An Elastic Collision-Based Rumor-Propagation Model in ... - IEEE Xplore

Received August 26, 2016, accepted September 16, 2016, date of publication September 21, 2016, date of current version October 15, 2016. Digital Object Identifier 10.1109/ACCESS.2016.2612298

ECRModel: An Elastic Collision-Based Rumor-Propagation Model in Online Social Networks ZHENHUA TAN1 , JINGYU NING1 , YUAN LIU1 , (Member, IEEE), XINGWEI WANG1 , GUANGMING YANG1 , AND WEI YANG2 1 College 2 Dalian

of Software, Northeastern University, Shenyang 110819, China NeuSoft Holdings Corporation, Dalian 116085, China

Corresponding author: Z. Tan ([email protected]) This work was supported in part by the National Natural Science Foundation of China under Grant 61402097 and Grant 61572123, in part by the National Science Foundation for Distinguished Young Scholars of China under Grant 61225012 and Grant 71325002, in part by the Natural Science Foundation of Liaoning Province of China under Grant 201602261, and in part by the Fundamental Research Funds for the Central Universities under Grant N151708005.

ABSTRACT With the rapid development of online social networks (OSN), the influence of rumor propagation on social life raises great concern. Traditional rumor-propagation models, which do not fully consider the features of OSN, are not suitable for use in OSN. In this paper, we focus on discovering a pattern of rumor-propagation phenomena in OSN, and propose a novel rumor-propagation model, inspired by a ball elastic-collision model, called the elastic collision-based rumor-propagation model (ECRModel). We investigate the dynamics of ball elastic collisions, which is similar to the dynamics of rumor propagation between nodes in OSN. We adopt the parameter relationships of the elastic collision model and apply them to rumor propagation in social networks. In the ECRModel, we do not directly adopt the node classification categories of ‘‘Ignorants, Spreaders, and Stiflers’’, but divide the user nodes into three types: 1) inactive and never spread rumors; 2) active and spread rumors forward; and 3) inactive but have previously spread rumors. We mathematically model node interaction attributes, and analyze the spreading probabilities and the steady state, considering both individual perspectives with detailed attributes and integral perspectives with node-state densities. At last, we conduct a series of simulations, and the results verify the correctness of the analytical results. We further investigate the effects of detailed properties on rumor propagation, such as average out-degree of OSN, rumor confusingness degree, and each node’s comprehensive influence. INDEX TERMS Rumor propagation, online social networks, social influence, information diffusion, kinetic energy.

I. INTRODUCTION

Social networks are social structures that consist of a set of social actors and social interactions between them. With the rapid development of internet and new media technologies, online social networking sites and applications have emerged into people’s lives [1]. Popular and highly visited online social networks include Facebook, Twitter, LinkedIn, Sina Weibo, and Pinterest. Information propagation in social networks has become a hot research topic in recent decades. In a social network, such as a microblogging system, a user is able to upload small pieces of information, and share them with his/her friends who follow him/her. The information would then be spread in a cascade mode by the active online friends

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of his/her friends. This phenomenon motivates researchers to discover the information diffusion patterns in social networks [2]–[6]. The diffusion of information or influence proceeds in discrete time steps. At each time step, each node has two possible states, inactive and active, where active means that the node adopts the new information, new idea, or new product being propagated through the network, while inactive means that node rejects it [5], [6]. A rumor is a special type of information with inaccurate/ misleading descriptions of problems or events, involving statements whose veracity is not quickly or ever confirmed [7], [8]. As the online users are often unaware that a given piece of information is a rumor, they may be easily

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affected by the rumor and become new spreaders, resulting in the quick propagation of the rumor in the social network. The impact of a rumor could be serious and bring harm to people’s lives and even political affairs. A practical rumor propagation analysis model can help to control rumor propagation, and it is also a necessary method for maintaining social stability. Researchers began to focus their attention on rumor propagation very early. The D-K model [7] and M-T model [8] were proposed to analyze rumor behaviors in the 1960s and 1970s, respectively, inspired by the epidemic model. They roughly divided the population into three types, Ignorants, Spreaders and Stiflers, and active probabilities between these divisions were designed. This classification method was adopted by most researchers, and they proposed many new or improved models for rumor propagation, such as [9]–[11]. Gradually, researchers investigated the limitations of these methods, and considered improving the rumor propagation model by expanding node states, to adapt to real social network patterns, such as [12]–[16]. Most researchers committed themselves to improving rumor propagation models based on other trans-disciplinary models, such as [17], [18], and [23]. Meanwhile, some researchers built specific rumor analysis tools or platforms, such as [19] and [20], based on relatively mature models. More details are introduced in section II. These models are mostly epidemic-like. However, the social network is different from epidemic virus relations, and the behavior diffusion in social networks is conducted by rational users who make more strategic choices rather than being randomly infected with certain probabilities [4]. Particularly, with the fast development of online social networks, the traditional epidemic-like models are not suitable for rumor spreading behavior analysis of online social datasets. References [12]–[15] observed this phenomenon, and proposed some improved models. Moreover, we observed that some significant facets are not considered in current models. Firstly, most of the models treated all social nodes equally, without taking into account the differences between them. For example, realistically, rumor spreaders with different interaction attributes would have quite different spreading tendencies. Secondly, the spreading probabilities between nodes would be quite different, rather than randomly selected from the same distribution, and the spreading probabilities should be determined by interactions in the social network. For example, if a user receives the same rumor from different users, the reactions should be different. Thirdly, it is necessary to consider the situation in which users receive rumors but do not spread them further. Meanwhile, in online social networks, the rumor propagation model should also cover the frequent phenomenon that a user receives the same rumor several times, and the rumor spreading probability should be influenced by this frequency. Considering the above issues, we focus on discovering a pattern of rumor propagation phenomena in online social networks, and propose a novel rumor propagation model inspired by the ball elastic collision model described in this article, 6106

the ECRModel (Elastic Collision-based Rumor propagation Model). We investigate the dynamics of ball elastic collisions, which is similar to that of rumor propagation between nodes in a social network, and adopt the parameter relationships in the elastic collision model and apply them to rumor propagation in a social network. In the ECRModel, we do not directly adopt the node classification of ‘‘Ignorants, Spreaders, and Stiflers’’ but divide the user nodes into three types, similar to elastic collisions. A user in an online social network is classified as (1) inactive and never spreads rumors, (2) active and spreads rumors forward, or (3) inactive but has spread a rumor in the past. Then, we mathematically model the nodes’ interaction attributes and analyze the spreading probabilities and the steady state. This work makes the following main contributions. (1) To our knowledge, this is the first attempt to map the ball elastic collision model into a rumor propagation model, trying to discover the rumor-spreading pattern. We consider both individual perspectives with detailed attributes and integral perspectives with node-state densities. It is more suitable for analyzing rumor propagation in online social networks than traditional epidemic-like models, which mainly consider the integral features. (2) Considering the forwarding behaviors in online social networks, we do not invariably adopt the node classification into ‘‘Ignorants, Spreaders, and Stiflers’’, but divide the user nodes into three types, similar to elastic collisions. In each time step, each node belongs to one of the three states: ‘‘inactive and never spreads rumors’’ (State 1), ‘‘active and spreads rumors forward’’ (State 2), and ‘‘inactive but has spread rumors’’ (State 3). Steady-state analysis and simulations prove the convergence of the rumor-propagation pattern. (3) Mathematical modeling of rumor propagation parameters in the ECRModel includes each node’s comprehensive influence, rumor-spreading breadth degree, rumor re-spreading rate, rumor confusingness degree, and each node’s resistance degree to a rumor. These definitions fully consider the differences between nodes, rather than treating all nodes the same. The rest of this article is organized as follows. We introduce the related work in Section II, and introduce ECRModel in Section III. Mathematical formalization of ECRModel is described in Section IV, and steady-state analysis for ECRModel is presented in Section V. Finally, we design simulations with facebook dataset to verify ECRModel and investigate its detailed properties in Section VI, with conclusions afterwards in Section VII. II. RELATED WORK

The earliest research on rumor-propagation models considered real social rumors in human lives. The first D-K model [7] was proposed by Daley and Kendall in 1964, which was a milestone for rumor-propagation research. They divided the humans into three types: those who are unaware of the rumor (Ignorants), those who spread the rumor (Spreaders), and those who are aware of the rumor but choose VOLUME 4, 2016

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not to spread it (Stiflers). In 1973, Maki and Thompson modified the D-K model and developed the M-T model [8], in which rumors propagate through direct contacts between spreaders and others. It is a spreader-centric model: once spreader i contacts an inactive node j, node j will be active with fixed probability. Gradually, researchers started to consider rumor spreading over networks. Zanette [9] firstly attempted to study rumor dynamics on small-world networks, and studied the evolution of an epidemic-like model evolving on small-world geometries based on a simplified D-K model, in which Ignorants would become infected once encountering Spreaders; and Spreaders would become Stiflers when encountering Stiflers or other Spreaders. Most of the existing models are based on epidemic-like models, such as the Susceptible-Infected-Removed (SIR) model. Zhao et al. [10] investigated an SIR rumorpropagation model that incorporates a forgetting mechanism into Barrat–Barthelemy–Vespignani (BBV) networks, and studied the dynamics of the rumor propagation model by accounting for the strengths of the nodes, and developed mean-field equations that account for the forgetting mechanism. The model represented differences in influence among the different spreaders using weights, to produce different results. They also proposed a SIR-based rumor-propagation model in new media networks [11], by modifying the flow chart of the rumor-spreading process to make it more realistic and apparent. In this model, they insist that an ignorant would inevitably change its status to either spreader or stifler, once it is contacted by a spreader. The probabilities that a spreader becomes a stifler by encountering a stifler or another spreader are denoted by η and γ , respectively. Some researchers also considered improving the rumorpropagation model by expanding the possible node states based on SIR, arguing that the traditional models were not appropriate for real online social networks, such as web forums and microblogging systems. Xia et al. [12] proposed a susceptible-exposed-infected-removed (SEIR) model by introducing the importance and fuzziness of the content of the rumor, and performed simulations to understand the dynamics of rumor spreading on web forums. Different from the standard or existing rumor spreading models, the novel model introduces a new user state (exposed state) and takes the characteristics of the rumor itself that can further influence users’ attitudes and behaviors toward rumors into account. In the model, at each time step, populations are divided into four parts, including people who had never heard the rumor (susceptible, similar to ignorants), those who resist the rumor and do not spread it (exposed), those who spread the rumor (infected, similar to spreaders), and the individuals who heard the rumor but had no interest in spreading it (removed, similar to stiflers). People in the exposed state who resist the rumor would spread the rumor under certain conditions. Liu et al. [13] studied the dynamics of rumor propagation in microblogging systems. To fit the for microblogs, they modeled five user states, including ignorant, infected, VOLUME 4, 2016

contacted, exhausted, and resistant, and added the coefficient of interest decay to the model, by assuming that an individual node/user can forward a rumor once at most. Zhang et al. [14] studied the dynamics of rumor propagation for information networks with eight influencing factors, including information attraction, objective identification of rumors, subjective identification of people, the degree of trust in the information media, spread probability, reinforcement coefficient, block value and expert effects, to pursue higher rumor-analysis accuracy. Bao et al. [15] noted that the traditional epidemiclike rumor control strategies could not meet the requirements of real social networks because the rumor propagation on social networks was largely different from the spreading of common diseases or viruses. They divided the infected states of rumors into the positive infected state and negative infected state, which are different from the states in common epidemic spreading models, and proposed and extended the SIR model to the SPNR model. Hong et al. [16] also noted that the traditional SIR model was not fully applicable to online social networks. They divided the infected states into credulous (C), neutrals (N) and denies (D), according to the characteristics of online social networks, and then proposed the SCNDR rumor-propagation model for online social networks. Different from the model-construction method based on expanding the possible states, some researchers designed trans-disciplinary rumor-propagation models. Han et al. [17] proposed an energy model for rumor propagation on social networks, considering that traditional rumor-propagation models designed for word-of-mouth processes may not be suitable for describing rumor spreading on social networks. In this model, the heat-energy calculation formula and metropolis rule were introduced and the amount of heat energy was used to measure a rumor’s impact on a network. Wang et al. [18] proposed a genetics-based rumor-diffusion model (GRDM), which considers an individual with multiple rumors in a network as a ’chromosome’ composed of a set of genes. The GRDM specified a rule for interactions between chromosomes to model the rumor interactions between individuals. Instead of focusing on the single influencing factor of diffusion, they reconstructed the key ingredients of the genetic algorithm—the crossover and fitness functions— to match the rumor spreading process in real-world social networks. Furthermore, some researchers focused on informationdiffusion problems involving specific microblogging systems, such as Twitter and Sina Weibo. Some of them built rumor analysis tools or platforms [19], [20], and some proposed specific rumor analysis models [21]–[23]. Finn et al. introduced an interactive, web-based tool (twittertrails.com) [19], which allows users to easily investigate the origin and propagation characteristics of a rumor and its refutation on Twitter. Liu and Chen [20] defined a follow-back rate to describe the true situation in Twitter-like microblog websites, and built a specific agent-based model using NetLogo based on SIR, assuming that the structure of these websites is a certain type of scale-free network. 6107

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Mendoza et al. [21] analyzed and characterized the social network of the community surrounding the topic, and focused on the propagation analysis of confirmed truths and false rumors on Twitter. Huang and Su [22] investigated the browsing behaviors of microblog users based on a Sina Weibo dataset, and proposed an improved SIR model of rumor spreading on microblogs, to differentiate rumor-refuting effects. Wu et al. [23] investigated the problem of automatic detection of false rumors on Sina Weibo, and proposed a randomwalk graph kernel to model the similarities of propagation trees, combining the graph kernel and a radial basis function kernel, together with other novel features, to build a hybrid SVM classifier. III. FROM ELASTIC COLLISIONS TO THE ECRModel

In this section, we firstly introduce the kinetic transformation in the ball elastic collision model, and propose a rumorpropagation analysis model named ECRModel (Elastic Collision-based Rumor-propagation Model) for online social networks, inspired by the dynamics of elastic collisions. A. KINETIC ENERGY TRANSFORMATION IN THE BALL ELASTIC-COLLISION MODEL 1) BEFORE COLLISIONS

The elastic collision is an encounter between two balls in which the total kinetic energy of the two balls after the encounter is equal to their total kinetic energy before the encounter; no energy is converted into other forms. Figure 1(a) shows the initial state of the ball elastic-collision model. The curve between balls i and j is a smooth track with no friction, while the plain track for ball j includes friction, with friction coefficient µ(µ > 0). Here, a ball is a spherical particle with mass m, radius r, and density ρ, where m = 34 π r 3 ρ. We assume that balls i and j are at rest (vi = vj = 0) at the very beginning, and ball i is located at the top of smooth trackij with height Hij , while ball j stays at the starting position of non-smooth plain trackj with length Lj . Ball i rolls down along the smooth trackij , and the elastic collision occurs between balls i and j on the connecting point between trackij and trackj , as shown in Figure 1(b). 2) ENERGY TRANSFORMATION

During the ball elastic collision, the conservation of the total momentum demands that the total momentum before the collision be the same as the total momentum after the collision. According to the kinetic energy transformation principle during elastic collision, ball i will be stationary at the collision point, while ball j will obtain the gravitational potential energy Energy(i) = mi gH ij of ball i. Therefore, the kinetic energyEnergy (j) = mi gHij after the collision. Ball j will undergo horizontal decelerated motion along the non-smooth plain Trackj with friction coefficient µ after collision. Then, the kinetic energy Energy (j) of ball j can support an ideal motion distance Dis(j) for ball j along the

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FIGURE 1. Ball elastic collision model.

Trackj : Dis(j) =

mi Hij . µmj

(1)

Now, let’s discuss the behavior of ball j after collision. (1) When the transformed kinetic energy Energy (j) is not enough to support an Lj -length horizontal motion along Trackj , such as in the situation shown in Figure 1(c), ball j stops on Trackj and all of its kinetic energy is consumed by friction. In this case, ball j has no opportunity for collision with the next ball. We say that ball j is in the state ‘‘receives energy but no further collisions’’ and that ball k in Figure 1(c) is the state ‘‘no collisions occur’’. (2) Otherwise, ball j arrives at the end of Trackj driven by the kinetic energy Energy (j) and the ideal motion distance Dis(j) ≥ Lj , as shown in Figure 1(d). This means that the kinetic energy transferred from ball i to ball j is enough to reach the end of Trackj and be ready for a new elastic collision along the same initial smooth track as ball i. In this case, ball j will undergo a new elastic collision with the next ball k, and we say that ball j is in the state ‘‘receives energy and ready for next ball collisions’’.

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B. ELASTIC COLLISION-BASED RUMOR-PROPAGATION MODEL

Inspired by the elastic collision model, we propose a new rumor-propagation analysis model ECRModel for social networks. An Online Social Network is a directed graph G =< V , E >, where V is a finite set of nodes, and E ⊆ V × V is the set of directed information-propagation relationships (edges) between pairs of nodes. A node u’s outgoing arc is denoted < u, v >∈ E, a node u’s incoming arc is denoted < v, u >∈ E, the set of node u’s out-neighbors is denoted N out (u), and the set of node u’s in-neighbors is denoted N in (u) [5]. As argued in [4] and [12]–[16], the dynamics of rumor propagation in online social networks is unlike that of traditional epidemic virus spreading. In online social networks, at any time step, users have three possible different attitudes for a piece of inaccurate online information, including (1) remain inactive and do not believe it, (2) remain active and spread it, and (3) return to being inactive and do not believe it, but have spread it in previous time steps. This phenomenon is very similar to the kinetic energy transformation in ball elastic collisions, where each ball has three possible states: ‘‘no collisions occur’’ corresponds to being inactive, ‘‘receive energy and get ready for the next collision’’ (active) corresponds to further spreading, and ‘‘receive energy but no further collisions’’ (inactive) corresponds to no further spreading. Therefore, in the ECRModel, intuitively, we can view the active state of a node u as the state in which u adopts the rumor information, while the inactive state means that u has not adopted the information. Rumor information propagation proceeds in discrete time steps in the ECRModel. At time t, each node v ∈ V usually has three possible states: ‘‘inactive and never spreads the rumor’’ (State 1), ‘‘active and spreads the rumor forward’’ (State 2), and ‘‘inactive but has previously spread the rumor’’ (State 3). State 1 and state 3 are both inactive, but a node in state 3 could have formerly been an active node in previous time steps. Let ASett ⊆ V be the set of active nodes at time t, referred to as the active set at time t. We call ASet0 the seed set. Nodes in the seed set are the initial nodes to propagate rumor information. Rumor propagation events from the initial state to the final state, along the arc in the social graph, are mutually independent [6], so each active node will be added into ASett . However, at last, these independent events will result in a wide, vicious influence. Before Propagation: At the very beginning, in the ECRModel several nodes in the seed set will be ready to spread original rumor information. Usually, these seed nodes will design confusingness information to mislead other user nodes. The rumor information will propagate along the relationship arc in the social network. This is similar to the initial conditions of the ball elastic collision model, where each seed ball would be at the top of Trackij ready for energy VOLUME 4, 2016

transformation. Rumor Propagation: In the ECRModel, once a node spreads a rumor, the subsequent nodes will receive the rumor information. The node’s influence on subsequent nodes and the degree to which the rumor is confusingness will determine the spreading distances. As we know from elastic collisions, the further motion of ball j is related to mi , mj , Hij , Lj and µ. Energy transformation in elastic collisions mainly depends on the driven ball (such as ball i in Figure 1) and its height position, which influences its gravitational potential energy. Once energy is transferred to the follower (such as ball j in Figure 1), whether another energy transformation will occur depends on the follower’s mass and its friction on the motion track. According to equation (1), the higher the value of mi Hij , the lower the value of µmj , resulting in a longer ideal distance Dis(j) of ball j, and higher possibility for additional collisions. This phenomenon is also very similar to rumor propagation in social networks. The seeds or spreading nodes’ comprehensive influences, each node’s in- or out-neighbors, each node’s influence on neighbors, and how confusingness the rumor is, will determine the rumor’s propagation to followers. The Followers’ characteristics and the rumor information’s confusingness characteristics will determine the next round propagation. Inspired by these parameters of energy transformation, we design parameters similar to those of the elastic collision model for the ECRModel according to characteristics of rumor propagation. Table 1 shows the parameter symbols of the ECRModel. TABLE 1. Parameters in proposed ECRModel.

In the ECRModel, we think the comprehensive influence (mRi ) of a seed or spreading node is similar to the mass of a driven ball, and the node’s comprehensive influence depends on the rumor’s possible spreading breadth (riR ) from the node out-degree in the social graph, and on the rumor’s re-spreading rate (ρiR ). The node’s influence on the following node (HijR ) is similar to the height of Trackij in the ball elastic collision model. Thus, higher mRi and HijR will result in a longer rumor-spreading distance. Meanwhile, higher rumor non confusingness degree (µR ) and higher node resistance degree (LjR ) to the rumor will prevent the rumor’s propagation. More discussion and mathematical formalization of these parameters will be introduced in the next section. 6109

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IV. MATHEMATICAL FORMALIZATION OF ECRModel A. FORMALIZATION OF RUMOR PROPAGATION PARAMETERS

In this section, we will model the propagation parameters in detail. Definition 1: Let mRi be the comprehensive influence value of node i, riR denote the rumor’s possible spreading-breadth degree and ρiR express the re-spreading rate of node out i’s rumor. N (i) denote the To model these parameters, let deg+ = i out-degree of node i, SPTimesRi denote the times of rumor spreading from node i, and ReSPTimesRi denote the times of rumor re-spreading via node i. We design an exponential function to compute riR , as ( η, if deg+ R i =0 ri = (2) + 1 − e−degi , else, where riR is a monotone increasing function of deg+ i , and riR ∈ (0, 1). Here, η is a small real value, such as 0.0001, that describes riR when deg+ i =0. Then, ρiR is designed via SPTimesRi , ReSPTimesRi and deg+ i , to express the re-spreading rate of node i’s rumor, as  +
R R  θ, if SPTimesi · ReSPTimesi · degi = 0 ReSPTimesRi ρiR =  , else,  SPTimesRi · deg+ i (3) where ρiR ≥ θ , and θ is a default positive small real value (such as θ = 0.0001). Based on the above design, we can obtain the mRi function. Inspired by ball elastic collisions, we retain the function feature as    3 4 R R R mi ∝ · π · ri · ρi . (4) 3 In fact, the constant 43 π is not a necessary coefficient for mRi , and we only need the function relations:  3 (5) mRi = riR · ρiR . Thus, we can obtain the final function for mRi to express the comprehensive influence of node i.   + 3   1 − e−degi · θ,   
R R if SPTimesi · ReSPTimesRi · deg+ mi = (6) i =0 R  3   ReSPTimes +  1 − e−degi i  · , else.  SPTimesRi · deg+ i HijR

Definition 2: Let be node i’s influence on node j. Let ReSPTimesRij denote the number of times that node j re-spread rumors of node i. In the ECRModel, HijR is determined by ReSPTimesRij and the total spreading times SPTimesRi . At the very beginning, the rumor-spreading time of node i is zero 6110

(SPTimesRi = 0), and we set HijR = 0.5.  0.5,    ε, HijR =  ReSPTimesRij   ,  SPTimesRi

if SPTimesRi = 0 elseif ReSPTimesRi = 0

(7)

else.

Here, ε is a default positive small real value (such as ε = 0.0001). Higher HijR means node j has re-spread more of i’s rumors. It is an intuitive mechanism of node influence in social networks. Definition 3: Let LjR be the resistance degree of node j to a rumor. Let RecTimesRj denote the number of times that node j received rumors from others. Thus, LjR is determined by SPTimesRj and RecTimesRj , as  if RecTimesRj = 0  0.5, SPTimesRj (8) LjR =  , else. 1 − R RecTimesj A node j with higher LjR will be more likely to resist spreading a rumor received from others. If node j never receives any rumor, we set LjR = 0.5. Definition 4: Let µR be the quality of the rumor information that express how non-confusingness the rumor is. HigherµR results in a rumor that is more difficult to spread. It is determined by the quality of the rumor itself. Let fEnergyRj be the resistance to influence (similar to friction energy) of node j. Inspired by the elastic collision model, we can obtain fEnergyRj ∝ mRj · g · µR · LjR .

(9)

Thus, since the coefficient g is not a feature of the rumor’s influence, we define fEnergyRj = mRj · µR · LjR .

(10)

Definition 5: Let REnergyRij be node i’s rumor influence energy propagated to node j, which is similar to the kinetic energy of elastic collision, REnergyRij ∝ mRi · g · HijR .

(11)

Thus we define REnergyRij = mRi · HijR .

(12)

B. FORMALIZATION OF RUMOR-SPREADING PROBABILITY

Assume node i is the spreader, with follower node j. When REnergyRij ≥ fEnergyRj , node j will re-spread the received rumor. In this situation, mRi · HijR ≥ mRj · µR · LjR , so mRj ·LjR mRi ·HijR

≤

1 . µR

Let condijR =

mRj ·LjR mRi ·HijR

be the re-spreading

condition of node j. In the ECRModel, the propagation events VOLUME 4, 2016

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FIGURE 2. Propagation rules in the ECRModel.

along the node’s relationships in the social graph are mutually independent. Therefore, each condition condijR in the ECRModel is independent.  We set x = condijR as a random variable, and let X be the set of all these random variables. Let D (X ) be the variance of X in a social network G =< V , E >, and E (X ) denote the expected value of the set X . (Here, the symbol ‘E (X )’ is quite different from the symbol ‘E 0 in G =< V , E >.) So, P condijR E (X ) =

∈E

P i∈V

deg+ i

P D(X ) =

∈E



,

(13)

condijR − E (X ) P

i∈V

2 .

deg+ i

(14)

  The distribution of x = condijR can be regarded as the normal distribution R

2

(condij −E(X )) 1 f (x) = √ · e− 2·D(X ) . 2π · D(X )

The density function of f (x) is  Z cond R  (t−E(X ))2 ij 1 ϕ(x) = · e− 2·D(X ) dt. √ 2π · D(X ) −∞

(15)

(16)

Finally, according to condijR ≤ µ1R , the spreading probability of a rumor is  Z 1  ))2 1 1 µR − (t−E(X 2·D(X ) dt. (17) ϕ( R ) = ·e √ µ 2π · D(X ) −∞ C. FORMALIZATION OF PROPAGATION RULES

Four propagation rules exist in the ECRModel, as shown in Figure 2. Rule 1 is the basis for propagation in a social network, and refers to the case in which rumor information is spread in a ‘‘one-to-one’’ style, whilerule 2 means a rumor is spread from one to many other nodes. Rule 3 is relevant in the situation in which multiple nodes spread a rumor to the same node, and rule 4 expresses a comprehensive propagation, in which a rumor is spread in a ‘‘many-to-many’’ fashion. Rule 1: In the situation shown in Figure 2(a), we call node u the spreader of the rumor, and node v is its follower (‘1-to-1’). Once node u spreads a rumor, node v will receive the rumor exactly (similar to how the ball collisions occur). VOLUME 4, 2016

Algorithm 1 RumorPropagationRule_1To1(G, u, v) Input: A Social Graph G =< V , E >; Two Nodes: u, v ∈ V , ∈ E; u is a spreader; µR is input by the analyst according to the rumor’s characteristics. Output: A Boolean flag indicating whether v spreads u’s rumor or not. (true or false) 1: Get SPTimesRu , SPTimesRv , ReSPTimesRu , ReSPTimesRv , R N out (u), N out (v) from graph G. ReSPTimesRuv , RecTimes v ,out N (u) , deg+ = N out (v) , r R = 2: Calculate: deg+ = u v u + + 1 − e−degu , rvR = 1 − e−degv . R = 0.5; 3: Initialization: ρuR = ρvR = 0.0001; Huv R Lv = 0.5. 4: if (SPTimesRu · ReSPTimesRu · deg+ u ) > 0 do 5: ρuR = ReSPTimesRu /(SPTimesRu · deg+ u ). 6: if (SPTimesRv · ReSPTimesRv · deg+ ) > 0 do v 7: ρvR = ReSPTimesRv /(SPTimesRv · deg+ ). v

3 3 8: Calculate: mRu = ruR · ρuR , mRv = rvR · ρvR . 9: if SPTimesRu > 0 do R = ReSPTimesR /SPTimesR . 10: Huv uv u 11: if ReSPTimesRu = 0 do R = 0.0001. 12: Huv 13: if SPTimesRu > 0 do 14: LvR = 1 − (SPTimesRv /RecTimesRv ). 15: Calculate: R ; fEnergyR = mR · µR · L R . REnergyRuv = mRu · Huv v v v 16: return flag = (REnergyRij > fEnergyRj ).

Whether node v will re-spread the rumor or not depends on the factual propagation attributes, but the rumor influence will be propagated to node v. Rule ‘‘1-1’’ is the basis for rumor propagation in a social network; all of the other rules are based on it. Algorithm (1) describes, in pseudocode, the process of rule 1, and it is also used for simulations of the ECRModel. Rule 2: In the situation shown in Figure 2(b), the spreader node u has more than one follower (‘1-to-many’). Once node u spreads a rumor in the social network, the propagation to each follower is independent and follows rule 1. Algorithm (2), in pseudo code, implements this propagation rule. Rule 3: In the situation shown in Figure 2(c), node v can receive rumors from many spreaders (‘many-to-1’). At time t, once these spreaders spread the same rumor to node v, the rumor will be propagated to node v from precursors sequentially. It is an accumulated propagation. Algorithm 3, with pseudocode, describes the propagation rule. Rule 4: This rule considers a comprehensive situation in which the propagation occurs in a ‘‘many-to-many’’ style, and makes use of the other three rules. In a real social network, it is a common phenomenon to spread rumors or other information, where each spreader may spread rumors to many followers and each follower may receive rumors from 6111

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Algorithm 2 RumorPropagationRule_1ToN(G, u,FV) Input: A Social Graph G =< V , E >; Spreader: u ∈ V ; follower set of node u: FV={v1 ,v2 ,v3 , . . . vn }, FV ⊆ V ; µR is an input data by the analyst according to the rumor’s characteristics. Output: A Boolean array FLAG[] whether vi in FV spreads u’s rumor or not. 1: Initialization: i = 1. 2: for each vi ∈ FV do 3: FLAG[i] = RumorPropagationRule_1To1(G, u, vi ); 4: i++; 5: return FLAG[]. Algorithm 3 RumorPropagationRule_NTo1(G,SV, v) Input: A Social Graph G =< V , E >; follower node: v ∈ V ; spreader set of node v: SV={u1 ,u2 ,u3 , . . . un }, SV ⊆ E; µR is an input data by the analyst according to the rumor’s characteristics. Output: A Boolean array FLAG[] whether node v spreads rumor of ui in SV or not. 1: Initialization: i = 1. 2: for each ui ∈ SV do 3: FLAG[i] = RumorPropagationRule_1To1(G, ui , v); 4: i++; 5: return FLAG[]. Algorithm 4 RumorPropagationRule_NToN(G,SV, FV) Input: A Social GraphG =< V , E >; Spreaders set: SV={u1 ,u2 ,u3 , . . . un }, SV ⊆ V ; followers set: FV={v1 ,v2 ,v3 , . . . vn }, FV ⊆ V ; µR is an input data by the analyst according to the rumor’s characteristics. Output: A Boolean array FLAG[]. 1: Initialization: i = 1,j = 1,k = 1. 2: for each ui ∈ SV do 3: for each vj ∈ FV do 4: if < ui , vj >∈ E do 5: FLAG[k] = RumorPropagationRule_1To1(G, ui , vj ); 6: k++; 7: j++; 8: i++; 9: return FLAG[].

FIGURE 3. States in the ECRModel.

never spreads rumor’’(State 1), ‘‘active and spreads rumor forward’’ (State 2), and ‘‘inactive but has previously spread rumor’’ ( State 3). Figure 3 shows the relations between these states. Let S1 (t), S2 (t) and S3 (t) denote the probability densities of nodes in these three states at time t, with S1 (t) + S2 (t)+ S3 (t) = 1, and S1 (t), S2 (t), S3 (t) ∈ [0, 1]. Let k¯ denote the average out-degree of all nodes in a social network G = and let P deg+ i i∈V . k¯ = |V | Proposition (1): In the ECRModel, the critical condition of rumor spreading in a social network is ϕ( µ1R ) · k¯ < 1. Proof: dS1 (t) dt dS2 (t) dt dS3 (t) dt dS3 (t) dS1 (t)

1 ) · k¯ µR 1 = S1 (t) · S2 (t) · ϕ( R )k¯ − S2 (t) µ = −S1 (t) · S2 (t) · ϕ(

= S2 (t) =

S2 (t) −S1 (t) · S2 (t) · ϕ( µ1R ) · k¯

=−

1 S1 (t) · ϕ( µ1R ) · k¯

.

Then, dS3 (t) 1 =− 1 dS1 (t) ϕ( µR ) · k¯ · S1 (t) ⇒ S3 (t) = −

1 ϕ( µ1R ) · k¯

· ln (S1 (t)) + c,

where c is a constant. Specially, at the very beginning, t = 0, S1 (0) = 1 and S2 (0) = S3 (0) = 0, so we can obtain the constant c = 0. Therefore, S3 (t) = −

1 ϕ( µ1R ) · k¯

· ln (S1 (t)) .

Since lim S1 (t) = 1 − S3 (t), we obtain t→∞

many spreaders. This rule repeatedly calls rule 2 and rule 3 to complete the propagation, while rule 2 and rule 3 are based on rule 1. Thus we design algorithm (4) with pseudo code based on rule 1.

lim S3 (t) = −

t→∞

6112

· ln (1 − S3 (t)) .

If rumors can spread in networks, lim S3 (t) must belong t→∞ to (0, 1], so

V. STEADY-STATE ANALYSIS

As we know from a previous section, at time t, each node v ∈ V usually has three possible states, ‘‘inactive and

1 ϕ( µ1R ) · k¯

S3 (t) = −

1 ϕ( µ1R ) · k¯

· ln(1 − S3 (t)). VOLUME 4, 2016

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Then, we can obtain the following equation. 1

f (S3 (t)) = S3 (t) +

ln(1 − S3 (t)) = 0.

ϕ( µ1R ) · k¯

Since f (0) = 0 and f (S3 (t) → 1− ) = 0 exist, f 0 (S3 (t)) = 1 −

1 ϕ( µ1R ) · k¯

· (1 − S3 (t))

.

Thus, f 00 (S3 (t)) = −

1 ϕ( µ1R ) · k¯

· (1-S3 (t))2

< 0.

1 ϕ( µ1R ) · k¯

> 0.

This means that rumors can spread in networks only when ϕ( µ1R ) · k¯ < 1. Proposition (2): In the ECRModel, ϕ( µ1R ) · k¯ is positively correlated with density S3 (t): higher ϕ( µ1R ) and k¯ will result in higher S3 (t), when all other conditions remain unchanged. Proof: From proposition (1), we know that f 0 (S3 (t)) is a monotone decreasing function. At the end of rumor propagation, density S3 (t) will be steady, and f 0 (S3 (t)) = 0, then f 0 (S3 (t)) = 1 − ⇒ S3 (t) = 1 −

1 ϕ( µ1R ) · k¯ 1 ϕ( µ1R ) · k¯

· (1 − S3 (t))

=0

.

Thus, when all other conditions remain unchanged, ϕ( µ1R ) · k¯ is positively correlated with density S3 (t), and higher ϕ( µ1R ) and k¯ will result in higher S3 (t). Proposition (3): In the ECRModel, the non-confusingness degree µR is negatively correlated with the spreading probability ϕ( µ1R ), when all other conditions remain unchanged. formulas (15) and (16), the distribution of Proof: From  x = condijR can be regarded as the normal distribution R

2

(condij −E(X )) 1 f (x) = √ · e− 2·D(X ) . 2π · D(X )

The density function of f (x) is  Z cond R  ))2 ij 1 − (t−E(X 2·D(X ) ϕ(x) = ·e dt. √ 2π · D(X ) −∞ So, ϕ(x) is a monotone increasing function. Thus, lower µR will result in higher µ1R and higher spreading VOLUME 4, 2016

2

(x−E(X1 )) 1 − · e 2·D(X1 ) , 2π · D(X1 ) (x−E(X2 ))2 1 − f2 (x) = √ · e 2·D(X2 ) . 2π · D(X2 )

f1 (x) = √

So, f 0 (S3 (t)) is a monotone decreasing function. Now, when S3 (t) = 0, we set hypothesis condition (1) as f 0 (0) ≤ 0. Under this hypothesis condition, f (S3 (t)) is a monotone decreasing function. We know that f (0) = 0, so f (S3 (t)) < 0. Therefore, in the range (0, 1], the equation f (S3 (t)) = 0 does not exist. Thus, hypothesis condition (1) is false. So f 0 (0) > 0, and f 0 (0) = 1 −

probabilityϕ( µ1R ). In other words, this proposition indicates that more confusingness rumors are more likely to be propagated. Proposition (4): In the ECRModel, the expected value E(X ) is negatively correlated with the spreading probability ϕ( µ1R ), when all other conditions remain unchanged. Proof: Given two sets X1 and X2 , D(X1 ) = D(X2 ), E(X1 ) > E(X2 ), and E(X1 ) = (E(X2 ) + z), where z is a natural number. According to equation (15),

So, 2

f1 (x) = √

((x−z)−E(X2 )) 1 − 2·D(X2 ) ·e . 2π · D(X2 )

Thus, f1 (x) = f2 (x −z), and ϕ1 (x) = ϕ2 (x −z). As we know ϕ(x) is a monotone increasing function, ϕ2 ( u1R − z) < ϕ2 ( u1R ), and ϕ1 ( µ1R ) < ϕ2 ( µ1R ). So, E(X ) is negatively correlated with the spreading probability ϕ( µ1R ). Proposition (5): In the ECRModel, D(X ) is positively correlated with ϕ( µ1R ) if E(X ) ≥ µ1R , and negatively correlated if E(X ) < µ1R , when all other conditions remain unchanged. Proof: Given two sets X1 and X2 , D(X1 ) > D(X2 ), E(X1 ) = E(X2 ). (1) When E(X1 ) = E(X2 ) ≥ µ1R , according to equation (15), we can draw the schematic curves of f (x), as in Figure 4(a). Obviously we can see that higher D(X ) will result in higher ϕ( µ1R ) under this condition. (2) For E(X1 ) = E(X2 ) < µ1R , we draw the schematic curves off (x) in Figure 4(b). Obviously, 1 − ϕ1 ( µ1R ) > 1 − ϕ2 ( µ1R ). Thus, ϕ1 ( µ1R ) < ϕ2 ( µ1R ). So when E(X ) < µ1R , higher D(X ) will result in lower ϕ( µ1R ). VI. SIMULATIONS AND ANALYSIS

In this section, we perform simulations to verify the analytical results presented in the previous section and to further investigate the properties of the ECRModel for social networks, especially the influence of the parameters, including µR , ¯ E (X ) and D(X ), on rumor propagation. We use a small k, Facebook dataset as an experimental social graph, which can be downloaded from snap of Stanford University [24]. This data set includes 333 nodes and 5038 slides, and its original average node degree is k¯ = 15.129. Figure 5 shows the topology (drawn by Gephi [25]) and degree distribution of the experimental social graph. We design the experiments using NetworkX [26], which is a Python-language software package for the creation, manipulation, and study of the structure, dynamics, and functions 6113

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TABLE 2. Experiments for ECRModel.

FIGURE 4. Schematic curves of f (x) for X1 and X2 , D(X1 ) > D(X2 ). (a) E (X1 ) = E (X2 ) ≥ 1R . (b) E (X1 ) = E (X2 ) < 1R . µ

µ

NetworkX, and record every interaction between any nodes u and v to calculate SPTimesRu , SPTimesRv , ReSPTimesRu , ReSPTimesRv , ReSPTimesRuv and RecTimesRv . For each experiment, we initialize two seed nodes to spread rumors. We initialized 2 seed nodes at random for each experiment, and the details of the parameters are described in the following subsections. The purpose of experiments (1) to (4) is to verify the analytical results of the ECRModel, and the purpose of experiment (5) is to verify the accuracy. A. EXPERIMENT (1): INFLUENCE OF µR ON RUMOR PROPAGATION

FIGURE 5. Experimental social graph. (a) Topology. (b) In-degree distribution. (c) Out-degree distribution.

of complex networks. Table 2 describes the experiments with the initialized parameters. Based on the propagation rules, we simulate more than 100,000 interactions in this social graph using 6114

The purpose of this experiment is to verify the negative correlation between non-confusingness degree µR and rumor propagation, which is analyzed in proposition (3). In this experiment, the average out-degree is k¯ = 15.129, expected value E(X ) = 1 and variance value D(X ) = 0.667. We perform the experiment for each µR in {0.01, 0.1, 1}. At the beginning, the inactive node density S1 (t) = 1, and we observe the change in the active node density S2 (t) and the change in the density S3 (t) of nodes that are inactive but have previously spread rumors. Figure 6 shows the result. As we can see, the density of the active nodes (S2 (t)) and the density of nodes that have previously been active (S3 (t)) are generally negatively correlated with the nonconfusingness degree µR . We obtain higher densities for VOLUME 4, 2016

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FIGURE 7. State scenarios at different time steps when µR = 0.1; state 1: green; state 2: red; and state 3: yellow. (a) t = 0. (b) t = 4. (c) t = 10. (d) t = 15.

B. EXPERIMENT (2): INFLUENCE OF k¯ ON RUMOR PROPAGATION

The purpose of this experiment is to verify the positive correlation between average out-degree and rumor propagation, which is discussed in proposition (2). We generate four different sub graphs with each V 0 = V and E 0 ∈ E based on the experimental G =< V , E >. The average out-degrees k¯ of these four sub graphs are k¯ = 15.129, 12.012, 9.063 and 6.25. Figure 8 shows the degree distributions of these sub graphs. We observe the densities of S2 (t) and S3 (t) for these four subgraphs, under the same parameters, µR = 0.4, E(X ) = 1, and D(X ) = 0.667. Figure 9 shows the results. FIGURE 6. Densities of nodes S2 (t ) and S3 (t ) over time for three values of µR , with k¯ = 15.129, E(X)=1 and D(X ) = 0.667. (a) S2 (t ). (b) S3 (t ).

lower values of µR . We also observe that the rumor propagation crest occurred at the early time steps (Fig. 6(a)), and nodes become inactive in later time steps (Fig. 6(b)), especially when µR = 0.01. In fact, we can judge the rumor propagation by (S2 (t) + S3 (t)) too, which is the scope of the infected nodes. At almost any time t, we can observe that (S2 (t) + S3 (t)) is highest when µR = 0.01, and lowest when µR = 1. Thus, this experiment proved that a rumor of lower non-confusingness degree µR will be more widely propagated. In fact, µ1R denotes the confusingness degree of a rumor, since µR is the non-confusingness degree in the ECRModel. Thus, we find that a rumor with higher confusingness degree ( µ1R ) has higher advantage in rumor propagation. Since the topology is of a small scale, all the propagations are convergent in 15 time steps. We trace nodes’ states of topology at t = 0, t = 4, t = 10 and t = 15 in different colors (state 1: green; state 2: red; and state 3: yellow). Figure 7 shows the scenarios when µR = 0.1. It intuitively describes the changes of node states during different time steps. As you can see, green nodes at t=0 are gradually changed into yellow nodes at the final time step.

VOLUME 4, 2016

FIGURE 8. Degree distributions of sub graphs. (a) k¯ = 15.129. (b) k¯ = 12.012. (c) k¯ = 9.063. (d) k¯ = 6.25.

As we can see from Figure 9, the densities of active nodes (S2 (t)) and densities of nodes that have previously been active (S3 (t)) are generally positively correlated with k¯ of the social graph. When k¯ = 15.129, we obtain the highest S2 (t) with the fastest

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FIGURE 10. State scenarios at different time steps when k¯ = 12.012; state 1: green; state 2: red; and state 3: yellow. (a) t = 0. (b) t = 4. (c) t = 10. (d) t = 15.

¯ with µR = 0.4, FIGURE 9. Densities S2 (t ) and S3 (t ) over time for four k, E(X ) = 1, and D(X) = 0.667. (a) S2 (t ). (b) S3 (t ).

propagation convergence among the four subgraphs, and the highest propagation crest occurs before t=5. On the contrary, we obtain the lowest S2 (t) of the nodes when k¯ = 6.25. With regard to S3 (t), there are clearer positive correlation tendencies. This proves that node degree distribution has significant positive influence on rumor propagation, and higher average out-degree k¯ will result in higher propagation density. Meanwhile, we trace the node states of topology at t = 0, t = 4, t = 10 and t = 15, respectively, with different colors (state 1: green; state 2: red; and state 3: yellow). Here we select the subgraph with k¯ = 12.012 to show the changes in node states. Figure 10 shows the scenarios intuitively. C. EXPERIMENT (3): INFLUENCE OF E(X) ON RUMOR PROPAGATION

The purpose of this experiment is to verify the negative correlation between expected value E(X ) and rumor propagation, which is described in proposition (4). Rumor propagation events along node relationships are independent, and each propagation condition condijR =

mRj ·LjR

mRi ·HijR

in the ECRModel is

a random variable. In this experiment, we want to investigate the influence of expected value E(X ) on rumor propagation, 6116

FIGURE 11. Densities of S2 (t ) and S3 (t ) over time for four values of E(X ), with µR = 0.4, k¯ = 15.129, and D(X ) = 0.667. (a) S2 (t ). (b) S3 (t ).

P

where E (X ) =

∈E

P i∈V

condijR

deg+ i

. We set µR = 0.4, k¯ = 15.129

and D(X ) = 0.667, and then observe the densities of S2 (t) and S3 (t) for different values of E(X ) in {0.01,0.1,1, 10}. Figure 11 shows the results. As we can see from the plots, the densities of S2 (t) and S3 (t) are higher for higher values of expected value E(X ). VOLUME 4, 2016

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The highest rumor propagation occurs under the lowest E(X ) = 0.01, while the lowest propagation occurs under the highest E(X ) = 10. This proves that lower expected value of rumor propagation results in higher scope of rumor spreading. It also proves that the rumor propagation crest occurs at the early time steps, and propagations are convergent in the later time steps.

FIGURE 12. State scenarios at different time steps when E(X ) = 1; state 1: green; state 2: red; and state 3: yellow. (a) t = 0. (b) t = 4. (c) t = 10. (d) t = 15.

Additionally, we trace the nodes’ states of topology at t = 0, t = 4, t = 10 and t = 15, with different colors (state 1: green; state 2: red; and state 3: yellow). Here we select scenarios when E(X ) = 1 as representative results of tracing, as shown in Figure 12. As we can see, the green nodes, which are inactive at t=0, are gradually changed into almost yellow at t=15. And the red nodes, which are active in spreading rumors, reach a summit at t=5. The tracing experiment also proved the analytical result in another form. D. EXPERIMENT (4): INFLUENCE OF D(X) ON RUMOR PROPAGATION

The purpose of this experiment is to verify the analytical results for D(X ), which are described in proposition (5). As we know from section III, D (X ) is the variance of X in a social network G=, where X is the set of random variables x = condijR . We perform the experiment under the conditions E(X ) < µ1R and E(X ) ≥ µ1R . (1) Firstly, we set parameters µR = 0.4, k¯ = 15.129 and E(X ) = 0.5 < µ1R , and then observe the S2 (t) and S3 (t) for different D(X ) in {0.0067, 0.0267, 0.1067}. Figure 13 shows the results under this condition. We observe that the rumor propagation is negatively correlated with D(X ) in this situation with E(X ) < µ1R . With regard to density S2 (t) shown in Figure 13(a), before convergence, the highest density occurs under the condition D(X ) = 0.0067, and the lowest occurs when D(X ) = 0.1067, while an intermediate density occurs with D(X ) = 0.0267. When given a higher D(X ), the rumor propagation scope VOLUME 4, 2016

FIGURE 13. Densities S2 (t ) and S3 (t ) over time for three values of D(X ) while E(X ) < 1R . (a) S2 (t ). (b) S3 (t ). µ

is more limited. We also observe that there are multiple propagation crests, especially when D(X ) is lower. This proves that spreaders are preferentially spreading rumors with lower D(X ). With regard to density S3 (t) shown in Figure 13(b), when D(X ) = 0.0067 (the lowest value considered), the density S3 (t) reaches a peak between these three curves. This proves that the nodes can more quickly return to the inactive state under lower D(X ). Meanwhile, we trace the nodes’ states of topology at t = 0, t = 4, t = 10 and t = 15. Figure 14 shows the scenarios in which D(X ) = 0.0067. It also intuitively describes the tendency of the node states. (2) Secondly, we observe the densities of S2 (t) and S3 (t) for different values of D(X ) in {0.6667, 6, 24} under the same parameters, µR = 0.4 and k¯ = 15.129. Meanwhile, we set E(X ) = 10 to ensure that E(X ) ≥ µ1R . Figure 15 shows the results. As we can see, the densities S2 (t) and S3 (t) are relatively negatively correlated with D(X ) in this situation. When given a higher D(X ), the rumor propagation is more drastic. We can also observe that the total propagation density (S2 (t) + S3 (t)) was always highest when D(X ) = 0.6667, and lowest when D(X ) = 24. 6117

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FIGURE 14. State scenarios at different time steps when D(X ) = 0.0067 and E(X ) < 1R ; state 1: green; state 2: red; and state 3: yellow. (a) t = 0. µ

(b) t = 4. (c) t = 10. (d) t = 15.

FIGURE 16. State scenarios at different time steps when D(X)=0.6667 and E(X ) ≥ 1R ; state 1: green; state 2: red; and state 3: yellow. (a) t = 0. (b) µ

t = 4. (c) t = 10. (d) t = 15. TABLE 3. Simulated Rumor Dataset.

FIGURE 17. Comparison between ECRModel and SIR Model.

FIGURE 15. Densities S2 (t ) and S3 (t ) over time for three values of D(X ), while E(X ) ≥ 1R . (a) S2 (t ). (b) S3 (t ). µ

Meanwhile, in this situation, we trace the nodes’ states of topology at t = 0, t = 4, t = 10 and t = 15. Figure 16 intuitively shows the scenarios when D(X ) = 0.6667. The two experiments in this subsection verify our previous analytical results that higher D(X ) will result in higher ϕ( µ1R ) when E(X ) ≥ µ1R , and higher D(X ) will result in lower ϕ( µ1R ) when E(X ) < µ1R . 6118

E. EXPERIMENT (5): COMPARISON WITH SIR MODEL

To verify the accuracy of the ECRModel, we perform this experiment comparing the ECRModel with the SIR model. Firstly, we generate interactions in the experimental topology with nine simulated rumors, and observe the parameters. Table 3 shows the results of the simulated rumor dataset. The simulations are performed over 10 rounds, from round one with rumor-1 to round ten with rumor-10. And all of the observed values of E(X ) and D(X ) are generated iteratively. Then, we perform an experiment to observe the propagation prediction accuracies of both SIR and the ECRModel, forrumor-10. Figure 17 shows the results. VOLUME 4, 2016

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As we can see from this chart, the ECRModel more closely models the propagation trends of rumor-10, while SIR has a drastic crest that differs from the actual propagation behavior of rumor-10. This proves that our ECRModel has higher accuracy than the traditional epidemic-like SIR Model because the ECRModel learns more experiences from past interactions. VII. CONCLUSIONS

The dynamics of rumor propagation in online social networks (OSN) is quite different from the dynamics of virus or disease transmission. Traditional epidemic-like models are not suitable for OSN. To find a novel pattern to analyze and discover rumor propagation phenomena in OSN, we propose a ball elastic collision-based rumor-propagation model (ECRModel), inspired by the kinetic energy transformations in elastic collisions. In the proposed model, we divide the users into three groups according to three states, including ‘inactive and do not spread rumors’, ‘active’, and ‘inactive but have previously spread rumors’. In the ECRModel, transmission behaviors are modeled with multiple parameters, and rumor propagation rules are designed by algorithms, considering both individual features with detailed attributes and integral features with node-state densities. We analyze the probability densities for different types of nodes, and determine the steady state both analytically and through simulations. The results prove that the ECRModel is rational and more suitable for analyzing rumor propagation in social networks. ACKNOWLEDGMENT

This manuscript was edited for proper English language, grammar, punctuation, spelling, and overall style by the highly qualified native English speaking editors at American Journal Experts. The authors also thank Lorretta Ayebazibwe for her help of language polishing to this manuscript. REFERENCES [1] Y. Jiang and J. C. Jiang, ‘‘Understanding social networks from a multiagent perspective,’’ IEEE Trans. Parallel Distrib. Syst., vol. 25, no. 10, pp. 2743–2759, Oct. 2014. [2] D. López-Pintado, ‘‘Diffusion in complex social networks,’’ Game. Econ. Behavior, vol. 62, no. 2, pp. 573–590, Mar. 2008. [3] Y. Jiang and J. C. Jiang, ‘‘Diffusion in social networks: A multiagent perspective,’’ IEEE Trans. Syst., Man, Cybern., Syst., vol. 45, no. 2, pp. 198–213, Feb. 2015. [4] Y. Wang, A. V. Vasilakos, J. Ma, and N. Xiong, ‘‘On studying the impact of uncertainty on behavior diffusion in social networks,’’ IEEE Trans. Syst., Man, Cybern., Syst., vol. 45, no. 2, pp. 185–197, Feb. 2015. [5] W. Chen, L. V. S. Lakshmanan, and C. Castillo, ‘‘Stochastic diffusion models,’’ in Information and Influence Propagation in Social Networks. San Rafael, CA, USA: Morgan & Claypool, 2014, pp. 9–35. [6] D. Kempe, J. Kleinberg, and E. Tardos, ‘‘Maximizing the spread of influence through a social network,’’ in Proc. SIGKDD, Wangshington DC, USA, 2003, pp. 137–146. [Online]. Available: http://www.cs.cornell.edu/home/kleinber/kdd03-inf.pdf [7] D. J. Daley and D. G. Kendall, ‘‘Epidemics and rumours,’’ Nature, vol. 204, no. 225, p. 1118, Dec. 1964. [8] D. P. Maki and M. Thompson, Mathematical Models and Applications: With Emphasis on the Social, Life, and Management Sciences. Englewood Cliff, NJ, USA: Prentice-Hall, 1973. VOLUME 4, 2016

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ZHENHUA TAN received the B.S., M.S., and Ph.D. degrees from Northeastern University, Shenyang, China, in 2003, 2006, and 2009, respectively, all in computer science. He is currently an Associate Professor with the College of Software, Northeastern University. He holds three U.S. patents about networking and security. He has published over 30 journal articles, books and book chapters, and refereed conference papers. His current research interests include networking behaviors analysis and information security. JINGYU NING received the B.E. degree in information security from Northeastern University, Shenyang, China, in 2014, where she is currently pursuing the master’s degree with the College of Software. She has authored several journal articles and conference papers. Her current research interests include networking behaviors analysis and information security.

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YUAN LIU (M’16) received the B.Sc. degree from the Honors School, Harbin Institute of Technology, China, in 2010, and the Ph.D. degree with School of Computer Engineering, Nanyang Technological University, Singapore, in 2014. She is currently an Associate Professor with Northeastern University, Shenyang, China. Her research interests include trust-based incentive mechanism design, multiagent system, trust management, human computation systems, and robustness evaluation. Her research papers have been published in top international conferences in the area of artificial intelligence, such as AAAI 2014, AAAI 2015, AAMAS 2013, and AAMAS 2014.

GUANGMING YANG received the B.S. degree in computational mathematics from the Mathematics Department, Liaoning University, Shenyang, China, in 1983. From 1983 to 2003, he was with the Institute of Computing Technology, Chinese Academy of Sciences. Since 2004, he has been a Professor with the College of Software, Northeastern University, China. He has authored over 40 journal articles, books and book chapters, and refereed conference papers. His current research interests include information security and computer operating system.

XINGWEI WANG received the B.S., M.S., and Ph.D. degrees in computer science from Northeastern University, Shenyang, China, in 1989, 1992, and 1998, respectively. He is currently a Professor with the College of Software, Northeastern University. He has authored over 100 journal articles, books and book chapters, and refereed conference papers. His current research interests include cloud computing, future Internet, and information security. He has received several best

WEI YANG received the B.S. degree in computer science from Northeastern University, Shenyang, China, in 1990. He was with Neusoft Corporation. In 1993, he was with Alpine Electronics, Inc., Japan, and also with NeuSoft as a Software Manager. He is currently the Director of the Educational System Development and Operation Department, Neusoft. He has developed over 10 related products. His current research interests include networking behavior analysis, e-learning,

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