# Influence Maximization in Social Network Considering Memory Effect and Social Reinforcement Effect
## 3 Influence Maximization under DCM4 VM
### Dependent Cascade Model for Viral Marketing  
Generally, the most commonly used propagation model is Independent Cascade Model (ICM) [13; 14]. However, there are only two states considered in ICM, namely active and inactive. In addition, the probability that a node changes from inactive to active is independent of the history of the process. This is insufficient for the actual viral marketing problem. In this paper, we propose a new propagation model, namely Dependent Cascade Model for Viral Marketing which is more suitable for the viral marketing problem, which also could be used for viruses spreading and any other living beings spreading.  
We stipulate that each node has five states: unknown, known, interested, accepted and exhausted. In viral marketing, unknown means that the node never heard about the product advertisement; \"known\" means that the node hears about the advertisement; \"interested\" means that the node is interested in the product but may or may not buy the product; \"accepted\" means that the node is interested in the product and purchases it; \"exhausted\" means that the node never purchases the product and also loses interest in the product.  
The state transition process is shown in Figure 1. Initially, most nodes are unknown about the product. And a small subset of initial known nodes tries to propagate the product information to their neighbors. Once known about the information, a node \(v\) becomes interested in it with probability _Pu,v_. The interested node will spread the information to its neighbors and become accepted with probability _Pv_. After spreading the information, the interested node becomes accepted or exhausted and will not spread it any more.  
As shown in Figure 1, it doesn't consider time delay [30], and time delays can be divided into two types, namely, communication delay and processing delay. But In this paper, the propagation problems we study are all concerned with the fact that there is no delay between each node of the network. The research is carried out in the case of average delay, and we will carry out further research in terms of time delay.  
We observed that whether a person will buy a product or not relies on two factors: internal factors and external factors. Internal factors reflect a person's inherent preference for the product. In other words, it is the probability that the person purchases the product without any influence from his/her friends. External factors reflect the effect of outside environment to user purchasing behaviors. For example, social reinforcement effect and memory effect are important external factors. That is, the more friends of a person have bought the product, the more likely the person will buy it.  
As shown in Figure 1, at the beginning, the user does not know the product. After the user knows it, he may become interested, and then the user may accept the product.  
Formally, for a given node \(v\), the probability \(Pv(t)\) that the node \(v\) purchases a product at time \(t\) is computed as follows [10]:  
\[P_{v}(t)=\left(P_{v}^{min}-P_{v}^{max}\right)exp^{-r1m1(v,t)-r2m2(v,t)}+P_{v} ^{max} \tag{2}\]  
Figure 1: Dependent Cascade Model for Viral Marketing.  
The components of Function (2) are summarized as follows: \(P_{v}^{min}\) is the purchasing probability reflecting the internal factors. \(m_{1}(v,t)\) and \(m_{2}(v,t)\) are the numbers of neighbors of node \(v\) at state interested and accepted until time \(t\) respectively, which reflect the memory effect. \(r1>0\) and \(r2>0\) are parameters which reflect the _social reinforcement effect_ for _interested_ and _accepted_ states respectively. \(P_{v}^{max}\) is the upper bound of the probability indicating maximal purchasing probability, which is used to control the maximal effect of _social reinforcement effect_ and memory effect when increasing \(m_{1}(v,t)\) and \(m_{2}(v,t)\), \(Pv(t)\) will gradually approach \(P_{v}^{max}\) and the speed is determined by the parameters \(r1>0\) and \(r2>0\).