' self-persistence evolution