# Edge Convolutional Networks for Style Change Detection in Arabic Multi-Authored Text
## 4 Proposed Solution: ECNN-ASCD
### GNNs  
In 2009, the GNN framework was introduced by applying a message-passing scheme to propagate information across nodes [12]. Figure 1 illustrates the general propagation framework, where the colors assigned to the nodes are utilized for visual separation to distinguish between individual nodes in the graph structure. The input graph represents different computation graphs. Each node defines its computation graph based on its neighborhoods. The computation graph of the target node \(A\) appears on the right side of Figure 1. The message-passing scheme can be summarized in three steps. First, each target node computes a message for its neighbors. The rectangles filled with slanted black lines in Figure 1 represent messages from one node to its neighbors within the computation graph. Second, the target node receives and aggregates the messages from its neighbors. Third, the target node updates its representation. The solid gray rectangle in Figure 1 symbolizesthe aggregation operation in the GNNs. These steps are applied to all nodes iteratively through multiple computation layers. Each node in the first layer uses information received from its immediate neighbors. By stacking two layers, the subsequent layer incorporates information from the neighbors of neighbors (2-hop neighborhood) derived from the first layer. Thus, stacking \(N\) layers enables nodes to utilize information with the \(N\)-hop neighborhood. This iterative nature enhances the capturing of dependence patterns in the graph data. Formally, let \(\mathcal{G}(\mathcal{V},E)\) be a graph, where \(\mathcal{V}\) and \(E\) are the sets of nodes and edges, respectively. \(X\in R^{n\times f}\) is a matrix containing representations of the length \(f\) for \(n\) nodes. Equation (1) shows the calculation of a representation of one node \(i\in\mathcal{V}\) in the \(k\)th layer. Different aggregation functions can be used. Choosing these functions is still critical and attracting significant attention from researchers [84]. For example, the first function, \(Agg_{1}\), can average a set of prior representations of neighbors \(\mathcal{N}(i)\) from the \(k-1\)th layer, while the second function, \(Agg_{2}\), can sum the result of the first function and the representation of the node \(i\) itself from the previous layer, \(k-1\). \[\mathbf{x}_{i}^{(k)}=Agg_{2}\Big{(}\mathbf{x}_{i}^{(k-1)},Agg_{1}\Big{(}\Big{\{} \mathbf{x}_{\mathcal{N}(i)}^{(k-1)}\Big{\}}\Big{)}\Big{)}. \tag{1}\]  
[PERSON] and [PERSON] [85] introduced GCNs as one of the early GNN models. Its message-passing schema involves a symmetric-normalized aggregation and an element-wise nonlinear transformation, as shown in Equation (2). Initially, the messages from all the neighbors \(\mathcal{N}(i)\) are symmetrically normalized by the degrees of both the neighbor \(j\in\mathcal{N}(i)\) and the target \(i\) nodes. Subsequently, these messages are aggregated using a weighted summation. Finally, target node representations are updated by combining the aggregated messages with the current target node representations by a neural network transformation. While GCNs are widely used for diverse tasks [16; 17; 18; 19; 20; 21; 24; 27; 28], they encounter limitations diminishing their abilities. First, GCNs can handle edge weights within the adjacency matrices as shallow edge representations, while edges can contain richer information. Second, the iterative propagation in deep GCN architectures results in receiving similar messages. This similarity leads to shared representations across most nodes, less distinctiveness, and high convergence, known as an over-smoothing issue. Third, GCNs update node representations based on the degree of neighbors and target nodes. Thus, GCNs are structure-dependent. GCNs may struggle with noisy graphs that have sparsely labeled edges. Figure 1: General propagation framework, where the colors are utilized for visual separation.