## 6 Performance Evaluation
### Selection of Threshold Value \(th_{cs}\)  
Figure 5 calculates the probability of revocation success of different users under different threshold conditions, which provides a reference for setting threshold \(th_{cs}\). We count the probability curve of voting schemes when the number of users is 100, 200, 300, and 500 with the C program. We assume that the value of votes \(VW_{1}\), \(VW_{2}\), \(VW_{3}\), and \(VW_{4}\) is 1, 2, 3, and 4. The voting success probability is defined as the proportion of the number of successful voting combinations, each of which reaches \(th_{cs}\), to the number of all possible voting combinations that also include those failing to reach \(th_{cs}\). Among them, voting user combination is the combination of the number of users at four different grades responding to each voting event. For example, when the threshold value is set to 100 and the total number of users is 200, a possible voting combination can be 30 users at the Basic Grade, 10 users of the Personal Grade, 10 users of the Professional Grade, and 5 users of the Top Grade. However, if only 10 Basic Grade users and 10 Personal Grade users respond to this vote, this voting event fails. The probability curve shows the proportion of changes with the increase in the preset threshold \(th_{cs}\). The numbers of users in the domain and the threshold value \(th_{cs}\) are the two important factors affecting the probability of successful revocation. The revocation success probability decreases with the increment of the threshold value \(th_{cs}\). With the increase in the threshold \(th_{cs}\), the number of approval votes required increases, while the number of users with high grades is a small proportion, so the probability of successful vote revocation decreases. In contrast, revocation success probability increases with the increment of user numbers, because the density of the high grades also increases. Figure 4: The number of voting users. Figure 5: Revocation success probability with different numbers of users. Meanwhile, according to the analysis in Section 4.4, we also give the setting standard of threshold \(th_{cs}\) in different conspirator ratios and users. The probability of successful attack is defined as the probability of revocation failure when there is a certain proportion of attackers. Figure 6 analyzes the collusion attack of condition one (voting against revocation) in Section 4.4. The threshold value \(th_{cs}\) satisfies Formula (13). The conspirator ratio is 0.1-0.5, and the number of users is 100, 200, 300, and 500 and the vote weights \(VW_{1}\), \(VW_{2}\), \(VW_{3}\), and \(VW_{4}\) is 1, 2, 3, and 4, respectively. The four figures represent the attack success probability with different user numbers. Figure 7 analyzes the collusion attack of condition two (approval voting) in Section 4.4. The threshold value \(th_{cs}\) satisfies Formula (15). Figure 8 analyzes the change curve of the successful attack probability in Formulas (13) and (15) when the vote weights \(VW_{1}\), \(VW_{2}\), \(VW_{3}\), and \(VW_{4}\) respectively increase from 1, 2, 3, and 4 to 10, 20, 30, and 40 and the conspirator ratios are 0.1, 0.3, and 0.5 with 100 users. We can have the following conclusions from the above figures:  
1. As the number of users in the domain increases, the threshold \(th_{cs}\) for the lowest successful attack probability also increases. 2. When the number of users in the domain is constant, the greater the voting weight, and the lower the probability of successful attack. That means we can reduce the success probability of an attack and improve the robustness of the system by setting the voting weight and threshold. It can be seen from Figure 8 that when the attacker ratio is 0.1 and the values of votes \(VW_{1}\), \(VW_{2}\), \(VW_{3}\), and \(VW_{4}\) are 2, 4, 6, and 8, the attack success probability has been reduced to 1%.