Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 204395, 11 pages

http://dx.doi.org/10.1155/2015/204395

## A Rumor Spreading Model considering the Cumulative Effects of Memory

^{1}Business School, Sichuan University, Chengdu 610064, China^{2}Department of Mathematics, Yibin University, Yibin 644007, China

Received 25 July 2014; Accepted 26 September 2014

Academic Editor: Antonia Vecchio

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

#### Abstract

This paper proposes a rumor spreading model which examines how the memory effects rate changes over time in artificial network and a real social network. This model emphasizes a special rumor spreading characteristic called “the cumulative effects of memory.” A function reflecting the cumulative memory effects is established, which replaces the constant rate of memory effects in the traditional model. Further, rumor spreading model simulations are conducted with different parameters in three artificial networks. The results show that all the parameters but the initial memory rate of memory effects function have a significant impact on rumor spreading. At the same time, the simulation results show that the final size of the stiflers is sensitive to the average degree when it is small but is not sensitive to when the average degree is greater than a certain degree. Finally, through investigations on the Sina Microblog network, the numerical solutions show that the peak value and final size of the rumor spreading are much larger under a variable memory effects rate than under a constant rate.

#### 1. Introduction

An old saying goes that rumors come true after being repeated a thousand times. In real life, if people are unable to distinguish authenticity, many rumors are deemed to be true after a large number of repetitions. When rumors are widely propagated, people tend to believe the rumor, especially if they lack timely real information. Because of the increased presence of online social networks, rumors are no longer spread by word of mouth over a small area but are spread amongst strangers in different regions and different countries, meaning that rumors are being spread faster and wider than ever before. This sustained and rapid spreading of rumors deepens people’s impression about the veracity of the rumor and thus improves the credibility. Rumor spreading, therefore, has the ability to shape public opinion and lead to social panic and instability [1]. For example, the 2011 Tohoku nuclear leakage accidents caused a number of rumors in China. Rumors said that taking materials containing iodine could help ward off nuclear radiation, which led to the fact that many people rushed to purchase iodized salt.

Rumor spreading has attracted significant attention from researchers and, as a result, a great deal of research has been done on rumor spreading models. In the early stages, scholars borrowed from epidemic models to describe rumor spreading process [2–4]. A classical model is the SIR model, which studies the dynamic behavior of rumor spreading using an epidemic dynamics SIR model [5]. Based on this research, there have been many more rumor spreading models, most of which have been a variant of the SIR model [6–8]. With the development of network technology, many novel models have appeared inspired by empirical discoveries about network topology [7, 9–11]. At the same time, researchers have started to consider the specific features of rumor spreading in their models. Dodds and Watts [12] studied the effects of limited memory on contagion. Zhao et al. [13, 14] proposed a rumor spreading model which considered remembering mechanisms in homogeneous and inhomogeneous networks.

However, most previous studies have discussed memory effects as a constant parameter in the model, but, in reality, people hear rumors many times and so have an accumulation of impressions about the rumors, which changes the probability as to when people become rumor spreaders. Therefore, memory effects have a strong time-dependency. Further, the remembering mechanisms can indicate repeatability, which affects the spreading characteristics of the rumor [13]. Even a small amount of memory can affect the rumor spread in small network sizes [15]. Lü et al. [16] proposed a model considered memory effects on information spread but did not construct a function of time and did not build spreading dynamics equations.

In the existing research on rumor propagation, there have been two main classification methods for the population. In the first, the population is divided into the three groups borrowed from the epidemic spreading classifications [5], those that are susceptible (have never heard the rumor), those who are infected (are spreading the rumor), and those who are recovered (have heard the rumor but do not spread it). These classifications have been adopted in most rumor spreading models. In the second classification, however, the population is divided into four groups. This classification adds a new group to the first classification, but this group has been interpreted differently in different studies. In some research, this additional group, which arises from the spreaders, is called the hibernators, those that have heard but forgotten the information [13]. In Lü’s research [16], the additional group is called the known, those who are aware of the rumor but are not willing to transmit it as they are suspicious of its authenticity. An incubation class was suggested in [17], where it was proposed that a susceptible individual first goes through a latent period after being infected before becoming a spreader or a stifler, which is a similar idea to that of Lü. In this paper, we follow a four-group classification including incubation class, the unaware, the spreaders, the lurkers, and the stiflers.

In this paper, we study a rumor spreading model with variable rate which considers cumulative memory effects as a function of time, which is more in line with reality. The remainder of this paper is organized as follows. In Section 2, first, the rumor spreading model with a variable rate varying over time is described. Then, the mean-field equations are derived, and the dynamic analysis of the model is conducted. In Section 3, numerical simulations demonstrating the dynamics of the established model are conducted on regular networks, random networks, BA networks, and online social networks. We compare our model using different parameters and analyze the model’s impact at different average degree of networks. Then, we make comparisons between a constant versus a variable rate on Sina Microblog. In Section 4, further discussion is presented. Finally, in Section 5, we conclude this paper and provide several avenues for further research.

#### 2. A Rumor Spreading Model with Variable Memory Rate

In this model, we construct a function of time which reflects the accumulation of memory, the speed of memory change, and the importance of an event which triggers rumors. Then, with this function added as a parameter to a differential dynamic spread model, a new rumor propagation model is proposed.

##### 2.1. Memory Effects Function

Consider a network with nodes and links representing the individuals and their interactions. At each time step, each individual is in one of the following four states:(1)the unaware: this individual has not yet heard the rumor;(2)the lurkers: this individual knows the rumor but is not willing to spread it because they require an active effort to discern the truth or falseness of the rumor;(3)the spreaders: this individual knows this rumor and transmits it to all their contacts;(4)the stiflers: this individual neither trusts the rumor nor transmits it.

People generally hear a rumor after many times, and therefore they get an accumulated impression about the rumor, which means that the probability that people become a spreader changes from “will never believe” to “believes.” This can be described as the cumulative effect of memory, which affects the probability that an individual becomes a spreader from a lurker in the rumor spreading process. In information spreading theory, a function was established which reflected the probability that a person would approve the information at time after having received the news times [16]. This function is , where is the approving probability of the first receipt of the information and is the upper bound of the probability indicating maximal approval probability. In fact, as is a function of time, we can transform this function into a function of time ; that is, the probability that a person approves the rumor at time can be denoted by a function of time . From the following analysis, we determine a specific form for the function of time* t*.

First, we analyze the changing process of the lurkers. From a microcosmic point of view, lurkers do not automatically change their states at time step . Some may become a stifler or a spreader, while others remain lurkers and may become stiflers or spreaders at a later time. We assume that the new lurkers at each time step have a part of the residuals which last until the end of the rumor spreading. This corresponds with the fact that there are always some people who take a long time to change their state in real life. From the above analysis, lurkers become spreaders at a variable probability, denoted by , and become stiflers at the rate of , so we can determine the lurkers’ process of change at each time step, as shown in Figure 1. In the following paragraph, we explain this function .