## 3 Statistical Modeling of Network and Parameter Estimation
### Hidden Markov Multi Linear Tensor Model  
Recently, dynamic modeling of longitudinal networks has gained much more attention. These networks have found applications in many fields, such as social, economical, biological, etc. One can model a longitudinal network with \(\mathbf{N}=\{1,\ldots,N\}\) nodes as a multilayer network in which there are \(T\) layers and relationships between nodes at time \(t\) are represented by a matrix \(\mathbf{Y}_{t}=\left\{y_{i\mu}|\dot{i},j\in\mathbf{N}\right\}\) with size \(N\times N\) for \(t\in\{1,\ldots,T\}\). Modeling the matrix \(\mathbf{Y}_{t}\), a hidden Markov multi-linear tensor model (HMTM) proposed by [PERSON] and [PERSON] [29] is used. [PERSON] and [PERSON] [29] extended the work of [PERSON] [30; 31], which presented a multi-linear tensor model (MTRM). According to [PERSON] [30; 31], network effects can be formulated as the following:  
\[\Pr\!\left(y_{i\mu}=1\big{|}\mathbf{x}_{i\mu},\mathbf{u}_{i},\mathbf{u}_{j}, \mathbf{v}_{t}\right)=\mathbf{x}_{i\mu}\beta+\left\langle\mathbf{u}_{i}, \mathbf{v}_{t},\mathbf{u}_{j}\right\rangle+\varepsilon_{i\mu} \tag{1}\]  
where, \(\mathbf{U}=\left(\mathbf{u}_{1},\ldots,\mathbf{u}_{N}\right)^{T}\) and \(\mathbf{v}_{t}=\left(v_{1t},\ldots,v_{Rt}\right)^{T}\), which are, respectively, R-dimensional latent node positions and node connection rules at time \(t\). Additionally, \(\varepsilon_{i\mu}\) follow \(N(0,\sigma^{2})\) and \(\mathbf{U}\) and \(\mathbf{V}=\left(\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{T}\right)\) follow matrix normal distributions where  
\[\mathbf{U} \sim\mathrm{MN}_{N\times R}(1\mu_{U},\mathbf{I}_{N},\varrho_{U}) \tag{2}\] \[\mathbf{V} \sim\mathrm{MN}_{N\times R}(1\mu_{V},\mathbf{I}_{N},\varrho_{V}) \tag{3}\]  
[PERSON] and [PERSON] [29] combined MTRM with a hidden Markov model and developed a hidden Markov multi-linear tensor model (HMTM). In a HMTM that involves \(k\) breaks, the probability distribution of network data, \(\mathbf{B}_{t}\), is modeled as the following:  
\[\mathbf{B}_{t}=\mathbf{Y}_{t}-\Omega_{t} \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Definition** \\ \hline \(i\) & Node index, for \(i=1,2,\ldots,N\) \\ \(j\) & Node index, \(j=1,2,\ldots,N\) \\ \(N\) & Number of nodes \\ \(t\) & Time periods, for \(t=1,2,\ldots,T\) \\ \(\mathbf{Y}_{t}=\left\{y_{i\mu}|\dot{i},j\in\mathbf{N}\right\}\) & Adjacency matrix \\ \(x_{i\mu}\) & Covariate vector for nodes \(i\) and \(j\) and time \(t\) \\ \(\beta\) & Coefficient vector of covariates \\ \(\mathbf{B}_{t}\) & Probability distribution of network \\ \(R\) & Number of latent variables \\ \(\mathbf{U}=\left(\mathbf{u}_{1},\ldots,\mathbf{u}_{N}\right)^{T}\) & Latent node position \\ \(\mathbf{V}=\left(\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{T}\right)\) & Node connection rule \\ \(\varepsilon_{i\mu}\sim N(0,\sigma^{2})\) & Error term for nodes \(i\) and \(j\) and time \(t\) \\ \(1_{N}\) & \(N\times N\) matrix with all one elements \\ \(\mathbf{v}_{t}\) & Hidden state variable \\ \(L_{t}=\left(\overline{\mu}_{1t},\overline{u}_{2t},\overline{v}_{1t},v_{2t}\right)\) & Vector of variables for monitoring \\ \(Z_{t}\) & MEWMA statistic \\ \(p\) & Number of variables for monitoring \\ \(\Lambda\ =\ (\lambda_{1},\ \lambda_{2},\ldots,\lambda_{p})\) & Vector of smoothing parameters \\ \(\alpha\) & Error type-I \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations list. \[\mathbf{B}_{t}=\beta 1_{N}+\mathbf{U}_{\mathbf{\Psi}_{t}}\mathbf{V}_{t}\mathbf{U}_{ \mathbf{\Psi}_{t}}^{T}+\mathbf{E}_{t} \tag{5}\]  
where, \(\mathbf{\Psi}_{t}\) is a hidden state variable, \(1_{N}\) is a \(N\times N\) matrix with all one element, and \(\mathbf{E}_{t}\) follows a matrix normal distribution with \(\text{MN}_{N\times N}(0,\sigma_{\mathbf{\Psi}_{t}}^{2}\mathbf{I}_{N},\mathbf{ I}_{N})\). We use the procedure and prior distributions adopted by [PERSON] and [PERSON] [29] for parameter estimation.