# An Evolution Model of Rumor Spreading Based on WeChat Social Circle

Lubang Wang* and Yue Guo**

## Abstract

Abstract: With the rapid development of the Internet and the Mobile Internet, social communication based on the network has become a life style for many people. WeChat is an online social platform, for about one billion users, therefore, it is meaningful to study the spreading and evolution mechanism of the rumor on the WeChat social circle. The Rumor was injected into the WeChat social circle by certain individuals, and the communication and the evolution occur among the nodes within the circle; after the refuting-rumor-information injected into the circle, subsequently, the density of four types of nodes, including the Susceptible, the Latent, the Infective, and the Recovery changes, which results in evolving the WeChat social circle system. In the study, the evolution characteristics of the four node types are analyzed, through construction of the evolution equation. The evolution process of the rumor injection and the refuting-rumor-information injection is simulated through the structure of the virtual social network, and the evolution laws of the four states are depicted by figures. The significant results from this study suggest that the spreading and evolving of the rumors are closely related to the nodes degree on the WeChat social circle.

Keywords: Evolution Equation , Rumor Spreading , Social Networking , WeChat Social Circle

## 1. Introduction

Network social, as a kind of virtual social, is considered as the expansion of the real social networking in the internet space. Multimedia-on-demand system in the network has increased the social people’s links by the virtual network [1] and the behavior of people on the network is a kind of mapping of the real behavior. As a part of their social activities, people play different roles and accept, inject and spread information in different social networking circles, by joining different social networking platforms. The current literature suggest that the social network structure has the small world effect [2], or the scale-free characteristic on the degree distribution [3]. The topology association between individuals is formed, due to information dissemination between the individuals, and the topology of different social networks has been studied by many scholars. The previous research [4-7] studied the structure of social network from the perspective of complex network. The topological structure of social network, including the property of degree distribution, the clustering coefficient, the vertex degree, and the correlation coefficient, have been subjected to study by many researchers.

The study of the rumor communication based on various indicators (i.e., the degree, the clustering coefficient, the distance, etc.) of network structure has a realistic basis, but the platform for the spreading rumors is the social medium, such as Twitter, Sina microblog, WeChat and other similar platforms. The communication law of information in the virtual communities and blogs is analyzed by name of authors [18]. The environment of the blog communication preserve some of the characteristics of a social network, therefore, we can see the blog network as a social network based on every blogger node. The literature [19] extracted characteristic factors from four aspects, including the publishing users, the receiving user, the user intimacy, and the timeliness of information, which affect the user’s forwarding behavior. On the basis of the dynamic model of infectious disease, the user behavior analysis and contact nodes were introduced, subsequently, a SCIR model was proposed, based on the user behavior analysis. The proposed model was used to analyze the association and evolution among nodes, in micro-blog social network. A two stage A-SC1 C2IR model was constructed by Song et al. [20] and introduced the dynamic equation of the model. The simulation results suggested that the prematurely intervention of authoritative information can effectively control the spread of micro-blog rumor. Lu et al. [21] studied the crowd of WeChat rumor spread and constructed the S1S2IR double S rumor spreading model, consequently, they provided the steady state value of the mean field equation of the model. Luo [22] compared the causes and governance schemes of the rumor spread between the micro-blog and the WeChat. Therefore, after the propagation research oriented to network topology structure, some literatures began to pay attention to the rumor spreading research, based on the real social background or the social platform.

The above literatures have carried out a more in-depth study on the related fields of the rumor, such as the social network, the communication model, and the rumor communication platform, as the main factors in this field. Due to the huge number of users and the huge information flow of WeChat, the research value of the rumor spread based on the WeChat platform is particularly prominent. In the study of WeChat rumors in [21], the two states of S1 S2 match the channels of rumors (i.e., the public number and the common node). However, in reality, most of the information published by the public number is the content of the common node transmission and the WeChat’s Circle, therefore, the display of this WeChat’s Circle is an important channel for the spread of rumors. Because common node transmission is a one to one transmission, and the spread of WeChat’s Circle is a display for all friends. This display is generally persistent and affects the friends’ nodes, repeatedly. Therefore, taking into account the particularity of WeChat’s Circle, this research creates a SLIR (Susceptible, Latent, Infective, Recovery) model for the rumor spreading. Subsequently, the rumor spreading law is deduced and the real data simulation is carried out based on the model.

## 2. Rumor Spreading Model SLIR

Tencent’s WeChat mobile dating software is the most popular and influential universal social application software. In this study, the rumor evolution model in the social network is based on the user functions and privileges of the WeChat’s dating software. Of course, the functions and user permissions of the WeChat and other social software will change throughout the time, but the basic functions of information sharing and information acquisition remains relatively unchanged. At present, the WeChat basically share and obtain the information through sending WeChat’s Circles, and information are released from friends group, and information transmission between the friends.

Based on the reality of WeChat social network, it constructs an undirected and unweighted attribute graph , in which V represents the set of nodes of the graph [TeX:] $$\mathrm{V}=\left\{V_{i} | i \in N\right\}$$. And E is a set of the edge of [TeX:] $$\mathrm{E}=\left\{\overline{{V}_{i} V_{j}}| i, j \in N \wedge i \neq j\right\}$$, where [TeX:] $$\overline{V_{i} V_{j}}$$ indicates that [TeX:] $$V_{i}$$ and [TeX:] $$V_{j}$$ have an edge connection, in other words, the information between the two nodes is connected ; and A is a collection of attributes of all nodes of the graph, and a set of attributes is defined as the four states in [TeX:] $$\mathrm{A}=\{S, L, I, R\}$$, which is corresponded to: S is a healthy and susceptible state[[TeX:] $$Susceptible$$], L is a latent state[[TeX:] $$Latent$$], [TeX:] $$I$$ is the state of spreading[[TeX:] $$Infective$$], and R is an immune state[[TeX:] $$Recovery$$], respectively.

According to the spreading law of rumor information in social network, the basic situation of the four kinds of nodes of [TeX:] $$S, L, I,$$ and [TeX:] $$R$$ in the network are as follows: susceptible node S indicate that no information about rumors is received in a social network; The infective node I indicates that the node accepts the rumor information from its neighbor nodes and implements activities to propagate the information, such as sending information in WeChat’s Circles. The latent node L indicates that the rumor information has been received or seen, but the nodes themselves do not share or disseminate information, however, they may share information or disseminate information at any time (now or later); recovery node R indicates that the node has accepted the message of refuting the rumor, which comes from the neighbor nodes (Refute the rumor by asking or looking at WeChat’s Circles).

The transfer between nodes in susceptible state, latent state, infective state, and recovery state, which depends not only on the state of the node itself, but also on the state of its neighbor nodes. On this basis, the following spreading rules are defined:

(1) If a susceptible Node S is contacted with an infective node I, the node S will convert to node L with the probability [TeX:] $$P_{1}$$. The latent node L will be converted to the infective node I with the probability [TeX:] $$P_{3}$$, and it is necessary to explain that the probability [TeX:] $$P_{3} : P_{3}$$ is no related to topology structure between nodes, the conversion [TeX:] $$\mathrm{V}_{\mathrm{j}}(L) \rightarrow \mathrm{V}_{\mathrm{j}}(I)$$ of the node [TeX:] $$\mathrm{V}_{\mathrm{j}}$$ state L to the I is completely subject to the subjective factors of the node itself (psychological factors);

(2) When the infective node [TeX:] $$I$$ meets the recovery node R, it converts to the node R by probability [TeX:] $$P_{2}$$; when the latent node L meets the recovery node R it will be converted to the node R by probability [TeX:] $$P_{2}$$;

(3) The rumor comes from outside and is injected into a point [TeX:] $$\mathrm{V}_{\mathrm{j}}$$ that makes [TeX:] $$\mathrm{V}_{\mathrm{j}}(S) \rightarrow \mathrm{V}_{\mathrm{j}}(I)$$. In other words, if there is no external injection, the closed social network has no so-called rumors. In the same way, the information of refuting the rumor also comes from the external injection.

Fig. 1 represents the rules of communication. It needs to be explained that the probability [TeX:] $$P_{3}$$ reflects the subjective state of the node itself. Some people see the rumor and believe it, hence they spread the rumor, while some people believe the rumor, but they reject to spread it. So, the probability [TeX:] $$P_{3}$$ is generally less than 1 in the real-world scenario. As for the “external injection” in Rule 3 is that the rumor does not come from the “social network circle”, but it comes from the acquisition of the individual outside the “social network circle”, perhaps, Injected into “the social network circle” by somebody after being coined.

The relation figure of node state conversion.

In the transformation rule of the spreading model (1), “if a susceptible node S is in contact with a infective node [TeX:] $$I$$, node S converts to the latent node [TeX:] $$L$$ with the probability [TeX:] $$P_{3}$$.” It can be understood as, [TeX:] $$\mathrm{V}_{\mathrm{j}}(\mathrm{S}) \stackrel{\mathrm{p}_{1}}{\longrightarrow} \mathrm{V}_{\mathrm{j}}(\mathrm{L})$$, and the converted result [TeX:] $$\mathrm{V}_{\mathrm{j}}(L)$$ will be converted to infective node [TeX:] $$I$$ with the probability [TeX:] $$P_{3}$$, and still is a latent node [TeX:] $$L$$ with the probability [TeX:] $$1 - P_{3}$$, as shown in Fig. 2.

Explanatory drawing of node S, L, I state conversion illustration.

The reality is that the conversion of [TeX:] $$\mathrm{V}_{\mathrm{j}}(\mathrm{L}) \stackrel{\mathrm{p}_{3}}{\longrightarrow} \mathrm{V}_{\mathrm{j}}(\mathrm{I})$$ is usually happened while [TeX:] $$\mathrm{V}_{\mathrm{j}}(\mathrm{S}) \stackrel{\mathrm{p}_{1}}{\longrightarrow} \mathrm{V}_{\mathrm{j}}(\mathrm{L})$$ changes. Since, a node [TeX:] $$S$$ is contacted with rumor mode [TeX:] $$I$$, if node [TeX:] $$S$$ trusts the rumor, and the rumor is forwarded in no time (node [TeX:] $$S$$ becomes [TeX:] $$I$$); if node [TeX:] $$S$$ does not trust the rumor, and the rumor is not forwarded (node [TeX:] $$S$$ will be latent node [TeX:] $$L$$), at that time and afterwards. So we can think of the approximation of the spreading model, as shown in Fig. 3 (the revised rule provides convenience for the subsequent equation description).

Revised drawing of node state transformation relationship.

In this way, the spreading model transformation rules (1) can also be expressed as: If a susceptible node S is contacted with a infective node [TeX:] $$I$$, the susceptible node will convert to the infective node [TeX:] $$I$$ with the probability [TeX:] $$p_{1} * \alpha$$, and will convert to the latent node L with the probability [TeX:] $$p_{1} * \beta$$ (among them, [TeX:] $$\beta=1-\alpha$$ ).

## 3. Dynamic Analysis of Spreading Model

##### 3.1 Average Transfer Probability of Node State

Assume that a node [TeX:] $$V^{j}$$ is a susceptible node, which is expressed as [TeX:] $$V^{j}(S) \cdot P^{j}_{S S}$$ indicates the probability that node [TeX:] $$V^{j}(S)$$ is still in a susceptible state in the time period of [TeX:] $$[t, t+\Delta t]$$. Accordingly, [TeX:] $$P_{S L}^{J}$$ represents the probability of the transition of node [TeX:] $$V^{j}(S)$$ into the latent node [TeX:] $$V^{j}(L)$$ in the time period of [TeX:] $$[t, t+\Delta t] \cdot P^{j}_{S I}$$ represents the probability that the node [TeX:] $$V^{j}(S)$$ is transformed into the infective node [TeX:] $$V^{j}(I)$$ in the time period of [TeX:] $$[t, t+\Delta t]$$. In similar tags, the probability expression of the node state invariance is also [TeX:] $$P^{j}_{L L} P^{j}_{I I}$$, and the probability expression of the node state change is [TeX:] $$P^{j}_{L I}, P^{j}_{L R}, P^{j}_{I R},$$ with the similar meaning to the above explanation. Then,

##### (1)
[TeX:] $$P^{j}_{S S}=(1-\Delta t * p 1)^{G}$$

Among them, [TeX:] $$G=G(t)$$ indicates the number of infective nodes in the neighbor of the [TeX:] $$V^{j}(S)$$ node at the t moment. It is assumed that the node [TeX:] $$V^{j}(S)$$ contains K edges, and [TeX:] $$G(t)$$ is a random variable with two distribution, as follows:

##### (2)
[TeX:] $$\prod(G, t)=\left(\begin{array}{l} {k} \\ {G} \end{array}\right) \omega(k, t)^{G}(1-\omega(k, t))^{(k-G)}$$

where, [TeX:] $$\omega(k, t)$$ represents the probability of connecting a susceptible node with a K edge to an infective node at [TeX:] $$t$$ time. [TeX:] $$\omega(k, t)$$ can be written as:

##### (3)
[TeX:] $$\omega(k, t)=\sum_{k^{\prime}} p\left(k^{\prime} | k\right) p\left(I_{k^{\prime}} | S_{k}\right) \approx \sum_{k^{\prime}} p\left(k^{\prime} | k\right) \rho^{I}\left(k^{\prime}, t\right)$$

where, [TeX:] $$\sum_{k} p\left(k^{\prime} | k\right)$$ is a degree correlation function, which represents the conditional probability of a node with a degree of [TeX:] $$k$$ and a node with a degree [TeX:] $$k^{\prime} ; p\left(I_{k^{\prime}} | S_{k}\right)$$ represents the probability of being in a infective state under the condition of node with a [TeX:] $$k^{\prime}$$ edges that are connected to a susceptible node with a degree of [TeX:] $$k \cdot \rho^{I}\left(k^{\prime}, t\right)$$ represents the density of the infective nodes of the scale of [TeX:] $$k^{\prime}$$ at the time of [TeX:] $$t$$.

Therefore, the average transfer probability [TeX:] $$\bar{p}_{s s}(k, t)$$ of a susceptible node with a degree of K, during the [TeX:] $$[t, t+\Delta t]$$ preiod can be calculated as follows:

##### (4)
[TeX:] \begin{aligned} \overline{p}_{s s}(k, t) &=\sum_{G=0}^{k}\left(\begin{array}{c} {k} \\ {G} \end{array}\right) \omega(k, t)^{G}(1-\omega(k, t))^{(k-G)}\left(1-\Delta t * p_{1}\right)^{G} \\ &=\sum_{G=0}^{k}\left(\begin{array}{c} {k} \\ {G} \end{array}\right)\left(\left(1-\Delta t * p_{1}\right) \omega(k, t)\right)^{G}(1-\omega(k, t))^{(k-G)} \\ &=\left(\left(1-\Delta t * p_{1}\right) \omega(k, t)+(1-\omega(k, t))\right)^{k}=\left(1-\Delta t * p_{1} \omega(k, t)\right)^{k} \\ &=\left(1-\Delta t * p_{1} \sum_{k^{\prime}} p\left(k^{\prime} | k\right) \rho^{I}\left(k^{\prime}, t\right)\right)^{k} \end{aligned}

## Acknowledgement

This research was supported by Zhejiang Province philosophy and social science planning project, China (No. 16NDJC274YB) and Key Research Institute of Philosophy and Social Sciences of Zhejiang Province—Modern Port Service Industry and Creative Culture Research Center.

## Biography

##### Lubang Wang
https://orcid.org/0000-0001-9104-7258

He was born in Anhui Province, China, February 16, 1974. Currently, he is an associate professor in Department of Electronic Commerce, College of Logistics and E-commerce, Zhejiang Wanli University, China. He received M.S. degree in School of Information from Sun Yat-sen University in 2004, and in July 2011, he received Ph.D. in School of Management from University of Shanghai for Science and Technology. His current research interests include knowledge management and public opinion spreading.

## Biography

##### Yue Guo
https://orcid.org/0000-0002-9472-9171

He received Ph.D. degree in School of Management from Wuhan University of Technology in 2011. He is a professor in the School of Economics and Management, Ningbo University of Technology, China. His current research interests include information management and enterprise management.

## References

• 1 W. C. Hou, Y. Pan, D. Che, "Utilizing fragmented bandwidth in a staggered striping multimedia system," Journal of Information Processing Systems, vol. 4, no. 1, pp. 1-8, 2008.doi:[[[10.3745/JIPS.2008.4.1.001]]]
• 2 D. J. Watts, S. H. Strogatz, "Collective dynamics of small-world networks," Nature, vol. 393, no. 6684, pp. 440-442, 1998.custom:[[[-]]]
• 3 A. L. Barabasi, R. Albert, "Emergence of scaling in random networks," Science, vol. 286, no. 5439, pp. 509-512, 1999.doi:[[[10.1515/9781400841356.349]]]
• 4 R. Kumar, J. Novak, A. Tomkins, "Structure and evolution of online social networks," in Proceeding of the 12th ACM SIGKDD International Conference on Knowledge Discovery And Data Mining, Philadelphia, PA, 2006;pp. 611-617. custom:[[[-]]]
• 5 H. Hu, X. Wang, "Evolution of a large online social network," Physics Letters A, vol. 373, no. 12-13, pp. 1p2p3p1p2p   1105-1110, 2009.doi:[[[10.1016/j.physleta.2009.02.004]]]
• 6 A. Mislove, M. Marcon, K. P. Gummadi, P. Druschel, B. Bhattacharjee, "Measurement and analysis of online social networks," in Proceedings of the 7th ACM SIGCOMM Conference on Internet Measurement, San Diego, CA, 2007;pp. 29-42. custom:[[[-]]]
• 7 H. Chun, H. Kwak, Y. Eom, Y. Ahn, S. Moon, H. Jeong, "Comparison of online social relations in terms of volume vs. interaction: a case study of cyworld," in Proceedings of the 8th ACM SIGCOMM Conference on Internet Measurement, V ouliagmeni, Greece, 2008;pp. 57-70. custom:[[[-]]]
• 8 J. N. Kapeerer, Rumor: The W orld's Oldest Media, China: Shanghai People’s Publishing House, Shanghai, 2008.custom:[[[-]]]
• 9 R. K. Garrett, "Troubling consequences of online political rumoring," Human Communication Research, vol. 37, no. 2, pp. 255-274, 2011.doi:[[[10.1111/j.1468-2958.2010.01401.x]]]
• 10 G. L. Cohen, J. Aronson, C. M. Steele, "When beliefs yield to evidence: reducing biased evaluation by affirming the self," Personality and Social Psychology Bulletin, vol. 26, no. 9, pp. 1151-1164, 2000.doi:[[[10.1177/01461672002611011]]]
• 11 D. J. Daley, D. G. Kendall, "Epidemics and rumours," Nature, vol. 204, no. 4963, pp. 1118-1118, 1964.custom:[[[-]]]
• 12 D. Maki, M. Thomson, Mathematical Models and Applications, NJ: Prentice-Hall, Englewood Cliffs, 1973.custom:[[[-]]]
• 13 B. Doerr, M. Fouz, T. Friedrich, "Social networks spread rumors in sublogarithmic time," Electronic Notes in Discrete Mathematics, vol. 38, pp. 303-308, 2011.doi:[[[10.1016/j.endm.2011.09.050]]]
• 14 D. H. Zanette, "Critical behavior of propagation on small-world networks," Physical Review E, vol. 64, no. 5, 2001.custom:[[[-]]]
• 15 D. H. Zanette, "Dynamics of rumor propagation on small-world networks," Physical Review E, vol. 65, no. 4, 2002.doi:[[[10.1103/physreve.65.041908]]]
• 16 Y. Moreno, M. Nekovee, A. Pacheco, "Dynamics of rumor spreading in complex networks," Physical Review E, vol. 69, no. 6, 2004.doi:[[[10.1103/PhysRevE.69.066130]]]
• 17 Z. F. Pan, X. F. Wang, X. Li, "Simulation investigation on rumor spreading on scale-free network with tunable clustering," Journal of System Simulation, vol. 18, no. 8, pp. 2346-2348, 2006.custom:[[[-]]]
• 18 A. Java, P. Kolari, T. Finin, and T. Oates, 2006;, https://ebiquity.umbc.edu/paper/abstract/id/300/Modeling-the-Spread-of-Influence-on-the-Blogosphere
• 19 Z. Y. Zheng, F. Guo, Z. F. Wang, D. Li, "Study on microblog propagation model based on analysis of user behavior," Computer Science, vol. 43, no. 12, pp. 41-46, 2016.custom:[[[-]]]
• 20 Z. J. Song, R. Shi, J. Wang, "Influence of authoritative information release on rumors microblog spreading in emergency events," Journal of Intelligence, vol. 35, no. 12, pp. 41-47, 2016.custom:[[[-]]]
• 21 D. Lu, J. L. Guo, H. Y. He, "Based on the double S model research in WeChat rumors spreading," Mathematics in Practice and Theory, vol. 2017, no. 16, pp. 157-163, 2017.custom:[[[-]]]
• 22 Y. X. Luo, "A comparative study of spread of Weibo and WeChat rumors from the perspective of public opinion," Journalism Knowledge, vol. 2014, no. 4, pp. 6-8, 2014.custom:[[[-]]]
• 23 W. Dou, Y. Jia, H. M. Wang, W. Q. Song, P. Zou, "A P2P approach for global computing," in Proceedings International Parallel and Distributed Processing Symposium, Nice, France, 2003;custom:[[[-]]]
• 24 Z. Liu, W. Don, W. M. Zhang, P. Zou, "Paradropper: a general-purpose global computing environment built on peer-to-peer overlay network," in Proceedings of the 23rd International Conference on Distributed Computing Systems W orkshops, Providence, RI, 2003;pp. 954-957. custom:[[[-]]]
• 25 W. Dou, H. M. Wang, Y. Jia, P. Zou, "A rumor-spreading analog on unstructured P2P broadcast mechanism," Journal of Computer Research and Development, vol. 2004, no. 9, pp. 1460-1465, 2004.custom:[[[-]]]
The relation figure of node state conversion.
Explanatory drawing of node S, L, I state conversion illustration.
Revised drawing of node state transformation relationship.
The virtual social network of 1,002 nodes of the construction.
degree distribution of virtual social network nodes.
Density variation of the three classes of infective nodes.
Density variation of the three classes of recovery nodes in a virtual social network
The average density changes after the injection of rumor and refuting-rumor-information.