## 3 Main Result
### Estimator Design
: We consider the situation when cyber-attacks and MMs occur; the occurrences of attacks and MMs are shown below. The occurrences of cyber-attacks and MMs are shown in Figures 5 and 6, respectively. Figure 4: True state and its estimated values based on our approach and other approaches from [11; 27; 55]. Figure 5: Occurrences of bias injection attacks. Figure 3: True state and its estimated values based on our approach and other approaches from [11; 27; 55]. Under the same parameters, the true state and the estimation values of the MNNs with MMs subject to cyber-attacks are shown below. The real values \(\omega_{u}^{(i)}\) and the estimation values \(\delta_{u}^{(i)}\)\((i=1,\,2)\) of the MNNs with MMs subject to cyber-attacks are shown in Figures 7 and 8, respectively. From Figures 7 and 8, we can see that the estimation framework we proposed in this paper can effectively estimate the true state of MNNs. When the system is threatened by cyber-attacks during transmission, the estimation error can increase accordingly. However, once the attack signal disappears, the estimation performance can recover immediately. In addition, when a sensor node exhibits an MM phenomenon, compared with the approaches based on the RR protocol in [27, 55], our proposed algorithm can also reduce the influence of MMs on estimation performance. Figure 8: True state and its estimated values based on our approach and other approaches from [11, 27, 55]. Figure 6: Occurrences of MMs on each sensor node. Figure 7: True state and its estimated values based on our approach and other approaches from [11, 27, 55]. The following figures show the sensor nodes selected to send the latest data at each sampling instant based on the WTOD protocol and RR protocol with MMs subject to cyber-attacks. Figures 9 and 10 show the situation of the sensor nodes sending data under the WTOD protocol and the RR protocol, respectively. From Figures 9 and 10, we can see that only one sensor node sends data to the estimator at each sampling instant, which means that the use of these two protocols can save communication resources and reduce the communication burden when the network bandwidth is limited, in turn avoiding possible data collisions. From Figures 5-10, we can obtain that, when using the approaches based on the RR protocol in [27; 55] to estimate the MNNs, the estimation performance can be greatly influenced once the sampling instants for the continuous occurrence of MMs exceeds the number of sensor nodes. In contrast, the state estimation results from the estimation framework proposed in this paper are more accurate, significantly less influenced by continuous occurrence of MMs. This is because, if we know that a certain sensor has a high occurrence probability of MMs, we can reduce the data transmission from this sensor to the estimator by changing the weight of the WTOD protocol, so as to reduce the sending of erroneous data to the estimator as much as possible. Finally, to further confirm the practicability of the proposed estimation framework, we apply this framework to a real engineering problem. **Case 3**: In this case, we consider a network-based test rig that consists of a plant (DC servo system) and a remote controller; the DC servo system is identified to be a third-order system, whose parameters are from [55]:  
Figure 10: Selected nodes to transmit the latest data under the RR protocol. Figure 9: Selected nodes to transmit the latest data under the WTOD protocol. \[A= \begin{bmatrix}1.12&0.213&-0.333\\ 1&0&0\\ 0&1&0\end{bmatrix} ,\quad B= \begin{bmatrix}0.8\\ 0\\ 0\end{bmatrix} ,\] \[D= \begin{bmatrix}-0.2193&0.0219&0.0844\\ 0.2177&-0.0032&-0.0662\\ 0.1298&-0.0087&-0.0381\end{bmatrix} ,\] \[C= \begin{bmatrix}1&0&0\\ 0&1&0\end{bmatrix} ,\quad E= \begin{bmatrix}0\\ 0.8\end{bmatrix}