# Reliability Evaluation for Clustered WSNs under Malware Propagation
## 6 Numerical Results  
### Illustrating Influence of \(\alpha\) and \(\beta\)  
With MATLAB R2010b, we explore how the optimal infection probability and the MTTF of a sensor node depend on the parameters of the true positive rate (_i.e._, the detection rate) and the false positive rate (_i.e._, the false alarm rate). The parameters of the strategic game are \(\omega=50\), \(c_{I}=20\), \(c_{D}=10\), and \(\lambda=0.5\). Note that we can attain similar trends if the parameters are changed. However, specific values will be correspondingly changed.  
Figures 3 and 4 demonstrate the changing optimal infection probabilities that player _malware_ adopts according to \(\alpha\) and \(\beta\), which reveals the _malware_'s intention. Obviously, a higher detection rate and a lower false-alarm rate can help a WSN IDS detect malware. Therefore, player _malware_ will choose its optimal strategy to lower the infection probability in order to minimize the loss arising from IDS detection. As shown in Figure 3, the optimal infection probability decreases gradually when the true positive rate increases slowly from 0.7 to 0.98. Moreover, a lower false positive rate results in a lower infection probability. For example, when \(\alpha=0.88\) in Figure 3, the optimal infection probabilities are \(\sim\)0.1667, \(\sim\)0.1984, and \(\sim\)0.2381 for \(\beta=0.01\), \(\beta=0.05\), and \(\beta=0.1\), respectively. We observe in Figure 4 the optimal infection probability increases gradually when the false positive rate increases slowly from 0.01 to 0.15. Furthermore, a higher detection rate leads to a lower infection probability. For example, when \(\beta=0.1\) in Figure 4, the optimal infection probabilities are \(\sim\)0.2727, \(\sim\)0.2308, and \(\sim\)0.2055 for \(\alpha=0.8\), \(\alpha=0.9\), and \(\alpha=0.98\), respectively. These experimental results indicate that the true positive rate should be increased and the false positive rate should be decreased in order to decrease the infection probability adopted by player _malware_ and to enhance a sensor node's reliability.  
Figure 5 shows the MTTF of a sensor node under the epidemic-malware propagation scenario in terms of \(\alpha\) and \(\beta\). As the true positive rate increases and the false positive rate decreases, it is easy for player _system_ to detect infected sensor nodes, increasing their MTTFs. From Figure 5, as we expect, the MTTF increases slowly when the true positive rate increases gradually from 70% to 98%. There is a similar tendency when the false positive rate decreases from 15% to 1%. Smaller decreases of the false positive rate increase the MTTF for a sensor node more than the same decreases of the true positive rate. For example, the MTTF of a sensor node increases from \(-4.6366\) to \(-5.7219\) (an increase of \(-23.41\%\)) as \(\beta\) drops from 15% to 1% when \(\alpha=88\%\). However, when \(\beta=10\%\), the MTTF of a sensor node increases from \(-4.2487\) to \(-5.0350\) (an increase of \(-18.51\%\)) as \(\alpha\) increases from 70% to 90%. These results indicate that we should further reduce the false positive rate while improving IDSes for WSNs in order to increase a sensor node's MTTF.