# Reliability Model Based Dynamic Multi-Level Trust Analysis
## 3 Multi-Level Trust Model and Problem Statement
### Multi-Level Trust Evaluation  
The trust evaluation between two users mainly includes two aspects. The first is how to evaluate the trust information of a path. The second is how to combine the trust information of multiple paths. Trust propagation and combination rules are keys to trust evaluation. In this paper, trust information includes trust, distrust and uncertainty. The propagation methods of these three types of information are different. We take an example of three parties \(A\), \(B\) and \(C\) (Figure 3) to illustrate the trust propagation rules in this paper. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Meaning** \\ \hline \(dir(A)=\big{(}t_{A,0},\ t_{A,1}\ldots t_{A,k-1},\ u_{A}\big{)}\) & The normalized trust information on the arc \(A\). \(u_{A}\) is used to represent the probability of unknown (uncertainty). \\ \(P_{p}^{\mathrm{t}}(i)=\big{(}P_{1,0},\ P_{1,1}\ldots,P_{i,k-1},\ U_{t}\big{)}\) & The trust information of the \(i\)th path from \(p\) to \(q\). \\ \(\textit{Total}(\mathrm{p},\mathrm{q})=\big{(}T_{(p,q),0},\ T_{(p,q),1}\ldots,T_{( p,q),k-1},\ U_{(p,q)}\big{)}\) & The aggregated trust information of all the paths going from node \(p\) to node \(q\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. Figure 3: Example of the three parties’ relationship. Figure 2: Multivalued decision diagram (MDD) for connection \(A\). Table 2 shows the propagation rules between these three parties, where T represents trust at any of levels 1 to \(k-1\), D is distrust or Level 0, U is level uncertainty. In the case of T, Table 3 shows the detailed propagation rules. For Situation 1 in Table 2, most of the literatures adopt the principle of \"friend's friend is a friend\". Trust is propagated to \(C\) through \(B\), then \(A\) trust \(C\) with a decaying degree. For multi-level trust, there are two cases in Table 3. When the trust level of a link is less than that of the other link in Figure 3, the smaller one is selected as the final trust level. For Situation 2 in Table 2, we adopt \"friend's enemy is enemy\", that is if \(A\) trusts \(B\), \(B\) distrusts \(C\), then A distrusts \(C\) with a decaying degree. For situation 3 in Table 2, if A has a clear opinion on \(B\), but \(B\) does not have a certain opinion on \(C\), then \(A\)'s trust in \(C\) is uncertain. The principle an \"enemy's friend is enemy\" is adopted in Situation 4, so \(A\) distrusts \(C\). However, we don't give a specific solution to the problem of whether or not the distrust can be propagated by means of the distrust, that is, how the enemy of the enemy should deal with it. Refs. [6; 34] treat enemy's enemies as friends and [3; 4] treat them as enemies. There is no certain result for this kind of propagation behavior. We cannot determine whether the enemy of the enemy can become our ally. As mentioned in [25], according to the results in the Epinion dataset, the enemy of the enemy has a 50% chance of being an ally, 50% chance of being an enemy. Therefore, our model conservatively chooses to treat the enemy of the enemy as uncertainty. It means that if a competitor whom you distrust tells you to distrust someone, you might be not able to decide which kind of attitude to take, so you likely choose to be neutral. That is, there are two or more links with distrust on a path, only uncertain trust information (uncertainty) can be generated. In addition, for the advice of strangers, we adopt a way that is consistent with our daily life: ignore it. Corresponding to our model, when there is a link in a path that uses uncertainty trust, this path can only generate neutral trust no matter how the other links on this path choose. That is, if a stranger tells you the attitude he holds towards someone, you might decide not to take into account his opinion and keep neutral because you do not know him sufficiently. In our model, trust can be propagated by trust and distrust, but distrust can only be propagated by trust. Based on the above situations, we make the following definitions:  
**Definition 1**.