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Social networks are an important infrastructure for information, viruses and innovations propagation. Since users’ behavior has influenced by other users’ activity, some groups of people would be made regard to similarity of users’ interests. On the
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  International Journal of Computer Networks & Communications (IJCNC) Vol.5, No.5, September 2013 DOI: 10.5121/ijcnc.2013.5512 161 A GENERAL STOCHASTIC INFORMATION DIFFUSION MODEL IN SOCIAL NETWORKS BASED ON EPIDEMIC DISEASES Hamidreza Sotoodeh 1 , Farshad Safaei 2,3 , Arghavan Sanei 3 and Elahe Daei 1   1  Department of Computer Engineering, Qazvin Islamic Azad University, Qazvin, IRAN {hr.sotoodeh, e.daei} 2   School of Computer Science, Institute for Research in Fundamental Sciences (IPM), P.o.Box 19395-5746, Tehran, IRAN 3  Faculty of ECE, Shahid Beheshti University G.C., Evin 1983963113, Tehran, IRAN,  ABSTRACT Social networks are an important infrastructure for information, viruses and innovations propagation. Since users’ behavior has influenced by other users’ activity, some groups of people would be made regard to similarity of users’ interests. On the other hand, dealing with many events in real worlds, can be justified in social networks; spreading disease is one instance of them. People’s manner and infection severity are more important parameters in dissemination of diseases. Both of these reasons derive, whether the diffusion leads to an epidemic or not. SIRS is a hybrid model of SIR and SIS disease models to spread contamination. A person in this model can be returned to susceptible state after it removed. According to communities which are established on the social network, we use the compartmental type of SIRS model. During this paper, a general compartmental information diffusion model would be proposed and extracted some of the beneficial parameters to analyze our model. To adapt our model to realistic behaviors, we use Markovian model, which would be helpful to create a stochastic manner of the proposed model.  In the case of random model, we can calculate probabilities of transaction between states and predicting value of each state. The comparison between two mode of the model shows that, the prediction of population would be verified in each state.  KEYWORDS  Information diffusion, Social Network, epidemic disease, DTMC Markov model, SIRS epidemic model 1. INTRODUCTION In recent decades, networks provide an infrastructure that economic, social and other essential revenues are depending on. They can form the physical backbones such as: transportation networks (convey vehicle flows from sources to destinations), construction and logistic ones (provide transforming the row material and presenting the ultimate products), electricity and power grid ones (consign required fuels) and Internet ones [1] (provides global public accesses and communications). These structures lead to thousands of jobs, social, politics, economics, and other activities. Moreover, complex physical networks are also established such networks, like financial, social and knowledge networks, and those ones are under development as smart grid [2]. Social network   has declared as a structure, that its entities can communicate with each other through various ways. These entities denote as users [3]. So, users play the main role in construction of social networks. Since, the social networks are abstraction of a real world; many social phenomena can be modeled at the level of social networks [4]. Dealing with all kind of diseases among population of a  International Journal of Computer Networks & Communications (IJCNC) Vol.5, No.5, September 2013   162   society, could be one major issue of them. Communication between infectious and susceptible users makes dissemination of these diseases. Outbreak of a disease and also, how people behave in the face of contagious can be the reason of modeling information diffusion in social networks [5].  Information diffusion  is a general concept and is defined as all process of propagation, which doesn’t rely on the nature of things to publish. Recently, the diffusion of innovations and diseases over social networks has been considered [6]. These models assume users in a social network are influenced by the others, in other words, they model processes of information cascades [7]. This process makes an overlay network on the social or information network. Context network and power of data influence effect on, how data spreads over the network [7, 8]. Many mathematical studies have been done on disease diffusion , assuming population in a society has totally homogeneous structure (i.e., individuals behave exactly the same as each other) [9, 10]. This assumption allows writing easier the nonlinear differential equations, which describe the individual’s behavior. But, this assumption is not realistic; because, the structures and features of individuals are not the same as each one [11]. For example, they don’t have the same capability of transferring and caching diseases. Therefore, population can be divided into some groups, that individuals in each community have similar abilities and structures; however, they have different capabilities in comparison with the other communities. This model refers to the compartmental epidemic models , which treat nearly real treatments [12]. By the thanks of difference equations, we can obtain the number of infected individual as a function of time. Also, it can be obtained the size of diffusion and be discussed about, whether the epidemic has occurred or not? Furthermore, epidemic behavior usually declares via a transition phase, which takes place whenever it can be jumped from epidemicless state to a condition which contains that. Basic reproduction number ( 0  R ) is a trivial concept in epidemic disease and determines this mutation from these two states [10]. This parameter has been defined as a threshold  , which if  0  R1 > , then the spreading of the contamination can be occurred through the infected individual. On the other hand, if  0  R1 < , then contamination cannot outbreak among all population, disease will be disappeared. So, the epidemic doesn’t occur [13]. Generally, epidemic models divided into two deterministic and stochastic categories [14]. The most important concern of deterministic models is their simplicity, which proposed in communities for large population. Usually in these models, we can find some questions as [15]: •   Will the entrance of a disease lead to an epidemic manner? Or when does a disease go into the epidemic? •   How many people are affected? Or what is the influence of immunization (vaccination) to the part of a community that has been incurred? •   ... However, as mentioned earlier, the oblivious fact is people are in different level of infections. If a group of people connected to one infected person, all those population could be affected. By grouping of individuals with their capacity, there are still some questions [15]: •   What is the probability, that individuals get the infection per each group? •   Or sometimes, the situations may arise that, how we can predict the probability of disease dissemination in a large population? Stochastic methods are usually suitable to response these questions. However, these ones have more complication than deterministic methods. Better solution is presenting a stochastic model based on deterministic one. Three approaches have been proposed of these methods [16], two of them are related to DTMC 1  and CTMC 2  Markovian models and the third one suggests using SDE 3  methods. 󰀱   Discrete Time Markovian Chain.   󰀲   Continuous Time Markovian Chain.    International Journal of Computer Networks & Communications (IJCNC) Vol.5, No.5, September 2013   163   In this paper, our goal is present a general model of information diffusion, which is based on epidemic diseases. In fact, this model actually is a result of developing SIRS deterministic model and including compartmental assumption. We aim to adapt our proposed deterministic model to stochastic one by using discrete-time Markovian model. It should be expressed that, the graph structure impacts to process of diffusion definitely; however, in this paper this affection is not be considered and assumed all structures have the same affection. The outline of the paper is as follows: we first introduce information diffusion models and related work of them especially about epidemic models in section 2;  then, propose a general model to diffusion based on SIRS deterministic epidemic model in section 3;  moreover, we get into the analytical model description and obtaining important threshold measurement for our model during this section. Another major aim of this section is leading our deterministic model to a stochastic one by the help of DTMC model. Finally, in section 4, we bring experimental results, in this section we are going to validate our measures that have gotten from our model. 2. RELATED WORK Researchers usually consider two approaches to model information diffusion, dependent and independent of graphs. Graph-based approaches focus on topology and graph structure to investigate of their impact on spreading processes. Two of the most important diffusion models in this class are  Linear Threshold   (LT) [8] and  Independent Cascade (IC) [17] models. These are based on a directed graph  ,  where each node can be activated or inactivated (i.e., informed or uninformed). The IC model needs probability to be assigned to each edge, whereas the LT needs an influence degree to be declared on each edge and an influence threshold for each node. Both of the models do the diffusion process iteratively in a synchronous way along a discrete-time, starting from a set of initially activated nodes. In the IC, newly activated nodes try to activate their neighbors with the probability defined on their edges. This activity has been done for per iteration. On the other hand, in LT, the active nodes join to activate sets by their activated neighbors when the sum of the influence degrees goes over their own influence threshold. This event is done at per iteration of this process. Successful activations are effective at the next iteration. In both models, the diffusion end while there is no neighboring node can be contacted. These two models rely on two different perspectives: IC refers to sender-centric and LT is receiver-centric approach. With the sake of both these models have the inconvenience to proceed in a synchronous way along a discrete time, which doesn’t suitable in real social networks. In order to establish more consistency on real networks, can referred to  ASIC   and  ASLT   models [18], which are asynchronous extension of these models. These mechanisms use continues time approach and for this, each edge would be equipped a time-delay parameter. In the case of independent graph models, there isn’t any assumption about graph structure and topology impaction on the diffusion. These models have been mainly developed to model epidemic processes. Nodes are organized to various classes (i.e. states) and focused on evaluation of proportion of nodes in each state. SIR  and SIS   are two famous instant of these models [9, 19]. Acronyms “S” is for susceptible state, “I” declare infectious (informed person) state. In both   models  ,  nodes in class “S” can exchange their state to “I” with a constant rate (e.g. α  ). Then  ,  in SIS model nodes can make a transition to “S” state again with the constant rate (e.g.  β  ). In case of SIR model nodes, can switch to “R” (stand for Removed/Recovery) situation permanently. The percent proportion of population determine by the help of difference equations. Both models assume that every node has the same probability to be connected to another; thus connections inside the population are made at random. But, the topology of the nodes relations is very important in social network; as a result, the assumptions made by these models are unrealistic. A good survey regarding to analysis, developing, vaccination and difference equations of populations growth has been done in [20], and this research is based on SIR  and SIS  , two famous epidemic models. 󰀳   Stochastic Differential Equation.    International Journal of Computer Networks & Communications (IJCNC) Vol.5, No.5, September 2013   164   But, as mentioned earlier, in these models assume, whole the network as a homogeneous and all the individuals is in equal statuses (i.e. similar node degree distributions, equal infection probability and …). In case of these models, each person has the same relation to another one and contagion rate is determined by density of infected peoples, which mean dependent to the number of infected individuals. The virus propagation on homogeneous networks with epidemic models has been analyzed by Kephart and Wiht [13]. One of the important things that could be obvious is real networks including social networks; router and AS 4  networks follow a power low structure instead. One study has done on non-homogeneous models and in there population divided into separate communities with each own features [13]. These models are part of compartmental models. The result can be deduced from [13], is using one important theorem, which declares the relation between threshold parameter and individual relationships as matrix Eigenvalues. Following the work of [20], many epidemic models have developed till now, which each one can be useful as their applications. A good reference that surveyed these models could be found in [21]. The compartmental models have been developed as a good manner in [22]; the authors’ work is based on SIR model and susceptible individuals decomposed to different groups with the same properties. In this model immunized people will permanently vaccinate and would not return to population. This proposed model, have been developed afterwards and been created a new differential model, which immunized people after a time can join to susceptible group [23]. Moreover, there are two time periods, first one, latent time period to appearance symptoms of disease that a person infected, and the second one is to start of transmission. Then, they present the extended SEIRS model for virus propagation on computer networks. Given the deterministic models represent the equations of population changes as well, but random behavior of users in real social networks, cause deterministic models to stochastic ones. One of the rich study have focused on three methods to obtain stochastic models, according to deterministic models [16]. Two of them related to  DTMC   and CTMC   Markovian models and the third one is suggested to use SDE methods for this aim. Epidemic threshold is one of the important parameters, which get considered in all studies. A survey has been investigated on this parameter as their major applications [24]. Pastor-Satrras and Vespignani [25] have studied virus propagation on stochastic networks with power low distribution structure. In such networks, have been showed that, threshold has a trivial value meaning; that even an agent with extremely low infectivity could be propagated and stayed on them. “Mean-field” approach is used by them, where all graphs should have similar behavior in terms of viral propagation in recent work; Castellano and Pastor-Satorras [26] empirically argue that, some special family of random power-low graphs have a non-vanishing threshold in SIR model over the limitation of infinite size, but provide no theoretical justification. Newman [10, 27] studied threshold for multiple competing viruses on special random graphs, accordingly mapping SIR model to a percolation problem on a network. The threshold for SIS model on arbitrary undirected networks has been given by Chakrabarti et al. [28] and Ganesh et al.; finally, Parkash et al. [29], focuses on the arbitrary virus propagation models on arbitrary, real graphs. 3. THE PROPOSED DIFFUSION MODEL The model, which we will propose for information diffusion in a social network, inspired of SIRS   deterministic model of epidemic disease [30]. The person who is susceptible  (i.e. waiting to get information) denoted by “S”, and the person who gets a new information will run into active , “A” state. Finally, rest of the population belongs to deactivate  state would be appeared with “D”. Deactivated users may return to networks after period of a time. A person who has returned, count for a susceptible one, since it can be influenced another reason and transfers to the same or different infection states. According to the nature of users’ behavior in social networks, the proposed model inclined toward a compartmental model. This means that, each state can be divided into several groups regarding to users’ features. All of 󰀴   Autonomous Systems    International Journal of Computer Networks & Communications (IJCNC) Vol.5, No.5, September 2013   165   the people in a group are similar to each other, whereas people in different groups are distinct. Figure 1 shows the state diagram of this model in general. Before analyzing the model in details, it is necessary to explain some assumptions, that the model is based on them. Figure 1.  A visual representation of our model for m  compartments in each state. Here is a probability of transmission a susceptible user to any of active state (who is informed) or deactivate one. Also, there would be possible that move to a deactivate state after it informed. All the arcs are included with the probability (rates), which these transmissions would occur . 3.1. MODEL ASSUMPTIONS The main assumptions of this model can be summarized as follows: 1.  Total population size is constant,  N (  ( ) ( ) ( ) mmmi1i1i1 iii  NStAtDt  = = = = + + ∑ ∑ ∑ ) . This assumes that the space is closed world, which have been considered internal constraints and relationships only. It would be impossible that information come from the external source in network. 2.  Each one of the susceptible , active  and deactivate  states divided into m different groups according to the user’s features. 3.  Any new node (i.e. user) has been added into the network; it would be susceptible and is belong to one of the i S  states. The rate of joining to each of  i S   groups would be constant i b . 4.  The natural death rate of the nodes for each group would be constant i d  . This rate explains the permanent disjoint according to the network; however, at deactivate state i  D , which exists a probability to return to the network. 5.  Different users of each groups, i S  can migrate to one of the active groups,  j  A with the rate ij λ .
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