## 3 Proposed Method
### Graph Convolutional Network Clustering (GCNC)  
The best of both worlds is achieved by the graph convolutional network clustering technique; this has the same memory complexity as conventional SGD and full gradient descent has the same temporal complexity per epoch [31]. Consider the scenario in which each batch's embeddings are determined for a collection of nodes B from layer 1 to layer L. Given that every layer of computation uses the same subgraph \(A_{B,B}(Links\)\(with\)\(B)\), we may assume the number of edges in this batch, \(\left\|A_{B,B}\right\|_{0}\). To indicate the embedding utilization, we need build a batch B that maximises within-batch edges in order to increase embedding utilisation. In order to link the effectiveness of SGD updates with graph clustering techniques, Figure 4 shows the complete graph G and the graph with the clustering partition G as examples of the community expansion. Cluster-GCN, as can be shown, can concentrate on the neighbours within each cluster rather than conducting a thorough neighbourhood search. A graph \(G=(V,\varepsilon,A)\) consists of \(N=|V|\) vertices and \(|\varepsilon|\) edges such that an edge among any two vertices \(i\) and \(j\) represents their similarity. The corresponding adjacency matrix \(A\) is an \(N\times N\) sparse matrix with \((i,j)\) entry equaling 1 if there is an edge between \(i\) and \(j\) and 0 otherwise. \(G\)'s partition its nodes into \(c\) groups: \(V=[V_{1},\ldots V_{c}]\), where \(V_{t}\) consists of the nodes in the \(t\)-th partition, \(L\) is the number of layers, \(F\) is the number of features, \(N\) is the number of nodes, and \(b\) is the batch size. As a result, we have \(c\) subgraphs  
\[\overline{G}=[G_{1},\ldots G_{c}]=[\{v_{1},\varepsilon_{1}\},\ldots,\{v_{c}, \varepsilon_{c}\}] \tag{1}\]  
where each \(\varepsilon_{t}\) is made up only of the connections connecting the nodes in \(v_{t}\). The adjacency matrix is divided into \(c^{2}\) submatrices after nodes are reorganised as  
\[A=\overline{A}+\Delta= \begin{bmatrix}A_{11}&1&A_{1c}\\ 1&0&1\\ A_{c1}&1&A_{cc}\end{bmatrix}  \tag{2}\]  
and  
\[\overline{A}= \begin{bmatrix}A_{11}&1&0\\ 1&0&1\\ 0&1&A_{cc}\end{bmatrix} ,\Delta= \begin{bmatrix}0&1&A_{1c}\\ 1&0&1\\ A_{c1}&1&0\end{bmatrix}  \tag{3}\]  
Figure 3: The main architecture of the data security and privacy model. The links within \(G_{t}\) are contained in each diagonal block \(A_{tt}\) is a \(|v_{t}|\times|v_{t}|\) adjacency matrix. The adjacency matrix for the graph \(G_{t}\). \(\overline{A}\) is the adjacency matrix for graph \(\overline{G}\); \(A_{st}\) contains the connections between its two partitions \(v_{t}\); \(\Delta\). Similar to this, we can divide the \(\widehat{X}\) and \(Y\) into \([V_{1},\ldots V_{c}]\) as \([X_{1},\ldots X_{c}]\) according to the partition \([Y_{1},\ldots Y_{c}]\) where \(X_{t}\) and \(Y_{t}\) are made up. The calculation \(\overline{G}\) allows for the decomposition of the objective function of GCN into many groups (clusters). The final embedding matrix is if we use \(\overline{A}\prime\) to represent the normalised version of \(\overline{A}\). \[Z^{(L)}=\overline{A}^{\prime}\sigma\Big{(}\overline{A}^{\prime}\sigma\Big{(} \ldots\sigma\Big{(}\overline{A}^{\prime}XW^{(0)}\Big{)}W^{(1)}\Big{)}\ldots \Big{)}W^{(L-1)} \tag{4}\]  
This is significantly less difficult to construct than the neighbourhood search method utilised in earlier SGD-based training methods because \(\overline{A}\) has a block-diagonal structure (remember that \(\overline{A}^{\prime}_{tt}\) is the matching diagonal block of \(\overline{A}\prime\)). Additionally, it has broken down into  
\[L_{\overline{A}\prime}=\sum_{t}\frac{|v_{t}|}{N}L_{\overline{A}^{\prime}_{tt}} \text{ {and} }L_{\overline{A}^{\prime}_{tt}}=\frac{1}{|v_{t}|}{\sum_{i\in v_{t}}}\text{{ loss}}(yi,z^{(L)}_{i}) \tag{5}\]  
The decomposition form in (3) and (4) serves as the basis for the Cluster-GCN (5).