\[\begin{aligned} &\left.\ell_{\text {atk }}\left(\theta, g_{t}\right)=\mathbb{E}_{G \in \mathcal{D}\left[y_{t}\right], G^{\prime} \in \mathcal{D}\left[y_{y}\right]} \Delta\left(f_{\theta}\left(m\left(G ; g_{t}\right)\right)\right), f_{\theta}\left(G^{\prime}\right)\right) \\ &\ell_{\text {ret }}\left(\theta, g_{t}\right)=\mathbb{E}_{G \in \mathcal{D}} \Delta\left(f_{\theta}(G), f_{\theta_{0}}(G)\right) \end{aligned}\] \[\begin{equation} \begin{aligned} &\left.\ell_{\text {atk }}\left(\theta, g_{t}\right)=\mathbb{E}_{G \in \mathcal{D}\left[y_{t}\right], G^{\prime} \in \mathcal{D}\left[y_{y}\right]} \Delta\left(f_{\theta}\left(m\left(G ; g_{t}\right)\right)\right), f_{\theta}\left(G^{\prime}\right)\right) \\ &\ell_{\text {ret }}\left(\theta, g_{t}\right)=\mathbb{E}_{G \in \mathcal{D}} \Delta\left(f_{\theta}(G), f_{\theta_{0}}(G)\right)\\ &\ell_{\text {sem }}\left(\theta, g_{t}\right)=\mathbb{E}_{G \in \mathcal{D}} \Delta\left(f_{\theta_{0}}(m\left(G ; g_{t}\right)), f_{\theta_{0}}(G)\right) \end{aligned} \end{equation}\] \[\begin{equation} \begin{split} PA + A^TP^T &= P(A + A^T)P^T \\ PA + A^TP^T &= PAP^T + PA^TP^T \\ PA(I-P^T) &= (P-I)A^TP^T \end{split} \end{equation}\] \[\begin{equation} \begin{split} \frac{1}{m}\sum_i^m \lvert \sigma_i - \sigma_i' \rvert \end{split} \end{equation}\] \[\begin{equation} \begin{aligned} X^{(l+1)} &= AX^{(l)}W^{(l)} \\ X^{(l+1)} &= A^{l+1}X^{0}\prod_i^{l+1}W^{i} = A^{l+1}X^{0}W^{'} \\ X^{(l+1)} &= Q\Lambda^{l+1} Q^{-1} X^{0}W^{'} \\ \end{aligned} \end{equation}\] \[\begin{equation} \begin{aligned} X^0 &= X+X^{'} \\ X^{(l+1)} &= A^{l+1}(X+X^{'})W^{'} \\ X^{(l+1)} &= A^{l+1}XW^{'}+A^{l+1}X^{'}W^{'} \\ \end{aligned} \end{equation}\] \[\begin{align} P_m(M; M_c, D_d) &= (f_M(D_d) == f_{M_c}(D_d)) \\ P_m(M; D_c, G) &= (f_M(D_c) == f_{M}(G(D_c))) \\ \end{align}\] \[\begin{equation} f_{M_d}(D_d) \equiv f_{M_c}(D_d) \end{equation}\]

my own gnn demo: \(\begin{equation} X^{(l+1)} = \sigma (AX^{(l)}W^{(l)}) \end{equation}\)

orginal:

\[\mathbf{x}^{\prime}_i = \mathbf{W}_1 \mathbf{x}_i + \sum_{j \in \mathcal{N(i)}} \mathbf{W}_2 \mathbf{x}_j\]

fixed:

\[\mathbf{x}^{\prime}_i = \mathbf{W}_1 \mathbf{x}_i + \sum_{j \in \mathcal{N(i)}} \mathbf{e}_{ij} \mathbf{W}_2 \mathbf{x}_j\] \[\mathbf{x}^{\prime}_i = \mathbf{W}_1 \mathbf{x}_i + \sum_{j \in \mathcal{V}} \mathbf{e}_{ij} \mathbf{W}_2 \mathbf{x}_j\] \[\mathbf{x}^{\prime}_i = \mathbf{W}_1 \mathbf{x}_i + \sum_{j \in \mathcal{N(i)} \cup \mathcal{N_{neg}(i)} } \mathbf{e}_{ij} \mathbf{W}_2 \mathbf{x}_j\] \[\begin{equation} \frac{\partial E}{\partial \mathbf{v}_{w_j}^{\prime}}=\frac{\partial E}{\partial \mathbf{v}_{w_j}^{\prime}{ }^T \mathbf{h}} \cdot \frac{\partial \mathbf{v}_{w_j}^{\prime}{ }^T \mathbf{h}}{\partial \mathbf{v}_{w_j}^{\prime}}=\left(\sigma\left(\mathbf{v}_{w_j}^{\prime}{ }^T \mathbf{h}\right)-t_j\right) \mathbf{h} \end{equation}\] \[\begin{equation} \varepsilon (x):= \begin{cases}0 & \text { if } x\le 0.5 \\ 1 & \text { if } x>0.5\end{cases} \end{equation}\] \[\begin{equation} e_{ij}' = \varepsilon( e_{ij}+ \sigma (XX^T)) \end{equation}\] \[\begin{equation} \frac{\partial \varepsilon }{\partial x}= 1 \end{equation}\] \[\begin{equation} \frac{\partial x_i^{\prime} }{\partial W_2}= \sum_{j \in \mathcal{N(i)}} \mathbf{e}_{ij} \mathbf{x}_j= \sum_{j \in \mathcal{V}} \mathbf{e}_{ij}\mathbf{x}_j = \sum_{j \in \mathcal{N(i)} \cup \mathcal{N_{neg}(i)} } \mathbf{e}_{ij} \mathbf{x}_j \end{equation}\] \[\begin{equation} \theta_{M_d}=\underset{\theta _{M_d}}{\arg \min } \max _{\theta_G} H(\theta_{M_d}, \theta_G):=\frac{1}{n} \sum_{i=1}^n L\left(M_d\left(G \left(d_i\right)\right), y_i\right) \end{equation}\]

True training process: \(\begin{equation} \theta_{G}=\underset{\theta_G}{\arg \min } H(\theta_{M_d}, \theta_G):=\frac{1}{n} \sum_{i=1}^n L\left(M_d\left(G \left(d_i\right)\right), y_t\right) \end{equation}\)

My training process: \(\begin{equation} \theta_{G}=\underset{\theta_G}{\arg \max } H(\theta_{M_d}, \theta_G):=\frac{1}{n} \sum_{i=1}^n L\left(M_d\left(G \left(d_i\right)\right), y_i\right) \end{equation}\)

\[\begin{equation} \mathbf{e}_{ij}:= \begin{cases}0 & \text { 不存在连边 } \\ 1 & \text { 存在连边 } \end{cases} \end{equation}\] \[\begin{equation} \text { Clean Accuracy }=\frac{\sum_{i=1}^n \mathbb{I}\left(M\left(G_i\right)=y_i\right)}{n} \end{equation}\] \[\begin{equation} \text { Attack Success Rate }=\frac{\sum_{i=1}^m \mathbb{I}\left(M\left(G_i\right)=y_t\right)}{n} \end{equation}\] \[(\frac{X'}{\lVert X' \rVert^2})(\frac{X'}{\lVert X' \rVert^2})^T\]

这个来自BACKDOOR DEFENSE VIA DECOUPLING THE TRAINING PROCESS \(\begin{equation} A S R \triangleq \operatorname{Pr}_{(\boldsymbol{x}, y) \in \mathcal{D}_{\text {test }}}\left\{C_{\boldsymbol{w}}(G(\boldsymbol{x}))=y_t \mid y \neq y_t\right\} \end{equation}\)

\[\begin{equation} PB \triangleq \operatorname{Pr}_{\mathit{g_c} \in \mathcal{D}_{\text {test }}}\left\{f_M(\mathit{g_c}) = f_{M}(G(\mathit{g_c}))\right\} \end{equation}\] \[\begin{equation} PB \triangleq \operatorname{Pr}_{\mathit{g_c} \in \mathcal{D}_{\text {test }}}\left\{f_C(G(\mathit{g_c})) = f_{M}(G(\mathit{g_c}))\right\} \end{equation}\]

第二篇论文

• NAD中attention计算的修改

  
    # Todo： meaning of norm's dim
        # image: B*C*H*W -> B*H*W  : B*H*W/H*W
        # graph: B*N*F -> B*N : B*N / N
        # graph(True): BxN * F -> BxN * 1 / BxN * 1
  

\[\begin{equation} split(losses) = [i \leq \max_{i=1}^{n-1} (losses_{i+1} - losses_i), \ \forall i \in [1, n]] \end{equation}\]