## 4 Methods
### Feature Learning Network Module (FLNM)
#### 4.2.1 Update of Node Features  
To enhance the accuracy of graph structure registration while reducing computational costs and improving efficiency, we propose a novel graph neural network architecture, namely EKNet. The network framework we designed mainly consists of two layers of graph neural networks: an KAN layer and EdgeConv layer. When the node features pass through each layer of the graph neural network, they will be aggregated and updated in sequence. This process strengthens the global feature representation capabilities of nodes, edges, and the entire graph, thereby laying a solid foundation for the subsequent feature matching stage. Subsequently, these features are processed through an EdgeConv layer and GraphKAN layer. The EdgeConv layer, derived from the Dynamic Graph CNN (DGCNN) [48], primarily addresses the issue that networks such as PointNet [37] fail to fully consider the geometric structure relationships in local information when extracting node features. By constructing a local neighborhood graph and applying convolution-like \begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature** & **Formula** & **Meaning** \\ \hline Clustering Coefficient & \(C_{i}=\frac{2\|\{e_{k}:v_{i},v_{k}\in N_{i},e_{k}\in E\}\|}{k_{i}(k_{i}-1)}\) & The coefficient of the degree of clustering between vertices in a graph. \\ Degree Centrality & \(C_{D}(v)=\frac{k_{i}}{N-1}\) & The degree to which a node is connected to all other nodes. \\ Eigenvector Centrality & \(C_{E}(v)=\frac{1}{\lambda}\sum\limits_{u\in N(v)}C_{E}(u)\) & Ranking of the likelihood of a node being visited during an infinite-length random walk in the graph. \\ PageRank & \(PR(a)_{i+1}=\sum\limits_{l=0}^{n}\frac{PR(T_{l})_{i}}{L(T_{l})}\) & Measures the importance of nodes through random walk models and transition probability matrices. \\ Load Centrality & \(C_{L}(v)=\sum\limits_{s\
eq\
eq i\in V}\frac{\sigma_{u}(v)}{\sigma_{u}}\) & Calculates the proportion of all shortest paths in the network that pass through this node. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complex network feature formulas and meanings. Figure 2: Overall structure of EKNet. operations on edges between adjacent point pairs to develop local geometric structures, its dynamic graph updating method enables the network to better capture the local geometric information of point clouds, thereby improving the accuracy of point cloud registration. We design the output feature dimension to be 64. The specific process is as follows:  
Since the feature extraction process of the two subgraphs is the same, one of the subgraphs is taken as an example. Firstly, take the initial node features \(F_{x}\) and adjacency matrix after geometrization as inputs. Then, construct a directed graph \(G\) as the \(k\)-nearest neighbor (\(k\)-NN) graph of \(x\) in \(\mathbb{R}^{F}\). The graph includes self-loops, meaning that each node also points to itself. Define the edge features of the paired subgraphs as \(E_{x_{ij}}=h_{\Theta}(x_{i},x_{j})\), where \(h_{\Theta}:\mathbb{R}^{F}\times\mathbb{R}^{F}\rightarrow\mathbb{R}^{F^{ \prime}}\) is a nonlinear function with a set of learnable parameters \(\Theta\). Here, \(F\) is 15-dimensional, and \(F^{\prime}\) is 64-dimensional. The processing of features in graph neural networks typically involves two steps: (a) Aggregation. The message function aims to combine the adjacent features of the target node, including its own features, the features of neighboring nodes, and the edge features connecting to neighboring nodes, into a single vector. This vector will represent the neighborhood information of the central node. (b) Update. The node update function is employed to integrate the node features of the current layer with the messages obtained through the message passing function, thereby updating the node features of the next graph layer. (a) EdgeConv Layer  
The EdgeConv layer employs an asymmetric edge feature calculation function. It explicitly combines global shape structures with local neighborhood information.