# Reliability Model Based Dynamic Multi-Level Trust Analysis
## 4 Dynamic Changes in Trust  
Trust information is continuously accumulated during the interaction process. Therefore, trust information will continuously change as time proceeds. Moreover, changes in the trust information  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Path 1** & **Path 2** & **Combined Trust** \\ \hline
1 & \(i\) (\(0<i<=k-1\)) & \(j\) (\(i<=j\)) & \(i\) \\
2 & D & D & D \\
3 & T & D & U \\
4 & D & T & U \\
5 & T/D/U & uncertainty & T/D/U \\
6 & uncertainty & T/D & T/D \\ \hline \hline \end{tabular}
\end{table}
Table 4: Trust combination rules of two paths from A to C.  
between two users may affect the trust information of other users related to them. Assume User \(A\) wants to check the trust level and probability of another User B. There are two different situations. The first one is \(A\) has no relation with \(B\) and \(A\)'s friends have no relation with \(B\) too; the second one is \(A\) has no relation with \(B\), but \(A\)'s friends have relations with \(B\). When some events happened between \(A\) and \(B\), then \(A\) will rate \(B\) on a trust level. The rate will dynamically affect the trust scores on each level of both \(A\) on \(B\) and \(A\)'s friends on \(B\).  
When \(B\) is a new user or \(A\) has no relation with \(B\), then \(A\) has uncertainty about \(B\). The initial trust information of \(A\) to \(B\) is \(dir((A,B))=(0,\ 0\ldots,0,\ 1)\), that is all the trust probabilities are 0 but uncertainty probability is 1.  
After a transaction or some events, \(A\) will have an impression on \(B\). \(A\) rates \(B\) on trust level \(i\). Then \(A\)'s direct trust scores on all levels to \(B\) will change. Assume there are \(k\) levels. \(t_{A,i}\) (\(i=0,\ldots\ k-1\)) represents the probability of trust level \(i\) of link \(A\). Generally, \"trust growing slower, declining faster\" and \"slow negative but fast positive\", if a rating/vote on level \(i\) happens, \(t_{A,i}\) is increased by Equation (6).  
\[t_{A,i}=\alpha_{1}.\mathit{sigmoid}(\beta_{i})+\alpha_{2}\ 0<\alpha_{1}\leq 1, \ 0\leq\alpha_{2}\leq 1-\alpha_{1} \tag{6}\] \[\beta_{i}=\beta_{i}+\delta\times\frac{(k-i)}{\delta n}\ 0<\delta\leq 5\]  
In Equation (6), \(\alpha_{1}\) and \(\alpha_{2}\) are constant values. \(\beta_{i}\) ranges from \(-5\) to \(5\), allowing for \(t_{A,i}\) within the range [0,1]. The incremental step of \(\beta_{i}\) is also shown in Equation (6). \(\delta\times\frac{i+1}{(k+1)n}\) represents the growth pace of \(\beta_{i}\). \(n\) is the numbers of transactions. The higher \(i\) is, the smaller the increase is. That is the trust information will converge to 1.  
The updates should guarantee \(t_{A,0}+t_{A,1}+\ldots t_{A,k}+u_{A}=1\). If \(t_{Ai}\) is increased by \(\Delta t_{Ai}\), the probabilities of other levels should be decreased by \(\Delta t_{Ai}\). \(u_{A}\) is the probability of uncertainty which means the transactions are not enough to eliminate uncertainty. We define the dynamic decrease of uncertainty as follows:  
1. if \(u_{A}>\Delta t_{Ai}\), then \(u_{A}=u_{A}-\Delta t_{Ai}\) and \(\Delta t_{Ai}=0\);
2. if \(u_{A}\leq\Delta t_{Ai}\), then \(u_{A}=0\) and \(\Delta t_{Ai}=\Delta t_{Ai}-u_{A}\).  
The lower level trust decreases faster than the higher level trust, and the uncertainty score turns to 0 faster with lower level ratings. Therefore, good transactions are encouraged.  
The changes of \(t_{Aj}(0\leq j<k,j\
eq i)\) depends on the distance between \(j\) and \(i\). For level \(i-1\) is closer to level \(i\) than level \(i-2\), the change in \(t_{A,\ i-2}\) will be greater than the change in \(t_{A,\ i-1}\). Therefore, \(t_{Aj}(0\leq j<k,j\
eq i)\) is changed as Equation (7).  
\[t_{A,j}=t_{i,\ j}-\frac{\Delta t_{Ai}}{i^{2}-i+\frac{k^{2}}{2}-k+i}\big{|}i-j \big{|},0\ \leq j!=i\leq k-1 \tag{7}\]