## 3 Means and Mathematical Formula of Trust
### Direct Trust Model  
Based on the above analysis, the direct trust computation function \(f_{\textit{dir}}\) can be defined as  
\[f_{\textit{dir}}(t)=f(T_{e}(t),T_{c}(t),T_{p}(t)) \tag{1}\]  
where \(T_{e}(t),T_{c}(t)\) and \(T_{p}(t)\) are trust value concerning environments, service clients, and the service provider, the values of which are limited in domain [0,1], and \(t\) is denoted as time. If a simple multiple function is used to derive the explicit form of Equation (1), the direct trust evaluation function \(f_{\textit{dir}}\) can be written as  
\[f_{\textit{dir}}(t)=T_{e}(t)*T_{c}(t)*T_{p}(t)) \tag{2}\]Following function \(T_{e}(t),T_{e}(t)\) and \(T_{p}(t)\) are defined separately based on the factors that influence values of trust. We notice that \(T_{e}(t)\) is related with the attitude of society towards techniques and societal factors. The change of attitude towards techniques and societal factors is very slow. So the values of \(T_{e}(t)\) is assumed to be some constant. \(T_{e}(t)\) is related to service clients. The different backgrounds of service clients, such as disposition to trust, attitude, and personal values can also be viewed as a constant. In our simulations, three classical types of service clients, which are risk taking, cautious, and conservative, are discussed. \(T_{p}(t)\) is related to the service provider, which includes the factors of websites, merchants, and online firms. The changes of values of dynamical direct trust are mainly dependent on the changes of values of \(T_{p}(t)\). Two situations those the values of \(T_{p}(t)\) change are simulated, and these are the procedures of improvement and descent. In the process of improvement of the values of \(T_{p}(t)\), there are three main stages: establishment stage, improvement stage, and stabilization stage. The derivative function of \(T_{p}(t)\) can be simulated by the following equation [52]  
\[\frac{dT_{p}(t)}{dt}=\alpha(1-(\frac{e^{\alpha}-e^{-\alpha}}{e^{\alpha}+e^{- \alpha}})^{2}) \tag{3}\]  
where \(\alpha\) is the parameter of the derivative function of \(T_{p}(t)\). From Equation (3), \(T_{p}(t)\) can be solved as  
\[T_{p}(t)=\frac{e^{\alpha}-e^{-\alpha}}{e^{\alpha}+e^{-\alpha}} \tag{4}\]  
We see that \(T_{p}(t)\) is the transformation of Hyperbolic Tangent. The images of \(T_{p}(t)\) and its derivative function shows that in the trust establishment stage, values of trust do not increase rapidly. In the improvement stage, however, values of trust do increase rapidly. In the stabilization stage, values of trust do not increase rapidly. Therefore, Equation (4) simulates the processes of the improvement of trust values. The derivative function of the process of the decrement of values of \(T_{p}(t)\) can also be similarly simulated by following equation  
\[\frac{dT_{p}(t)}{dt}=-\alpha(1-(\frac{e^{\alpha}-e^{-\alpha}}{e^{\alpha}+e^{- \alpha}})^{2}) \tag{5}\]  
The corresponding function \(T_{p}(t)\) can be solved as  
\[T_{p}(t)=\frac{e^{-\alpha}-e^{\alpha}}{e^{-\alpha}+e^{\alpha}} \tag{6}\]  
There are two stages of the decrement processes. In the first stage, values of \(T_{p}(t)\) fall quickly. Then, the values of \(T_{p}(t)\) will be relatively low slowly. Concrete numerical simulations of the trust computation model will be discussed in Section 5. The reputation computation model based on trust will be further discussed.