# RGCLN: Relational Graph Convolutional Ladder-Shaped Networks for Signed Network Clustering
## 3 The Proposed Model
### Multi-Relational Graph Convolutional Ladder-Shaped Networks  
Due to the large variability between non-Euclidean data and data such as images, in this way, many excellent frameworks in the image domain cannot be used for graph-structured data. In [17], the authors proposed a graph convolutional ladder network (GCLN) to solve the over-smoothing problem by borrowing the well-known U-Net structure from medical image segmentation. The GCLN is a symmetric architecture with a GCN layer consisting of a systolic path and an extended path. Three contextual feature channels allow the contextual features captured from the systolic path to be fused with the localization information learned through the extended path. However, the GCLN framework is only applicable to general networks. We extend its design idea to signed networks and design the RGCLN architecture using multi-relational graph convolution instead of base convolution to address the over-smoothing problem under signed network clustering.  
Next, we will focus on the multi-relational graph convolution ladder-shaped network framework proposed in this paper. We set the underlying graph convolution as multi-relational graph convolution, which differs from the standard graph convolution shared parameter settings by using different weights for two different types of edges: positive and negative edges. We do not make any assumptions about the positive and negative relations and only use the network to fit the deep structure information. The main part of our RGCLN model is shown in Figure 1, where it can be found that there are also contraction paths and expansion paths. However, in the contraction path, the number of neurons needs to be reduced to one-third of the previous layer. We define the message-passing process. Specifically, the node \(v\) of the \(l^{th}\) layer is denoted as \(Z_{v}^{(l)}\), and its formula is defined as follows:  
\[\begin{split} h_{v}^{(l)}=\left[z_{u}^{(l-1)},\Sigma_{u\in N_{v} ^{+}}Z^{(l-1)},\Sigma_{u\in N_{v}^{-}}Z^{(l-1)}\right]\\ Z_{v}^{(l)}=\sigma\left(h_{v}^{(l)}\cdot w^{(l)}+b^{(l)}\right) \end{split} \tag{1}\]  
where \(w^{(l)}\) denotes the weight matrix of the \(l^{th}\) layer, and \(b^{(l)}\) denotes the deviation vector of the \(l^{th}\) layer.  
Due to the convolution of multi-relational graphs, each convolution process has three parts: the aggregation of node features, positive domain information, and negative domain information, which makes the dimensionality of nodes expand to three times the original one after this process. Therefore, the settings of neurons on the expanding path need to refer to the settings of neurons on the contracted path and keep the principle of dimensionality consistency. This design is due to the specificity of the multi-relational graph convolution on the one hand and the need to aggregate the information on the corresponding side of the contraction path for the expansion path on the other. This makes the structure a non-strictly symmetric ladder-shaped network. We use Tanh as the activation function because it has been experimentally shown to work better when it is used.  
Figure 1: Framework of RGCLN.