## 3 Methods
### Substructure Signature Learning  
The purpose of learning substructure flags is to map a large number of substructure instances onto \(K\) representative subgraph patterns so that any subgraph can be reconstructed by this global \"topological dictionary\". This approach not only preserves the similarity between subgraphs but also provides a unified modular platform for the representation of different attribute graphs. This is also the basis for modeling individual interactions (projection pooling mechanism) by effectively preserving the identity of subgraph individuals and avoiding information loss caused by traditional methods. We use clustering or dictionary encoding to segment \(n\) attribute graphs into \(N=\sum_{i=1}^{n}k_{i}\) subgraph instances and learn \(K\) flag vectors, the criteria of which are as follows: (1) high coverage, subgraph flags should faithfully reflect the distribution of subgraphs; (2) high discriminability, the graph representation formed by subgraph flags should significantly distinguish samples of different categories. The former is achieved through dictionary reconstruction, while the latter is ensured by an end-to-end learning framework. The subgraph flag matrix \(\mathbf{U}\) can directly represent the subgraph flags. In addition, clustering can also be implemented through nonnegative dictionaries. Let \(\mathbf{U}=\mu_{1},\mu_{2},...,\mu_{k}\in\mathbf{R}^{c\times K}\) be the subgraph flag matrix and require that all subgraph instances within the attribute graph can be reconstructed by the column vectors of \(\mathbf{U}\), i.e., minimizing the following reconstruction error:  
\[\text{min}\sum_{l=1}^{n}\sum_{i=1}^{k}\sum_{j=1}^{K}||\mu_{j}-\alpha_{ilj}| \mathbf{g}_{l,[;l]}||^{2},s.t.\alpha_{ilj}\leq 0 \tag{10}\]  
where \(\alpha_{ilj}\) is the reconstruction coefficient of the \(i\)-th subgraph in the attribute graph \(G_{i}\) with respect to the subgraph symbol \(\mu_{i}\). In addition, subgraph symbols can also be calculated through deep clustering. We use a clustering indicator matrix \(\mathbf{W}_{i}\in\mathbf{R}^{n_{i}\times K}\) to cluster the subgraphs of each attribute graph \(G_{i}\), where the \(jk\)-th element represents the probability that the \(j\)-th substructure of \(G_{i}\) belongs to the \(k\)-th subgraph symbol of \(G_{k}\), and the form of this probability can be determined by a Gaussian or \(t\)-distribution function, for example:  
\[\mathbf{W}_{i}(j,k)=\frac{(1+||\mathbf{g}_{i}(j,:)-\mu_{k}||^{2}/\alpha)^{- \frac{\alpha+1}{2}}}{\sum_{k^{\prime}}(1+||\mathbf{g}_{i}(j,:)-\mu_{k}^{\prime }||^{2}/\alpha)^{-\frac{\alpha+1}{2}}}, \tag{11}\]Kullback-Leibler divergence can be trained to learn the above. The objective function is:  
\[ \begin{split}\min_{\mathbf{U},\mathbf{H}_{i}^{\prime}\approx i} \sum_{l}KL(\mathbf{W}_{i},\widetilde{\mathbf{W}}_{i})\\ s.t.\widetilde{\mathbf{W}}_{i}(j,k)=&\frac{\mathbf{W }_{i}^{2}(j,k)/\sum_{l}\mathbf{W}_{i}(l,k)}{\sum_{k^{\prime}}[\mathbf{W}_{i}^ {2}(j,k^{\prime})/\sum_{l}\mathbf{W}_{i}(l,k^{\prime})]}.\end{split}  \tag{12}\]  
Here, \(\widetilde{\mathbf{W}}_{i}\) is the sparse (self-sharpening) version of \(\mathbf{W}_{i}\). Minimizing the KL divergence above ensures that each subgraph instance is only assigned to a small number of subgraph signature points, thus forming the dictionary reconstruction code for the coefficients. The choice of the number of subgraph signatures \(K\) is controlled by the complexity of the topology. For attribute graphs with large sizes and numbers of nodes, \(K\) should be chosen to be a sufficiently large value, also known as \"structural resolution\". The pooling operator of the topology-preserving graph pooling algorithm projects the local structure of each graph onto common structural signatures \(\mu_{j},\mu_{2},...,\mu_{K}\), and the resulting graph representation dimension depends only on the number of structural signatures \(K\) rather than the number of nodes in the attribute graph.