Rumor spread in social network
Rumor is an important form of social communications, and the spread of rumors plays a significant role in a variety of human affairs. There are two approaches to investigate the rumor spreading process: the microscopic models and the macroscopic models. The macroscopic models propose a macro view about this process are mainly based on the widely used Daley-Kendall and Maki-Thompson models. Particularly, we can view rumor spread as a stochastic process in social networks. While the microscopic models are more interested more on the micro interactions between individuals. Rumor propagation Models. In the  last few years, there has been a growing interest in rumor propagation in Online social networks problems where different approaches have been proposed to investigate it. By carefully scrutinizing the existing literature, we categorize the works into macroscopic and microscopic approaches. Macroscopic models. The first category is mainly based on the Epidemic models where the pioneering research engaging rumor propagation under these models started during the 1960s. Epidemic models. A standard model of rumor spreading was introduced by Daley and Kendall. Assume there are N people in total. And those people in the network are categorized into three groups: ignorants , spreaders and stiflers, which are denoted as S, I, and R respectively hereinafter: The rumor is propagated through the population by pair-wise contacts between spreaders and others in the population. Any spreader involved in a pair-wise meeting attempts to “infect” the other individual with the rumor. In the case this other individual is an ignorant, he or she becomes a spreader. In the other two cases, either one or both of those involved in the meeting learn that the rumor is known and decided not to tell the rumor anymore, thereby turning into stiflers. One variant is the  Maki-Thompson model. In this model, rumor is spread by directed contacts of the spreaders with others in the population. Furthermore, when a spreader contacts another spreader only the initiating spreader becomes a stifler. Therefore, three types of interactions can happen with certain rates. Of course we always have conservation of individuals: The change in each class in a small time interval is: Since we know formula_5, formula_6 and formula_7 sum up to formula_8, we can reduce one equation from the above, which leads to a set of differential equations using relative variable formula_9 and formula_10 as follows which we  can write Compared with the ordinary SIR model, we see that the only difference to the ordinary SIR model is that we have a factor formula_15 in the first equation instead of just formula_16. We immediately see that the ignorants can only decrease since formula_17 and formula_18. Also, if which means the rumour model exhibits an “epidemic” even for arbitrarily small rate parameters. Epidemic models in social network. We model the process introduced above on a network in discrete time, that is, we can model it as a DTMC. Say we have a network with N nodes, then we can  define formula_21 to be the state of node i at time t. Then formula_22 is a stochastic process on formula_23. At a single moment, some node i and node j interact with each other, and then one of them will change its state. Thus we define the function formula_24 so that for formula_25 in formula_5,formula_27 is when the state of network is formula_25, node i and node j interact with each other, and one of them will change its state. The transition matrix depends on the number of ties of node i and node j, as well as the state  of node i and node j. For any formula_29, we try to find formula_30. If node i is in state I and node j is in state S, then formula_31; if node i is in state I and node j is in state I, then formula_32; if node i is in state I and node j is in state R, then formula_32. For all other formula_34, formula_35. The procedure on a network is as follows: We would expect that this process spreads the rumor throughout a considerable fraction of the network. Note however that if we have a strong local  clustering around a node, what can happen is that many nodes become spreaders and have neighbors who are spreaders. Then, every time we pick one of those, they will recover and can extinguish the rumor spread. On the other hand, if we have a network that is small world, that is, a network in which the shortest path between two randomly chosen nodes is much smaller than that one would expect, we can expect the rumor spread far away. Also we can compute the final number of people who once spread the news, this is given by formula_36 In networks  the process that does not have a threshold in a well mixed population, exhibits a clear cut phase-transition in small worlds. The following graph illustrates the asymptotic value of formula_37 as a function of the rewiring probability formula_38. Microscopic models. The microscopic approaches attracted more attention in the individual's interaction: "who influenced whom." The known models in this category are the information cascade and the linear threshold models, the energy model, HISBmodel and Galam's Model. HISBmodel model. The HISBmodel is a rumor propagation model that can reproduce a trend of this phenomenon and provide indicators to assess  the impact of the rumor to effectively understand the diffusion process and reduce its influence. The variety that exists in human nature makes their decision-making ability pertaining to spreading information unpredictable, which is the primary challenge to model such a complex phenomenon. Hence, this model considers the impact of human individual and social behaviors in the spreading process of the rumors. The HISBmodel proposes an approach that is parallel to other models in the literature and concerned more with how individuals spread rumors. Therefore, it try to understand the behavior of individuals, as well as their social interactions in  OSNs, and highlight their impact on the dissemination of rumors. Thus, the model, attempts to answer the following question: ``When does an individual spread a rumor? When does an individual accept rumors? In which OSN does this individual spread the rumors?". First, it proposes a formulation of individual behavior towards a rumor analog to damped harmonic motion, which incorporates the opinions of individuals in the propagation process. Furthermore, it establishes rules of rumor transmission between individuals. As a result, it presents the HISBmodel propagation process, where new metrics are introduced to accurately assess the impact of a rumor  spreading through OSNs.