The Research on Social Networks Public Opinion Propagation In fluence Models and Its Controllability

Lejun Zhang, Tong Wang, Zilong Jin, Nan Su, Chunhui Zhao,, Yongjun He＊

1 College of Information Engineering, Yangzhou University, Yangzhou 225009, China

2 College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China

3 School of Computer and software, Nanjing University of Information Science and Technology, Nanjing 210044, China

4 School of Computer Science and Technology, Harbin University of Science and Technology, Harbin 150080, China

Abstract: Public opinion propagation control is one of the hot topics in contemporary social network research. With the rapid dissemination of information over the Internet, the traditional isolation and vaccination strategies can no longer achieve satisfactory results.A positive guidance technology for public opinion diffusion is urgently needed. First,based on the analysis of influence network controllability and public opinion diffusion, a positive guidance technology is proposed and a new model that supports external control is established. Second, in combination with the influence network, a public opinion propagation influence network model is designed and a public opinion control point selection algorithm (POCDNSA) is proposed. Finally,An experiment veri fied that this algorithm can lead to users receiving the correct guidance quickly and accurately, reducing the impact of false public opinion information; the effect of CELF is no better than that of the POCDNSA algorithm. The main reason is that the former is completely based on the diffusion cascade information contained in the training data, but does not consider the speci fic situation of the network structure and the diffusion of public opinion information in the closed set. thus, the effectiveness and feasibility of the algorithm is proven. The findings of this article therefore provide useful insights for the implementation of public opinion control.

Keywords: social network; public opinion propagation control; in fluence network

I. INTRODUCTION

With the rapid development of the Internet as a new form of information diffusion, network media now have a great effect on the daily lives of the people. Active netizens’ comments have reached unprecedented levels. Whether the important events of the day are domestic or international, online public opinion can be immediately formed, with individuals expressing views and spreading ideas through the Internet. Pressure brought about by public opinion cannot merely be ignored by organizations or agencies. The Internet has become a distributing center of ideological and cultural information and an ampli fier of public opinion.In recent years, research into network public opinion has been given increasing attention by governments of all countries. Experts and scholars from many fields have participated in the studies on public opinion. Current topics of research include the spread and control of network public opinion and related issues.[1-3]

The generation and dissemination of false information presented as public opinion can have serious impacts on society and nations.On the other hand, however, there were also malicious users using microblogs to spread rumors and deceive the public to make people anxious and panic, with very negative consequences. The rapid diffusion of comments on immediate situations, current hot events,and adverse social trends has become a major issue in social network research. It also has signi ficance in public opinion monitoring and the maintenance of public safety.

Our contributions are as follows: (1) a positive guidance technology is proposed based on the analysis of in fluence network controllability and public opinion diffusion; (2) a public opinion propagation in fluence network model is designed and a public opinion control point selection algorithm (POCDNSA) is proposed;(3) Experiment veri fied that this algorithm can lead to users receiving the correct guidance quickly and accurately, reducing the impact of false public opinion information

Based on the controllability of the in fluence network and the analysis of public opinion diffusion influence,a positive guidance technology is proposed, and the SEIR model, which supports the incorporation of external control, is constructed.

II. RELATED RESEARCH WORKS

In the literature on public opinion diffusion,some models are proposed such as Sznajd model [4], the cellular automata for the public opinion diffusion [5], the small world model[6-7], and the public opinion diffusion situation that emerged from the interaction of the main model [8]. The inadequacy of these methods is that they cannot simulate real world situations accurately. Leskovec et al. [9]used the SIS(S and I are the states of nodes)model to simulate public opinion diffusion networks.S(susceptible individuals) corresponds to individuals who have not heard a particular public opinion in the diffusion process; andI(infectious individuals) corresponds to individuals who have heard and diffused the public opinion. Some individuals, however,are neither in stateSnor stateI. Gruhl et al.[10] introduced the notion of immune individuals into the process, using the SIR (SIR add a R state from SIS) model to simulate network public opinion diffusion, whereR(recovered,individuals) corresponds to individuals who have heard the public opinion, but have no interest in diffusing it. The main disadvantage of the above mentioned methods is that the processing of the individuals is too simplistic and cannot re flect the in fluence of external public opinion control on diffusion.

In the recent years, researchers have also studied the influence of social networks. In 2010, Cha et al. [11] compared and analyzed three measures of user in fluence according to forwarding number, the number of references(@mention), and the number of followers; and the rules of how the influence changed over time were analyzed. It was found that users with more followers may not be the ones who lead in the forwarding of messages. This implies that we cannot measure user influence simply from the characteristics of network topology, such as the number of followers.This research also focuses mainly on measures of user influence at a point in history; there is no algorithm proposed to estimate social in fluence at current or future times from historical data. In 2011, Bakshy et al. [12] used a regression tree to estimate the global diffusion scale of microblogs with short links published by the users. The average diffusion scale was also used as an index of user social in fluence.The method is proven effective. DeGroot [13]proposed the general DeGroot model based on the research of French [14] and Harary [15],which describes information diffusion, the formation of opinions, consensus reaching, and other network interactions. The concern of this approach is network consensus. Liu et al. [16]studied network controllability based on control and graph theory. Wang et al. [17] used the pinning control strategy to control a subset of the nodes to drive the network from any initial state to the desired target state. Xiong et al [18-20] put forward a lot of good new ideas,including the analysis and application of opinion model with multiple topic interactions, the research the predicting opinion formation with trust propagation in online social networks.However, for networks with arbitrary topology, choosing the pinning nodes effectively is still an open problem.

For the management and control of network public opinion, the traditional approach is to transform the nonlinear irregular network diffusion system into a linear system to obtain a given result. The common control methods mainly comprise vaccination and isolation.The vaccination approach includes promoting a site’s self-restraint and prompting its users through the use of a real name system to consider the risks of diffusion and to restrain their behavior. The isolation approach is to place gatekeepers in the network to strengthen daily monitoring of web content, take control of public opinions if there are problems, and delete or block undesired opinions; related technologies include topic detection and tracking[21-22], text tendentiousness analysis [23-24],and so on. Management technologies such as vaccination and quarantine can be partly applied in practice. However, these approaches tend to deprive people of the freedom to express their views and are very difficult to implement. Therefore, a positive guidance technology of public opinion is proposed. By providing positive information for diffusion,Internet users can receive the correct guidance information quickly and accurately to achieve the goal of reducing the in fluence of false information expressed as public opinion.

III. PUBLIC OPINION PROPAGATION INFLUENCE NETWORK MODEL

3.1 Public opinion propagation in fluence network

Based on the controllability of influence networks and an analysis of network public opinion diffusion in fluence, a new SIR model should be proposed. The work of this section includes: (1) the de finitions and related formulas of in fluence network are given; (2) the control of in fluence network is studied and some theoretical basis is given; (3) public opinion diffusion process is described and control point selection algorithm is designed.

The public opinion propagation influence networkis an extension of the in fluence network, in whichVis the set of nodes andEis the set of weighted links.Zis the set of the states of nodes, includingis the set of false public opinion information access nodes that continues to diffuse negative and false information to the network.DVis the set of public opinion control drive nodes that lead the network to the desired direction and try to protect the network from the influence of false public opinion.

For convenience of discussion, some definitions underlying the network public opinion diffusion model are given below:

Definition 1:The in fluence network nodes may be in one of the following four states:S(susceptible state) represents the state in which the nodes do not know the public opinion;E(exposed state) represents the state in which the nodes know the public opinion, but do not diffuse it;I(infectious state) represents the state in which the nodes know the public opinion and diffuse it; andR(recovered state)represents the state in which the nodes know the public opinion, but have no interest in diffusing it.

Definition 2:The direct immunity rateδrepresents the probability that susceptible nodes are converted into immune nodes under the in fluence of external driving nodes.

Definition 3:External driving control aims to control the diffusion of public opinion and reduce its in fluence.

According to figure 1, the external driving nodes mainly achieve the following functions:(1) take parameterδas a benchmark for the case in which nodes in stateSare converted directly into the immune stateR; (2) take parameterγas a benchmark indicating the increased probability that nodes in stateEare converted into the immune stateR; (3) take parameterηas a benchmark indicating the reduced probability that nodes in stateEare converted into stateI; and 4) take parameterγas a benchmark indicating the increased probability that nodes in stateIare converted into the immune stateR.

In the end, the goal of public opinion diffusion is to convert more nodes to the stateI,whereas the goal of public opinion control is to convert more nodes to the stateRwithin the shortest time and to cut the diffusion path quickly and effectively from public opinion to the unaffected stateS.

3.2 In fluence network

An influence network [25] is a directed networkG=(V,E), whereVis a set of nodes andEis a set of directed weighted links. The attribute value of the node is defined as the attitude value associated with the proposition.The link weight is de fined as the value of the node influence.the attitude value of network nodes for the public opinion,nis the number of nodes in the network, andrepresents the attitude value ofviat timet. The edgee:v→urepresents the in fluenceφvuof nodevon nodeu, the range ofeis [0,1], where the number “0” represents no effect, and the number “1” means complete effect. The de fi-nition of the matrixis a matrix ofn×norder. For any network node, the weighted sum of all the neighboring nodes’ in fluence is equal to or less than “1.” The in fluence network can be used to analyze social, economic,and policy decision-making processes, predict the possible outcome of negotiations among social members, and determine the influence and degree of importance of network nodes[26, 27, 28].

In the in fluence network, the change rate of one node’s attitude values is determined by the attitude values of all of its neighboring nodes.So, the attitude value of the nodeiat timet+1 is:

To facilitate the discussion later in the article,is defined as the in fluence matrix, whererepresents the in fluence of nodevionvj.

The attitude value of a node in the in fluence network depends on the average weight of its neighboring nodes, and the update rules public opinion of the attitude value can also be described by the DeGroot model[13].

3.3 In fluence controllability

Strongly connected components (SCC) may exist in a weakly connected directed graph.If an SCC has no outward directed edge, it is called a sink SCC. Similarly, if an SCC has no inward directed edge, it is called a source SCC. In addition, in a Markov chain, if a node set has no outward edge, the set is called a closed set [29]. In this article, if one node set in a Markov chain is called a closed set,which refers to the minimal closed set, then it is strongly connected. Obviously, an SCC is also a strongly connected closed set. In an influence network, as shown in figure 2(a),the directed edgerepresents the fact that nodevihas in fluence on nodevj, whereasφvi vjis the in fluence weight of nodevionvj. The influence matrixTcan be regarded as the “one-step transfer probability” matrix of a Markov chain. In the Markov chain corresponding to the influence network, such as the Markov chain officgure 2(b) corresponding to figure 2(a), the direction of the edge in the Markov chain is opposite to the one in the influence network.

Fig. 1. State transfer of SEIR model including external driving.

In the influence network, the initial valueavi

(0) of nodeviis between –1 and +1.As time goes by, nodeviis affected by its neighboring nodes and the attitude value of nodevimay be changed. Intuitively, through controlling a subset of nodes, the network can be guided to achieve the desired state. A node that can influence and change the attitude values of other nodes while being immune to in fluence itself is called an external controller.It is assumed that the initial attitude value of the external controller is “+1.” Furthermore,we hope that the final attitude values of all other nodes in the network is “+1” or near“+1.” With an external controller, if the final attitude values of all the nodes are the same as the external controller, the network possesses in fluence controllability [30].

Fig. 2. A closed set and its corresponding Markov chain.

Definition 4:If the attitude values of all nodes in the network converge to the same attitude values as the external controller, then the network possesses in fluence controllability.

Theorem 1: In a networkG=(V,E), for a given drive node setDV, if there is a directed path from at least one node inDto each node, the network possesses in fluence controllability.

Proof: If the nodevjis the external controller, there is a directed edge from nodevito all nodes in setD. If there is a path from at least one drive node to each node, the external controller can reach every node in the network. Because there is a directed path from nodevito all other nodes in the influence network in the corresponding Markov chain,other nodes have a directed path to the nodevj(absorbing state). Therefore, the probability that any nodevireaches the absorbing state. This Markov chain is then an absorption chain. After a finite number of time steps, each of the transients eventually converges to the absorbing statevj. Therefore,the node state will be the same as the external controller (nodevj).

The intuitive explanation of Theorem 1 is as follows: if there are nodes that the external controller cannot reach, these nodes will not be influenced by the external controller. The in fluence network is, therefore, uncontrollable.If an in fluence network needs to be controlled,the set of drive nodes first needs to be selected, and then a positive effect is applied from the drive nodes. Theorem 2 provides a simple way to find the drive nodes.

Theorem 2:In the situation where the networkG=(V,E) is strongly connected, the minimum number of drive nodes required to in fluence and control networkGis “1.” In the situation where the network is weakly connected, the minimum number of drive nodes required to in fluence and control networkGis equal to the number of minimum closed sets of Markov chains corresponding to the in fluence network.

Proof: In the situation where the networkGis strongly connected, there is a directed path from any node inGto any other node.From Theorem 1, any node that is chosen as the drive node can in fluence and control networkG. In the situation where networkGis weakly connected, if there is only one closed set in the network, according to the de finition of the closed set in the Markov chain, there is a path from any node out of the closed set to the closed set. If a node in the closed set is directly influenced by the external controller in the Markov chain corresponding to the influence network, each node can reach the external controller. From Theorem 1, the influence network is de fined as having in fluence controllability. If there are multiple closed sets in the Markov chain corresponding to the network, then a drive node is chosen optionally in each closed set; there being a directed path from each such node to the external drive controller; this Markov chain converts to an absorbing chain with only one absorbing state and the attitude values of all nodes will converge to a consensus, that is, the attitude value of external controller.

In a strongly connected influence network or a closed set, any node can be chosen as the drive node. However, different nodes have different in fluences in the network. In the case of the convergence of attitude values, the final attitude values of the network are the average weighted values of the initial attitude values of all nodes. IfB={bi} is de fined as the nodes’in fluence in a given in fluence matrixT, a simple method for calculatingBis to calculate it directly usingB?T=B.

In figure 2(a), node 2 is selected to drive. At the same time, node 3 can be driven excluding the driving cost, as shown in figure 3.

Fig. 3. Graph of driving other nodes in the in fluence network.

Using formulaB?T=B, driving node 3 achieves the stable state with fewer steps than driving node 2 because node 3 has greater influence than node 2. Therefore, the influence weight of the external controller for the drive node can theoretically be arbitrarily small.However, if the drive node has a bigger in fluence weight, its in fluence will be stronger and the network will converge faster.

3.4 Public opinion diあ usion process and control point selection

The description of the public opinion diffusion process and the control process is as follows:In the initial state of the public opinion propagation in fluence network, the attitude values of all nodes for the public opinion is “0” and in a vulnerable stateS. All the nodes in setXare always in the stateIin the public opinion diffusion process and their attitude values are“–1.” The public opinion control nodes are in the stateRand their attitude values are“+1” in the public opinion diffusion process.The responses of each node to public opinion information and public opinion control are different, as shown in table 1.

For better understanding, table 2 shows the parameters used in POCDNSA

When public opinion diffusion starts,a false public opinion information node influences its neighboring nodesviat timet, withviin a vulnerable state. This can be represented as; the attitude valueofchanges to negative and nodevichangesfrom stateSto stateEwith probabilityα.If the node is in stateEand its attitude value is in fluenced by its neighbors, but its attitude value does not influence its neighbor nodes,then with each time step it changes to stateIwith probabilityβand a node in stateIchanges to stateRwith probabilityλ. The whole process proceeds according to the SEIR model state transfer graph in figure 1.

Table I. Responses of nodes to public opinion propagation information and public opinion control.

Table II. Definitions of parameters.

The course of the diffusion and control of public opinion begins with public opinion diffusion. When the public opinion diffusion reaches a certain degree, the control strategy is adopted, and the public opinion control node starts to work.

When public control starts, the public opinion control drive node influences its neighboring nodevjat time. Ifvjis in stateS,vjis directly immune with probabilityδand the attitude valueofvjis changed positively. Ifvjis in stateE,vjis directly immune with probabilityχ+γ, or it changes into stateIwith probabilityβ–η,and updates the attitude value of public opinion. Ifvjis in stateI,vjchanges into stateRwith probabilityλ+γ, and updates the attitude value of public opinion.

According to the above description, the selection of the public opinion control driving node is the key to the extent and intensity of public opinion propagation influence. In setting the public opinion control driving node,there is a need to consider synthetically the whole situation of public opinion diffusion in the closed set; and equation (3) is used to sort the closed set.

The number of control nodes is relative to the number of closed sets in the network and the number of nodes with different states in each closed set, in which the number of nodes in stateSrepresents the future situation of public opinion diffusion in the closed set, the number of nodes in stateEand stateIin the closed set represents the current situation of public opinion diffusion, and the number of nodes in stateRin the closed set represents the previous situation of public opinion diffusion. In equation (5),represents the number of nodes in stateSin the closed setc; parametersf,g,hare the respective ratios representing the future, present, and past of public opinion; parameterzis the number of existing control nodes in the closed set; and the logarithmic function is used to guarantee smooth calculation results.

The selection algorithm of the control driving node in the public opinion propagation in fluence network is proposed as algorithm 1.

IV. EXPERIMENT AND ANALYSIS

4.1 Introduction of data set and evaluation criteria

The Enron email dataset and Sina microblog dataset are selected for the test of the algorithm in this article.

1) The Enron email dataset is the first largescale real email communication database of its kind, the data having been collected in the first place by the Federal Energy Regulatory Commission. MIT purchased the dataset and a group of researchers at SRI Laboratory started to collect and complete the dataset for the CALO project [29]. This article uses the April 2011 version of the Enron dataset. In the dataset, if userAsends an email to userB, a weighted edge is built fromAtoB; the weight of the edge is the ratio of the number of emails from userAtoBand the number of all emails from userA.

(2) Sina microblog dataset. Lichman [31]grabbed the web user message contents and their followers. The data in total contains 221,579 microblog users. If userAfollows userB, a weighted edge is built from userBto userA; the weight of the edge is the ratio of the number of microblogs forwarded byAand the number of microblogs forwarded by all ofB’s neighbors.

The coverage of public opinion diffusion in the social network and the time of public opinion diffusion in fluence are used to evaluate the effect of diffusion control[32, 33, 34].The users set that public opinion diffusion can reach before control is designatedUm, and after control it isUn. The formula of public opinion diffusion coverage rate is:

4.2 Enron experiment and analysis

The parameters of the communication network in Enron are regarded as a benchmark to generate the Small World network [35,17] and the Scale Free network [36]. The establishment of Scale Free network parameters includes the following: node size, 158; density, 0.12; initial node count = 10; initial density, 0.06. Establishing the parameters of the Small World network includes the following: node size,158; number of neighbors, 9; probability of removing neighbor, 0.2; probability of adding far neighbor, 0.2; power law exponent, 0.05.Table 3 shows the main parameters of the network generated and of the Enron communication network.

The public opinion information diffusion nodes from three kinds of network were selected for public opinion diffusion. Statistical data of the diffusion of the three kinds of network are analyzed, which were taken from 300 groups of experiments to get the average value. The parameters areα=0.3,β=0.3,λ=0.2,χ=0.1,δ=0.2,η=0.2, andγ=0.2.

Table III. Basic information of three networks.

The experimental results are shown in figure 4. From figure 4(a), the flow of public opinion in the three kinds of network has experienced the process of “spreading, diffusion,and disappearing.” From section 4.1, we can know that the Enron dataset is a fully connected network; therefore, the strategy of selecting public opinion controlling nodes is to select the nodes with high influence value in the influence network. Public opinion control is included at each stage of public opinion diffusion. The public opinion diffusion results after controlling are shown in figure 4. Figure 4(b)is the result of the controlling in step 1, figure 4(c) shows the results following step 3, and figure 4(d) follows step 9.

The coverage and average influence time are calculated respectively as shown in table 4.

From table 3, the public opinion diffusion without any controlling will occur at the end and the average influence time is longer, but the extent of influence is very high (near 100%). If the controlling strategy is implemented at the end of diffusion, although the average coverage does not change, the average influence time is shorter. If the controlling strategy is implemented at the middle of the diffusion, both the diffusion coverage and average influence time become better. Control at the beginning of the diffusion has the most satisfactory results. Through the above experiments, it can be seen that the earlier the control is implemented, the better is the controlling effect obtained for public opinion diffusion.

Fig. 4. Comparison of public opinion diffusion and controlling point.

Table IV. Three kinds of network public opinion control situation.

4.3 Microblog experiment and analysis

In the pretreatment process of microblog data,the edges with lower attention weight (less than 0.1) are deleted. There are 21 closed sets contained in the generated microblog public opinion propagation influence network. The biggest closed set contains 132,372 nodes,whereas the smallest closed set contains 45 nodes. One thousand nodes are selected as public opinion diffusion nodes from the microblog public opinion propagation in fluence network and 2000 nodes are selected as public opinion control driving nodes. Our algorithm is compared with CELF [37], which is a greedy algorithm based on submodularity for information outbreak detection in the blog network. The problem of information detection is abstracted as a group objective functionR(A),which needs to be maximized. ParameterArepresents the set of observation nodes that are selected as deployment nodes. The cascade information of 100 microblog topics is selected as a diffusion hotspot to train the CELF algorithm and obtain the set of diffusion nodes chosen.

This article imitates the real process of public opinion diffusion and control. When the number of diffusion nodes reaches 10% of all the nodes, the implementation of public opinion control strategy is triggered. One hundred groups of experiments are taken and the average value is calculated. The other parameters are the same as in section 4.3. The contrast experiment results are shown in figure 5.

Fig. 5. Contrast experiment of microblog dataset.

From figure 5, the two selection algorithms of public opinion diffusion control nodes both reduce the coverage and time of public opinion diffusion in fluence to a certain extent. However, the effect of the CELF algorithm on public opinion detection and public opinion propagation in fluence reduction is no better than that of the POCDNSA algorithm. The main reason is that the former is completely based on the diffusion cascade information contained in the training data, but does not consider the speci fic situation of the network structure and the diffusion of public opinion information in the closed set.

The direct immunization SEIR model proposed in this article has two characteristics compared with SIR epidemic models:

1) Latency period:In traditional SIR ep-idemic models, there is only one way to treat individuals in latency, which is to convert them into infected individuals with one specific probability. However, this does not accommodate the differences among individuals in the process of public opinion diffusion. There will be different outcomes when individuals are in the latent period; some may be converted into immune individuals under the intervention of external driving nodes, while others may become infected individuals.

2) External control environment:External driving nodes are used to influence the diffusion of network public opinion in order to intervene effectively and realize control of network public opinion.

Summary:

The proposed algorithm has a good fit with the Enron dataset, Small World network, and Scale Free network. If we are able to control public opinion earlier, we get a better effect.The POCDNSA algorithm also obtains a satisfactory fit with the microblog dataset and provides an effective method for the selection of the controlling nodes. The time complexity is mainly decided by the calculation of node influence, that is., the calculation of the Laplace transform of the influence matrix. The influence network graph has been proven to be a sparse graph, and the corresponding Laplacian matrix is a sparse matrix. The Lanczos algorithm is an accepted and effective method for calculating the large-scale sparse matrix.Its time complexity isO(m×n),nbeing the number of nodes, whereasmdepends on many parameters. It can be considered thatmandnare of the same order of magnitude;therefore, the time complexity of the whole algorithm isO(n2).

V. CONCLUSION

With the rapid dissemination of information over the Internet, the traditional isolation and vaccination strategies can no longer achieve satisfactory results. A positive guidance technology for public opinion diffusion is urgent needed. Based on the controllability of the influence network and the analysis of public opinion diffusion influence, a positive guidance technology is proposed, and the SEIR model, which supports the incorporation of external control, is constructed. The concept of the influence network is combined with the design of the public opinion propagation in fluence network model to propose the Public Opinion Control Driving Node Selection Algorithm (POCDNSA). Experiments verify the feasibility and effectiveness of the POCDNSA. The findings of this article provide useful insights for public opinion control. Future research will include topics such as the selection of control nodes combined with the range of public opinion diffusion, the technology of false public opinion information identi fication,and related issues such as the effect of public opinion diffusion on collaborative manufacturing and so on.

ACKNOWLEGEMENTS

The authors would like to thank the reviewers for their detailed reviews and constructive comments, which have helped improve the quality of this paper. This work is sponsored by the Natural Science Foundation of Heilongjiang Province of China under Grant No.LC2016024. Natural Science Foundation of the Jiangsu Higher Education Institutions Grant No. 17KJB520044 and 16KJB510024.