# Dynamic Node Privacy Feature Decoupling Graph Autoencoder Based on Attention Mechanism
## 5 Method Details
### Privacy Decoupling Based on Node Privacy Ranking  
In order to achieve more accurate privacy information decoupling, based on the node privacy importance scores obtained from Algorithm 1, we sort them in descending order and select the top-\(k\) nodes' indices and their privacy latent representations to construct the privacy distribution of the graph data, which can be formulated as:  
\[Z_{p}^{k}=\{z_{p1}^{(1)},z_{p2}^{(2)},\ldots,z_{pk}^{(k)}\}, \tag{16}\]  
where \(Z_{p}^{k}\) represents the privacy distribution and \(pi\) represents the index of selected nodes. Correspondingly, the latent representations of the selected \(top-k\) index nodes in the graph embedding generated by the utility encoder constitute the distribution of non-privacy information:  
\[Z_{u}^{k}=\{z_{n1}^{(1)},z_{n2}^{(2)},\ldots,z_{uk}^{(k)}\}, \tag{17}\]  
where \(Z_{u}^{k}\) represents the non-privacy distribution and \(ui\) is the index of the corresponding nodes.  
To eliminate the dependence between the two potential distributions learned and protect sensitive information, we adopt the Hilbert-Schmidt Independence Criterion (HSIC) as the independence measurement method. HSIC maps two random variables or distributions to the regenerating kernel Hilbert space (RKHS) through kernel functions, thereby quantifying the dependence between them. Specifically, HSIC is capable of evaluating the independence between two embedded distributions in a non-parametric manner, thereby avoiding the assumption of specific distribution patterns. From a mathematical perspective, HSIC measures independence through the relationship between two kernel matrices, and the formula is as follows:  
\[HSIC(Z_{u},Z_{s})=\frac{1}{(n-1)^{2}}\text{tr}(RK_{u}RK_{s}), \tag{18}\]  
\[R=I-\frac{ee^{T}}{k}, \tag{19}\]  
where \(I\) is the identity matrix, and \(e\) is a column vector of ones. \(K_{u}\) and \(K_{s}\) represent the Gram matrices of \(Z_{x}\) and \(Z_{s}\), respectively, where the elements are defined as \(k_{u,ij}=k_{u}(z_{ix},z_{jx})\) and \(k_{s,ij}=k_{s}(z_{is},z_{js})\). The kernel functions are derived from inner products. \(R\) is the centering matrix used to remove the influence of the distribution means, and \(\text{tr}()\) denotes the computation of the trace of a matrix. A smaller HSIC value indicates a lower dependency between the two distributions, thereby achieving the goal of decoupling the potential representations. Compared with mutual information, the HSIC independence criterion does not require probability density estimation. It directly calculates independence through the kernel matrix, avoiding errors and complexity in high-dimensional scenarios. Moreover, it can capture linear and nonlinear dependencies through multiple kernel functions and adapt to more complex data distributions.  
Therefore, the overall loss function \(L_{tot}\) of the utility autoencoder can be expressed as the multi-objective optimization function:  
\[\begin{split} L_{tot}=L_{adj}-\beta L_{dec}-L_{reg}& =\mathbb{E}_{q_{\phi_{f}}(Z|A,X)}\Big{[}\log p_{\theta_{f}}(A|Z) \Big{]}-\beta HSIC(Z_{u},Z_{s})\\ &-\text{KL}\Big{[}q_{\phi_{f}}(Z|A,X)\parallel p(Z)\Big{]},\end{split} \tag{20}\]  
where the first term \(L_{adj}\) is the utility loss, measuring the representational capability of the latent representations. The second term \(L_{dec}\) is the privacy decoupling loss, quantifying the dependence between privacy and non-privacy distribution. The third term \(L_{reg}\) is the regularization term, constraining the distribution of \(q_{\phi_{f}}(Z|A,X)\) to minimize its distance from the predefined prior distribution \(p(Z_{u})\). The KL divergence \(\text{KL}[\cdot]\) measures the distance between these two distributions. Additionally, \(\beta\) is a trade-off factor controlling the strength of privacy decoupling. Higher values of \(\beta\) increase privacy decoupling strength but may lead to greater utility loss, while lower values reduce utility loss but weaken privacy protection effectiveness.