## 2 Problem Formulation
\(\bar{T}=[\bar{T}_{1}\quad\bar{T}_{2}\quad\cdots\quad\bar{T}_{n_{x}}]\), \(\bar{\bar{X}}_{i}=[\bar{X}^{\mathsf{T}}_{i\  
According to the derivation above, we can verify that the matrices \(F_{i,u}\) are time-varied and satisfy \(F_{i,u}^{\mathrm{T}}F_{i,u}\leq I\). The schematic diagram of measurement saturation, MMs, WTOD protocol and cyber-attacks is shown in Figure 2.  
As shown in Figure 2, MMs can prevent the sensor from sending data to the network, which means that the network will not receive any information from this sensor if the MMs occur in this sensor. Hence, the measurement data that sent by sensors during MMs occur can be expressed as:  
\[\tilde{J}_{u}=\alpha_{u}\tilde{q}_{u}, \tag{9}\]  
where \(\alpha_{u}=\mathrm{diag}\{\alpha_{u}^{1},\,\alpha_{u}^{2},\,...,\,\alpha_{u}^ {n_{y}}\}\) is the known matrix, \(\alpha_{u}^{i}\in\{0,1\}\) is used to indicate whether the MM occurs in the \(i\)-th sensor at sampling instant \(u\), \(\alpha_{u}^{i}=0\) represents the occurrence of the MM at sampling instant \(u\) and, on the contrary, \(\alpha_{u}^{i}=1\) represents that the MM does not occur at sampling instant \(u\). To alleviate data conflicts caused by limited communication network bandwidth, the WTOD protocol is utilized to schedule the communication between the estimator and the sensor, which only allows a sensor node to transmit the latest data to the estimator during each sampling instant. Zero-order-holders (ZOHs) are used to maintain the information transmitted from other nodes at the previous sampling instant to compensate the loss of real-time data. The node \(l_{u}\in\{1,\,2,\cdots,n_{y}\}\) that sends data to the network at sampling instant \(u\) is selected according to  
\[\bar{D}\triangleq(v^{+}+v^{-})/2,\ \ \bar{E}\triangleq(E^{+}+E^{-})/2.\]0  
where \(\hat{q}_{i,u-1}\) represents the latest data sent by the sensor node \(i\) before sampling instant \(u\), and \(\partial_{i}>0\) is a known matrix that represents the weight coefficient of the sensor node \(i\). \(\hat{q}_{u}=(\hat{q}_{1,u},\,\hat{q}_{2,u},\cdots,\hat{q}_{n_{y},u})^{\mathrm{ T}}\) represents the actual output after the output of the sensors are scheduled by the WTOD protocol, and the update rule for \(\hat{q}_{i,u}\) (\(u\in N^{+},\,i\in n_{y}\)) can be formulated as  
\[\hat{q}_{i,u}= \begin{cases}\tilde{q}_{i,u}&i=l_{u}\\ \hat{q}_{i,u-1}&i\
eq l_{u}\end{cases} \]  
Define \(\Xi(l_{u})=\mathrm{diag}\{\Re_{1-l_{u}},\,\Re_{2-l_{u}},\,\cdots,\Re_{n_{y}- l_{u}}\}\); then, \(q_{u}\) can be rewritten as  
\[\hat{q}_{u}=\Xi_{l_{u}}\tilde{q}_{u}+(1-\Xi_{l_{u}})\hat{q}_{u-1}, \tag{10}\]  
where \(\Re_{i-l_{u}}\in\{0\,1\}\) is the Kronecker delta function. Figure 2: Schematic diagram of measurement saturation, MMs, WTOD protocol and cyber-attacks. **Remark 1**. : _During the state estimation of MNNs, the sensors send data to the estimator using communication networks at each sampling instant. Nevertheless, the transmission capacity of the network is typically constrained in practical scenarios, which causes network-induced phenomena such as transmission delay or out-of-order. To reduce the occurrence of these phenomena, utilizing the WTOD protocol to reduce the number of data packets transmitted is important. The process of using the WTOD protocol to select sensor nodes is divided into two stages at each instant: the initial stage is to calculate the difference between \(\tilde{q}_{i,u}\) and \(\hat{q}_{i,u-1}\), and the subsequent stage is to prioritize the nodes through combining the established weight coefficients and the maximal difference acquired during the initial stage. Noteworthy is the fact that the node which has that smallest number of indexes is usually selected when multiple nodes have the same priority._  
As one of the most destructive cyber-attacks, bias injection attack refers to the malicious injection of an attack signal into the measurement data transmitted to the estimator, which makes the estimator receive false information.