Shuo Han, Fuzhen Zhuang, Qing He, Zhongzhi Shi, Xiang Ao

Transcription

1 Accepted Manuscript Energy model for rumor propagation on social networks Shuo Han, Fuzhen Zhuang, Qing He, Zhongzhi Shi, Xiang Ao PII: S (13)968- DOI: Reference: PHYSA To appear in: Physica A Received date: 3 May 213 Revised date: 4 September 213 Please cite this article as: S. Han, F. Zhuang, Q. He, Z. Shi, X. Ao, Energy model for rumor propagation on social networks, Physica A (213), This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Highlights: We propose a novel model for rumor propagation based on physical theory. The proposed model shows rumor propagation experiences three evolutionary stages. We investigate why some weakening rumors can get resurgence. The proposed model shows different people have different effects on rumor propagation. We study the impacts of some influencing factors on the dynamics of rumor propagation.

3 *Manuscript Click here to view linked References Energy Model for Rumor Propagation on Social Networks Shuo Han a,b,, Fuzhen Zhuang a, Qing He a, Zhongzhi Shi a, Xiang Ao a,b, a The Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, Beijing 119, China b University of Chinese Academy of Sciences, Beijing 149, China Abstract With the development of social networks, the impact of rumor propagation on human lives is more and more significant. Due to the change of propagation mode, traditional rumor propagation models designed for word-of-mouth process may not be suitable for describing the rumor spreading on social networks. To overcome this shortcoming, we carefully analyze the mechanisms of rumor propagation and the topological properties of large-scale social networks, then propose a novel model based on the physical theory. In this model, heat energy calculation formula and Metropolis rule are introduced to formalize this problem and the amount of heat energy is used to measure a rumor s impact on a network. Finally, we conduct track experiments to show the evolution of rumor propagation, make comparison experiments to contrast the proposed model with the traditional models, and perform simulation experiments to study the dynamics of rumor spreading. The experiments show that 1) the rumor propagation simulated by our model goes through three stages: rapid growth, fluctuant persistence and slow decline; 2) individuals could spread a rumor repeatedly, which leads to the rumor s resurgence; 3) rumor propagation is greatly influenced by a rumor s attraction, the initial rumormonger and the sending probability. Keywords: Rumor propagation, Energy Model, Social networks, Heat calculation formula, Metropolis rule Corresponding author. Tel.: address: hans@ics.ict.ac.cn (Shuo Han) Preprint submitted to Physica A September 4, 213

4 1. Introduction A rumor, as a form of social communication, widely exists in human lives. As it always involves influential events such as political and economic issues or public figures, a rumor can easily shape public opinion, cause panic, harm others and has significant impacts on society. For example, last year, there was a rumor that the earth would experience three days of darkness during the end of world spreading in some districts of China. Some people believed it and stocked up on candles, which led to a temporary supply shortage of candles. Traditionally, a rumor spreads by word of mouth. However, as information technology develops, computer-mediated communication is becoming a major way of information dissemination. There are some new characteristics of rumor propagation emerging on social networks such as fast velocity, wide extent and liable to repetition, that bring new challenges to this problem. Rumor propagation has attracted many researchers great attention and there are several classical models on it found in the literatures. Since rumor spreading could be regarded as a social contagion process [1], early scholars borrowed epidemic models to describe this phenomenon [2, 3, 4]. However, epidemic models are not specially designed for rumor propagation and the mean-field rate equations are too simple to depict its complex evolution accurately. Subsequently, Zhao et al. improved epidemic models and applied them to rumor spreading on social networks [5, 6]. Nonetheless, they still didn t overcome the natural flaws of epidemic models. Galam proposed a novel model called Galam s Model to simulate rumor propagation [7]. And then Ellero et al. introduced a new scheme to improve Galam s Model[8]. Nevertheless, these models are confined to word-of-mouth information exchanging and are not suitable for describing rumor spreading on large-scale social networks. Zhang et al. investigated the interplay between rumor propagation and emergency development [9, 1, 11]. However, their work focused on studying the strategy to minimize negative impacts of rumor propagation instead of modeling the evolution. Reviewing the existing work, we find that the mechanisms of rumor propagation in most of the current models are designed by macromethod but are lacking in the following important details. Firstly, a rumor s attraction to people always presents a downtrend and such characteristic should be described in the model. Secondly, different individuals have different capabilities to transmit a rumor to others. In large-scale networks, high-degree 2

5 nodes usually have more authorities to influence other nodes [12]. Thirdly, the probability that an individual transmits a rumor to others is not constant. When people first hear a rumor, they are active to share it with others. However, as people lose interest in the rumor gradually, the transmit probability correspondingly decreases. Based on these observations, we analyze this problem from five aspects, including the characteristics of a rumor, the impact of a rumor on an individual, the accumulation of an individual s energy, the impact of a rumor on a network and the rules of rumor transmission. Along this line, we introduce the physical theory to analogize and formalize rumor propagation. Specifically, we find a rumor s influence involves three factors, i.e. the rumor s attraction, the infected individual s authority and the infected individual s discriminability of the rumor. Then we adopt the heat energy calculation to formalize their relationship. We also use simulated annealing schedule to describe the characteristics of a rumor and apply the property of Metropolis rule to formulating the rumor transmit probability. Finally we integrate these physical theories to propose a novel model called Energy Model for simulating rumor propagation. In addition, we systemically conduct experiments on both synthetic and real-world data sets to evaluate the performance of the proposed model for simulating rumor spreading and investigate the dynamics of rumor propagation. The rest of the paper is organized as follows. In Section 2, we state the problem and introduce some preliminary knowledge related to our work. In Section 3, we expound our model including the design methodology and algorithmic details. In Section 4, we perform experiments on both synthetic data sets and real data sets to visualize the evolution of rumor propagation and study its dynamic characteristics. Finally we conclude this paper in Section Problem Statement and Preliminary Knowledge To better understand the proposed model and comprehend how we apply the physical theory to analogize and formalize rumor propagation, we introduce the problem statement and preliminary knowledge here Problem Statement The problem statement of rumor propagation could usually be defined as follows [13]. Consider a population formed by N individuals where some of them may have closed relationships and each individual can be in one 3

6 of three different possible states: an ignoramus who has not yet heard the rumor, a spreader who is trying to spread the rumor to his neighbors, a stifler who no longer believes or spreads the rumor. Initially, only one individual is rumormonger and the remainder are ignoramuses. And in the subsequent time steps, the rumor spreads in the network by some certain mechanisms. Our work is to build a model to simulate this phenomenon and discuss its dynamic characteristics Preliminary Knowledge Heat Energy Calculation Substances absorb or release thermal energy when the temperature of surrounding changes. In physics, the amount of heat energy transfer is determined by three factors, i.e., mass, heat capacity and temperature change. Their relationship is defined by heat calculation formula as follows. E = cm T, (1) where E is the amount of heat energy transfer, m is mass, c is heat capacity and T is temperature change. In this paper, we analogy that rumors bring effects to individuals is similar to substances absorb or release thermal energy. Their correlation and how we incorporate heat energy calculation into the proposed model will be detailed in Section Metropolis Rule As we mentioned above, the probability that an individual transmits a rumor to others is not constant. A rumor is a kind of misinformation or disinformation and has always not been verified in the beginning of rumor propagation [14]. And it is also eye-catching so that can easily have people believe it and irrationally spread it as soon as they hear it [15]. Therefore, the rumor transmit probability is usually large in the beginning of rumor propagation. However, as time goes on, a rumor may be officially refuted or swallowed by other information. People no longer believe it or gradually lose interest in it. The transmit probability thus decreases. Based on these analyses, in our paper, we assume that the rumor transmit probability for an individual is large at the beginning and then decreases gradually. And we apply the form of Metropolis rule to formulating this probability. The 4

7 Metropolis rule can be expressed as follows. { 1 if E(x new ) < E(x old ), P = exp( (E(x new ) E(x old ))/T ) if E(x new ) E(x old ), (2) where P is the probability of making the transition from state x old to state x new, E(x) is the energy of state x, and T is the current temperature of system [16]. The expression of Metropolis rule has two properties. Firstly, if E(x new ) < E(x old ), the probability P is 1. Otherwise, the probability P is a value less than 1. Secondly, the probability P decreases with the decrease of the temperature T. And in simulated annealing algorithm, the temperature T is represented as a decreasing function called annealing schedule, that guarantees the probability P remains in a large value at the initial stage and then decreases gradually [17]. In our model, we use E(x) to indicate a rumor s impact on an individual, and use T to indicate a rumor s attraction which is also defined as a decreasing function. Based on the property of Metropolis rule, the transmit probability is in accord with the above assumption that it is a large value in the beginning of rumor propagation, and then decreases as the rumor s attraction fades away. The detailed rule design and explanation will be elaborated in Section Model of Rumor Propagation In this section, we analyze rumor propagation from five aspects and propose a novel model to describe this phenomenon based on the physical theory, called Energy Model. We first detail the design mechanisms of the proposed model, and then analyze its advantages compared with the traditional models Energy Model Motivated by the observations mentioned in Section 1, there are five aspects need to be studied in our model: the characteristics of a rumor, the impact of a rumor on an individual, the accumulation of an individual s energy, the impact of a rumor on a network and the rules of rumor transmission. 5

8 The characteristics of a rumor A rumor is often viewed as an unverified account or explanation of events spreading in population and pertaining to an object or issue in public concern [14]. It is usually eye-catching and has not been verified in the beginning of rumor propagation so that it has much attraction to people when they first hear it [14, 15]. At the meantime, most rumors have their timelines, they will extinct if they are officially refuted or swallowed by other information. Thus, as time goes on, a rumor s attention usually fades away gradually. Consequently, we think the attraction of a rumor to an individual is large initially and then exhibits a downtrend. Such characterization of a rumor is similar to the annealing in physics. Referring to the simulated annealing schedules in simulated annealing algorithm [16], we could define a rumor s attraction as T (t) = T / lg(1 + t), (3) where T is the initial temperature and t is the time step. T indicates a rumor s initial attraction, that is, the larger the value is, the more attractive the rumor is. According to the expression, the attraction of a rumor declines as the time step goes on. In addition, to depict different rumors, the form of a rumor s attraction expression could be various as long as it is monotonously decreasing. Note that, in our paper, we model the characteristics of a rumor from a microcosmic view. For example, individual A accepts a rumor at time step t, the attraction of the rumor to individual A at time step t is a large value and it decreases in the subsequent time steps. Individual A transmits the rumor to individual B at time step t+k, the attraction of the rumor to individual B at time step t+k is a large value and then it decreases. In other word, the attraction of a rumor to different people is asynchronous The impact of a rumor on an individual In our model, considering a rumor brings effects to an individual is similar to substance absorbs thermal energy, we introduce heat energy calculation to analogize and formalize the impact of a rumor on an individual. Specifically, we use the amount of heat energy transfer to indicate a rumor s impact on an individual, use the mass to indicate an individual s authority, use the heat capacity to indicate an individual s discriminability of a rumor, and use the temperature to indicate a rumor s attraction. Moreover, an individual s 6

9 authority can be expressed as the number of its neighbors (also referred as the degree in undirected networks and the out-degree in directed networks), and the larger the value is, the stronger transmission capability the individual has. An individual s discriminability of a rumor means how much degree he trusts the rumor, in this paper, the value is sampled from the range [, 1]. The rumor s attraction is formalized as Eq.(3), which is a function of time step t. To sum up, the energy transfer expression for an individual could be defined as E(t) = cm T (t). (4) For an ignoramus, his temperature is. He gets energy as long as he accepts the rumor. Therefore, we can rewrite the energy expression as E(t) = cmt (t), (5) where E(t) is a rumor s impact on an individual at time step t, c is an individual s discriminability of a rumor, m is an individual s degree, T (t) is a rumor s attraction at time step t. The definition shows the impact of a rumor on an individual changes over time. From the expression, we find that hub nodes (with large connection degree) infected by a rumor can bring huger effects than common nodes (with small connection degree), which is called celebrity effect The accumulation of an individual s energy Energy Model permits individuals spread and accept rumor repeatedly. Each acceptance can increase the individual s energy and such accumulation satisfies diminishing returns property [18]. Supposed an individual has accepted a rumor j times before time step t j, then the energy accumulation function can be defined as E(t) = j E(t j )/j, (6) where E(t j )/j denotes the energy increment brought by the j-th rumor acceptance. Eq.(6) indicates that an individual may become more active if more of his neighbors send the rumor to him. And the marginal effect of each acceptance decreases as the number of the acceptances increases. Note that, although Eq.(6) expresses the accumulation of an individual s energy, it doesn t mean that it is a monotonically increasing function. When an individual accepts a rumor at some time, its energy will increase. 7

10 However, when it does not accept the rumor at other time, the energy will decrease, since both T (t j ) and E(t j ) are decreasing functions. Moreover, a spreader s transmit probability decreases over time, thus an individual is impossible to keep accepting the rumor from others forever. Finally, the accumulation satisfies diminishing returns property, that is, with the increase of the acceptances, the marginal increase of the energy accumulation decreases. Consequently, although Energy Model permits individuals to spread and accept rumor repeatedly, the impact of a rumor on an individual is not infinite The impact of a rumor on a network At a certain time step, we sum all the individuals energy values and get the current impact of a rumor on the whole network, E net (t) = N E i (t), (7) i=1 where E i (t) is the accumulated energy value of individual i at time step t The rules of rumor transmission In this model, we describe the interaction between individuals through two probabilities: sending probability and acceptance probability. Sending probability determines whether a spreader sends a rumor to his neighbors. In our model, this probability has two properties. Firstly, it is judged by a rumor s impacts on a spreader in current and previous two moments. At a certain time step, if the rumor s current impact on the spreader is larger than the previous one, the spreader will transmit the rumor with a high probability, otherwise, he will do with a low probability. Secondly, since a rumor is popular initially and then diminishes over time, the sending probability for a spreader should be large at the beginning and then decreases gradually. We describe these two properties with Metropolis rule and define the sending probability as follows. { p if E(t) E(t 1), P send = (8) p exp[(e(t) E(t 1))/T (t)] if E(t) < E(t 1), where E(t) indicates a rumor s impact on an individual at time step t, and T (t) indicates a rumor s attraction at time step t. 8

11 For the first property, (E(t) E(t 1)) implies the spreader accepted the rumor at the last time step, which makes him transmit the rumor with a high probability of p. On the contrary, (E(t) < E(t 1)) means the spreader has less interest in the rumor than before and transmits the rumor with a low probability of (p exp[(e(t) E(t 1))/T (t)]). For the second property, we have modeled the attraction of a rumor T (t) being a decreasing function. As we elaborated in Section 2.2.2, the property of Metropolis rule can guarantee that the transmit probability remains in a large value initially, and then decreases gradually. Acceptance probability determines whether an individual accepts a rumor from his neighbors. In the real world, high-degree nodes have more authorities to influence other nodes, but are not easy to be influenced by others. Kempe et al. propose an influence model called Weighted Cascade Model to describe this issue [12]. In this model, the acceptance probability depends on the connection degree of the accepter. And its definition is P acc = 1/d acc, (9) where P acc is the acceptance probability and d acc is the connection degree(also referred as the degree in undirected networks and the in-degree in directed networks) of the accepter. Most of the social networks have scale-free property [19], and the degree difference between hub nodes and common nodes is very large. The definition of acceptance probability implies that when high-degree nodes transmit a rumor to common nodes, the common nodes will accept the rumor with high probability. In contrast, when common nodes transmit a rumor to highdegree nodes, the high-degree nodes accept the rumor with small probability. Based on the above analyses and analogy, we could simulate the evolution of rumor propagation using Energy Model. It starts with only one spreader existing in the network and progresses as spreaders transmit the rumor to their neighbors. The interaction between individuals follows the rules of rumor transmission. During the process, if an ignoramus hears a rumor and doesn t accept it, his local energy is still zero, otherwise he turns to be a spreader and has energy calculated by Eq.(5) and Eq.(6). When a spreader s energy reduces to, he becomes a stifler. We regard the whole network as an energy system and use the amount of energy to measure the impact of a rumor on a network. As a rumor emerges, spreads and fades away, the energy of a network rises, sustains and decays. 9

12 3.2. Model Innovation Comparing Energy Model with the other traditional models, we think the proposed model could have the following three advantages: Most traditional models for rumor propagation [2, 5] neglect the complex microcosmic mechanisms of rumor spreading, including the characteristics of a rumor, the rules of rumor transmission and so on. While in our model, we analyze this problem from five microcosmic aspects and incorporate them into one integral model. Hence, the simulation of Energy Model is more similar to the real rumor propagation. Most existing models for information diffusion, including voter models [21, 22] and linear threshold models [12, 23, 18], treat all the nodes in the network as equals, and ignore the distinction between high-degree nodes and lowdegree nodes. While in our model, we introduce the connection degree of individuals to distinguish them, and set different individuals to have different heat capacities. Almost all the current models are discrete, which use a binary variable to record an individual whether or not get infected, and adopt infection rate to evaluate a rumor s influence. However, in reality, different people have different degrees of identifications to the same rumor. In addition, the spreaders who have different authorities usually have different impacts on society. Consequently, in our model we use energy value, which is a continuous quantity to measure a rumor s influence on social networks. At the meantime, since the energy expression is a continuous function of time step, the conversion from spreaders to stiflers in our model is a continuous process. We think a reasonable measure can help us monitor the rumor more accurately. 4. Experiments In this section, we first give a brief introduction of the data sets. Secondly we perform a track experiment to display the evolution of rumor propagation applying the proposed model. And then we make comparison experiments between Energy Model and other models referring to real rumor propagations. Finally, we study the dynamics of rumor propagation by discussing the impact factors of rumor spreading. 1

13 4.1. Data Set There are four data sets used in the experiments, summarized in Table 1. The first one is a synthetic data set, which is generated using BA model [24, 25]. These data could construct a small undirected scale-free network shown at the top left corner of Figure 1, where the vertexes denote individuals and the lines represent the closed relationships of two individuals. Through this data set, we can conveniently follow the track of rumor propagation. The second one is crawled from Sina Weibo 1, which is a Chinese microblog service with Twitter-like unidirectional following relationships. It consists of two subsets and each one records the data of a real rumor spreading. From this data set, we can observe the trend of real rumor propagation. The third one is also a synthetic data set, which is generated using KR model [26]. This data set has a comparative scale to that of the second data set and can construct a directed scale-free network. We adopt it for comparison experiments to show the simulation results of different rumor models. Finally, the fourth data set crawled from Twitter, contains users and their following relationships. It is a real-world directed social network. We run the proposed model on it to study the dynamics of rumor propagation. Table 1: Introduction of Data Sets No. Scale Direction Source Usage 1 2 nodes undirected BA model Section nodes; 595 nodes directed Sina Weibo Section nodes directed KR model Section nodes directed Twitter Section A Track for Rumor Propagation To display the evolution of rumor propagation by the proposed model, we make a track experiment on a small synthetic data set. We set the rumormonger to be the one with medium degree in the network, the initial temperature as 1, and the rumor expression as an approximate form of classical annealing schedule. The track of rumor propagation over time steps is illustrated through the second subgraph to the last subgraph in Figure

14 Network step 1 step 2 step 3 step 4 step 5 step 6 step 15 step step 17 step 18 step 19 step 2 step 21 Figure 1: A Track for Rumor Propagation at Each Time Step The asterisks indicate spreaders and the lines indicate the propagation paths from spreaders to accepters. From these figures, we can observe the track of rumor propagation: - Initially the rumormonger spreads the rumor to two other individuals. At this time the rumor still spreads locally. - At time step 2, one of the current spreaders is the hub node of the network, who spreads the rumor to more individuals. - Due to the celebrity effect, at time step 3 more individuals become spreaders and further spread the rumor. Along this trend, large numbers of individuals are infected and the rumor breaks out at time step 5. The dense asterisks and lines mean that the rumor has become a hot topic at this time, people are glad to discuss it and exchange their opinions with each other. - There are no eternally popular rumors. From time step 6 to time step 16, people may no longer be interested in the rumor, and the number of spreaders gradually decreases. 12

15 - However, the hub node spreads the rumor again at time step 17, which leads to the rumor s reflourish. And in the subsequent steps, the rumor grows slightly and declines again. Until time step 21, the rumor eventually dies out. Normalization of Energy Time Step Spreaders Infected Individuals Energy Number of Spreaders(Infected Individuals) Figure 2: Evolution of Normalized Energy, Spreaders and Infected Individuals We also record the evolutions of three important variables, i.e., the number of current spreaders, the number of current infected individuals, and the current energy at each time step in Figure 2. The spreaders are the people who are transmitting a rumor to others and the infected individuals are the people who are accepting a rumor. From the figure, we can get some interesting observations. 1) the evolution of the energy lags behind the change of the numbers of spreaders and infected individuals. When the numbers of spreaders and infected individuals turn to decrease, the energy value still grows. When the former two died out, the latter still lasts for a long period. This is because whether a spreader transmits a rumor to others depends on his sending probability which decreases over time. If the sending probability reduces to a small value, the individual may not transmit the rumor any more. When all the individuals sending probabilities reduce to a small value, there may not be any infected individuals or spreaders in the network. However, at this time, the individuals energies still exist (E(t) = j E(t j)/j = j cmt (t j)/j has not yet reduced to ). 13

16 2) The evolution of rumor propagation goes through a process of three stages, including rapid growth, fluctuant persistence and slow decline. 3) The energy value oscillates at time step 17. Such oscillation points that authoritative individuals repetitive transmissions may lead to a rumor s reflourish. Note that, the simulation result is particular to the proposed model, since the Energy Model permits individuals spread a rumor repeatedly, which can lead to the rumor s resurgence. Moreover, rumor s decay function designed in Energy Model guarantees the energy value can ultimately converge to. These characteristics are coincident with the real-world phenomenon of rumor propagation Model Comparison Normalization of the Number of Spreaders Rumor 1 Rumor 2 Normalization of the Number of Spreaders Galam s Model Epidemic Model Energy Model Time Step (a) Trends of Two Real Rumor Propagations Time Step (b) Comparison between Energy Model and Two Traditional Models Figure 3: Comparison among Energy Model, Traditional Models and Real Rumor Propagations It has been demonstrated that the rising and falling pattern of rumor propagation does not present a simple or stable shape [27, 28]. A rumor s popularity always grows and fades in a periodic and fluctuant trend [15, 27]. In order to further confirm this conclusion, we collect two real rumors the United State formally declares war on Iran (Rumor 1) and Bai Yansong (who is a famous host) resigns his post (Rumor 2) spreading in Sina Weibo and analyze their propagation trends with the relevant data. We first subdivide the span of rumor propagation into equal time intervals and count the number of spreaders in each interval. And then we normalize the count 14

17 values through dividing them by the maximum value of all the intervals. The trends of rumor propagation for these two rumors are shown in Figure 3(a). To evaluate the performance of the proposed model, we make the simulations of three models, i.e., Energy Model, Galam s Model and Epidemic Model on a synthetic network and compare their simulation results with real rumor propagations. We first construct a synthetic directed network using KR model which is an approach to generate directed scale-free networks and is widely used to simulate web hyperlinks and social networks [26, 29]. And then we run the three models on it and observe the simulation results. To avoid the randomness, we make simulations 5 times for each model and take the average of them. We also normalize the simulation results through dividing the number of spreaders in each time step by the maximum one of the whole evolution. In this experiment, we set the proportion of initial rumormongers for Galam s Model to be.2 and set the probability that people changed from ignoramuses to spreaders for Epidemic Model to be.8 and the probability that people changed from spreaders to stiflers to be.1. These parameters are set according to the previous researches [8, 2]. For Energy Model, we set the initial temperature of the rumor to be 5 and the probability parameter p to be.7. The trends of rumor propagation for the three models are shown in Figure 3(b). The experiment results show that 1) Galam s Model only simulates the decay stage of rumor propagation and the number of spreaders monotonously diminishes in a short time. This conclusion is in line with the experimental results in Ellero s research [8]. 2) Epidemic Model describes rumor propagation going through growth and decline in two stages. The trend of rumor propagation for Epidemic Model exhibits a spiky shape. In the previous research, Moreno et al. performed extensive simulation experiments and obtained the same results [2]. 3) Energy Model depicts rumor propagation experiencing rapid growth, fluctuant persistence and slow decline in three stages. Differing from the above two models, the evolution is not stable, but periodic and fluctuant. To examine the simulation results in different parameter settings for Energy Model, we make simulations by perturbing the parameters. We first set the probability parameter p to be.7 and perturb the initial temperature of a rumor. The experiment results are shown in Figure 4(a). We then set the initial temperature to be 5 and perturb the probability parameter p. The experiment results are shown in Figure 4(b). The experiment results demonstrate that Energy Model can depict the 15

18 Normalization of the Number of Spreaders Initial Temperature = 5 Initial Temperature = 1 Initial Temperature = 2 Normalization of the Number of Spreaders Probability Parameter p =.1 Probability Parameter p =.4 Probability Parameter p =.7 Probability Parameter p = Time Step (a) Trends of Rumor Propagation in Different Initial Temperatures Time Step (b) Trends of Rumor Propagation in Different Probability Parameters Figure 4: Trends of Rumor Propagation for Energy Model in different Parameter Settings periodic and fluctuant trends of rumor propagation in different initial temperatures and probability parameters. We also find that the higher the initial temperature is or the larger the probability parameter p is, the more popularity the rumor has. This is because initial temperature and probability parameter p are respectively the indirect acting factor and the direct acting factor of sending probability. The two large parameters can make the sending probability keep large for long and lengthen the duration of rumor propagation. From the above experiments, we could conclude that Energy Model simulates rumor propagation more accurately than the other two models Dynamics of Rumor Propagation The reach of rumor spreading and the peak value of a rumor s influence are two important practical measures to reflect the dynamics of rumor propagation. In this section, we study the relationship between these two measures and their influencing factors on a large-scale data set crawled from Twitter Reach of Rumor Propagation The reach of rumor propagation measures whether a rumor reaches a high number of individuals, and is defined as the final density of stiflers r = N stiflers /N [2]. If r is large, the rumor spreading within the network is threatening; oppositely, the rumor has less impacts. In this section, we investigate how the two factors, i.e., the degree of initial spreader d init and the sending probability p, affect the measure r. 16

19 We run the experiments on Twitter data set by disturbing the two factors and observe the final density of stiflers r. The parameter p of sending probability is set from to 1 by step.1. And each user is in turn to be the initial rumormonger. In this network, the number of users followers (the degree) ranges from 1 to 93. Due to the randomness of rumor propagation, we make the simulations 5 times for each parameter setting and calculate the average value. The experimental results are shown in Figure 5(a). As expected, the number of infected individuals increases as the rumormonger s degree and probability parameter p increase. Density of Infected Individuals Degree of Initial Spreader Probability Parameter p (a) r affected by d init and p p =.7 degree = Density of Infected Individuals.2.1 Density of Infected Individuals Degree of Initial Spreader Probability Parameter p (b) Relationship between r and d init (c) Relationship between r and p Figure 5: Investigation for Reach of Rumor Propagation We also set the probability parameter p as.7 and the degree of initial 17

20 rumormonger as 9 respectively, to get two sectional drawings. The subgraph(b) in Figure 5 describes the relationship between the reach of rumor propagation r and the degree of initial rumormonger d init. When the degree is low, the rumor spreads locally. Because the rumormonger has less authority in the network and is difficult to push his message to other people. With the increasing of rumormonger s degree, the rumor spreads widely. And when the degree reaches 2, the growth begins to flatten, which means at this stage, the reach of rumor propagation doesn t obviously scale with the degree of rumormongers. This is because social networks usually have some local groups. In these groups, even the local hub individuals are set to be the initial spreaders, the rumor still remains locally and is difficult to spread among the groups. When the degree is up to 8, the reach of rumor increases rapidly. At this time, the individuals with such high degrees are usually the hubs of the whole network, and they can push their information extensively to all the groups. The subgraph(c) in Figure 5 describes the relationship between the reach of rumor propagation r and the parameter p of sending probability. When the parameter p is less than.2, the rumor can not spread. However, when the parameter p is higher than.2, the reach of rumor propagation is approximately linear growth with the increase of p. Based on the above discussion, we could conclude that the reach of rumor propagation increases with the growth of the initial rumormonger s degree and the sending probability. On the one hand, hub individuals always play important roles in information dissemination. Therefore, it is an effective strategy to prevent rumor spreading by controlling the high-degree nodes. On the other side, sending probability measures the activeness of social networks. Thus, frequent communication would facilitate the rumor propagation Peak Value of Rumor s Influence The peak value of rumor s influence is given by the maximum energy value of the network E max during the whole evolution. A large value of E max means the rumor has an explosive influence at one point. Here we consider it is relevant to the initial temperature of a rumor T and the parameter of sending probability p. In the experiment, the parameter p of sending probability is set from to 1 by step.1 and the initial temperature is set from to 1 by step 1. The experiment results in Figure 6(a) show that the peak value of a rumor s influence increases as the initial temperature and the probability parameter 18

21 x Maximum Energy Initial Temperature Probability Parameter p (a) E max affected by T and p 4 x 15 p = x 15 initial T = Maximum Energy Maximum Energy Initial Temperature Probability Parameter p (b) Relationship between E max and T (c) Relationship between E max and p Figure 6: Investigation for Peak Value of Rumor s Influence p increase. We respectively set the probability parameter p to be.7 and set the initial temperature of a rumor to be 5, to get two sectional drawings. The subgraph(b) in Figure 6 describes the relationship between E max and T. The experimental results show that the peak value is approximately linear growth with the increase of the initial temperature, because an attractive rumor would have more impacts on society. The subgraph(c) in Figure 6 describes the relationship between E max and the parameter p of sending probability. From this figure, we observe that when p is less than.2, the peak value E max is close to, since under this condition the rumor is difficult to spread. 19

22 When p is higher than.2, the peak value E max increases with the growth of the sending probability, because high sending probability facilitates people to spread the rumor more actively. And this increase has the property of diminishing marginal utility, because although there are new infected ones emerging, the previous spreaders keep losing their interests in the rumor as well. Consequently, for a given rumor, whose peak value and length are finite. 5. Conclusions In this paper, by the inspiration of the physical theory we propose a novel model called Energy Model to simulate the rumor propagation on social networks. Comparing with the other traditional models, the proposed model can discover some important characteristics of rumor propagation, i.e., 1) rumor propagation experiences three stages; 2) hub nodes and common nodes have different effects on rumor spreading; 3) individuals may spread rumor repeatedly to result in a rumor s resurgence; 4) the dynamics of rumor propagation are affected by several important influencing factors. The extensive experiments and convincing analyses show that the proposed model could depict the evolution of rumor propagation more accurately. 6. Acknowledgment This work is supported by the National Natural Science Foundation of China (No , ,69334,61353),National High-tech R&D Program of China(863 Program)(No.213AA1A66,212AA113),National Program on Key Basic Research Project (973 Program)(No.213CB32952) References [1] K. Kawachi, M. Seki, H. Yoshida, Y. Otake, K. Warashina, H. Ueda, A rumor transmission model with various contact interactions, Journal of theoretical biology 253 (1) (28) [2] D. J. Daley, D. G. Kendall, Epidemics and rumours. [3] R. M. Anderson, R. M. May, B. Anderson, Infectious diseases of humans: dynamics and control, Vol. 28, Wiley Online Library, [4] H. W. Hethcote, The mathematics of infectious diseases, SIAM review 42 (4) (2)

23 [5] L. Zhao, Q. Wang, J. Cheng, Y. Chen, J. Wang, W. Huang, Rumor spreading model with consideration of forgetting mechanism: A case of online blogging livejournal, Physica A: Statistical Mechanics and its Applications 39 (13) (211) [6] L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng, H. Cui, Sihr rumor spreading model in social networks, Physica A: Statistical Mechanics and its Applications 391 (7) (212) [7] S. Galam, Modelling rumors: the no plane Pentagon French hoax case, Physica A: Statistical Mechanics and Its Applications 32 (23) doi:1.116/s (2) [8] A. Ellero, G. Fasano, A. Sorato, A modified Galams model for word-ofmouth information exchange, Physica A: Statistical Mechanics and Its Applications 388 (29) doi:1.116/j.physa [9] Z.-L. Zhang, Z.-Q. Zhang, An interplay model for rumour spreading and emergency development, Physica A: Statistical Mechanics and Its Applications 388 (29) doi:1.116/j.physa [1] L.-A. Huo, P. Huang, X. Fang, An interplay model for authorities actions and rumor spreading in emergency event, Physica A: Statistical Mechanics and Its Applications 39 (211) doi:1.116/j.physa [11] L. Zhao, Q. Wang, J. Cheng, D. Zhang, T. Ma, Y. Chen, J. Wang, The impact of authorities media and rumor dissemination on the evolution of emergency, Physica A: Statistical Mechanics and its Applications 391 (15) (212) [12] D. Kempe, J. Kleinberg, É. Tardos, Maximizing the spread of influence through a social network, in: Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, 23, pp [13] D. Zanette, Dynamics of rumor propagation on small-world networks, Physical review E 65 (4) (22) [14] W. A. Peterson, N. P. Gist, Rumor and public opinion, American Journal of Sociology (1951)

24 [15] F. Zhang, G. Si, P. Luo, A survey of rumor propagation models, Complex systems and complexity science 6 (4) (29) [16] S. Kirkpatrick, M. P. Vecchi, C. D. Gelatt, kb, Optimization by Simulated Annealing, in: IBM Germany Scientific Symposium Series, Vol. 22, 4598, 1983, pp doi:1.1126/science [17] P. J. Van Laarhoven, E. H. Aarts, Simulated annealing, Springer, [18] E. Mossel, S. Roch, On the submodularity of influence in social networks, in: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, ACM, 27, pp [19] R. Albert, A. Barabási, Statistical mechanics of complex networks, Reviews of modern physics 74. [2] Y. Moreno, M. Nekovee, A. Pacheco, Dynamics of rumor spreading in complex networks, Physical Review E 69 (6) (24) [21] C. Castellano, D. Vilone, A. Vespignani, Incomplete ordering of the voter model on small-world networks, EPL (Europhysics Letters) 63 (1) (27) 153. [22] C. Schneider-Mizell, L. Sander, A generalized voter model on complex networks, Journal of Statistical Physics 136 (1) (29) [23] D. Kempe, J. Kleinberg, É. Tardos, Influential nodes in a diffusion model for social networks, Automata, Languages and Programming (25) [24] A. Barabási, R. Albert, Emergence of scaling in random networks, science 286 (5439) (1999) [25] Y. Moreno, R. P. Satorras, A. Vespignani, Critical Load and Congestion Instabilities in Scale - Free Networks. [26] P. Krapivsky, G. Rodgers, S. Redner, Degree distributions of growing networks, Physical Review Letters 86 (23) (21) [27] J. Yang, J. Leskovec, Patterns of temporal variation in online media, in: Proceedings of the fourth ACM international conference on Web search and data mining, ACM, 211, pp

25 [28] Y. Matsubara, Y. Sakurai, B. A. Prakash, L. Li, C. Faloutsos, Rise and fall patterns of information diffusion: model and implications, in: Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, 212, pp [29] H. Zhang, A. van Moorsel, Fast generation of scale free networks with directed arcs, Computer Performance Engineering (29)