# Robustness Analysis on Graph Neural Networks Model for Event Detection
## 4 Model
### Edge-Enhanced Graph Convolution Networks  
Due to previous GNN-based ED models usually ignoring dependency label information, which conveys rich and useful linguistic knowledge for ED, [PERSON] et al. proposed EE-GCN [21], which embeds the edges of the syntactic dependency graph into a vector space to obtain an edge representation tensor \(\mathbf{EM}=[\mathbf{em}_{i,j\in]}\in\mathbb{R}^{n\times n\times p}\), where \(\mathbf{em}_{i,j}\in\mathbb{R}^{p}\) is the vector representation of the corresponding edge in the syntactic dependency graph, it contains more semantic information than 0 or 1 of the traditional adjacency matrix.  
Denoting \(\mathbf{H}^{0}=(\mathbf{h}_{1},\mathbf{h}_{2},\ldots,\mathbf{h}_{n})\in \mathbb{R}^{n\times d_{l}}\). After each node (word) representation was converted to the dimension of \(d_{g}\), \(\mathbf{H}^{0}\) is the input state of layer 1 in EE-GCN. The vector of each node in layer \(l\in[1,L]\) updates by aggregating the information from its neighbor nodes through the edge representation tensor, so the output state of each layer \(l\) (\(\mathbf{H}^{l}\in\mathbb{R}^{n\times d_{g}}\)) is as follows:  
\[\mathbf{H}^{l}=\sigma(\text{Pool}(\mathbf{H}_{1}^{l},\mathbf{H}_{2}^{l}, \ldots\mathbf{H}_{p}^{l})), \tag{2}\]  
Specifically, the aggregation is conducted channel by channel in the adjacency tensor as follows:  
\[\mathbf{H}_{k}^{l}=\mathbf{EM}_{i,j}^{l-1}\mathbf{H}_{k}^{l-1}\mathbf{W}_{N}, k\in[1,p], \tag{3}\]  
where Pool is the mean-pooling operation to compress information from all channels, \(\mathbf{W}_{N}\in\mathbb{R}^{d_{g}\times d_{g}}\) is a learnable parameter, and \(\sigma\) is the ReLU activation function.  
The vector of each edge in layer \(l\) updates as follows:  
\[\mathbf{em}_{i,j}^{l}=\mathbf{W}_{E}[\mathbf{em}_{i,j}^{l-1}\oplus\mathbf{h}_ {i}^{l}\oplus\mathbf{h}_{j}^{l}],i,j\in[1,n], \tag{4}\]  
where \(\mathbf{W}_{E}\in\mathbb{R}^{(2*d_{g}+p)\times p}\) is a learnable parameter and \(\oplus\) is the join operation.  
Figure 3: Example of syntactic dependency parsing.  
We feed the final representation of each word (node) \(\mathbf{h}_{i}^{L}\) to a fully-connected network, which is followed by a softmax function to compute the probability distribution over all event types as follows:  
\[p(y_{i}|\mathbf{h}_{i}^{L})=\text{softmax}(\mathbf{W}_{C}\mathbf{h}_{i}^{L}+ \mathbf{b}_{C}), \tag{5}\]  
where \(\mathbf{W}_{C}\) maps the word representation \(\mathbf{h}_{i}^{L}\) to the feature score for each event type, and \(\mathbf{b}_{C}\) is a bias term. The event label with the largest probability is chosen as the classification result.  
The bias loss function is used to enhance the influence of event labels during training:  
\[J(\theta)=-\sum_{i=1}^{N_{\text{s}}}\sum_{j=1}^{n_{i}}\log p(y_{j}^{t}|s_{i}, \theta)\cdot I(O)+\alpha\log p(y_{j}^{t}|s_{i},\theta)\cdot(1-I(O)), \tag{6}\]  
where \(N_{\text{s}}\) is the number of sentences, \(n_{i}\) is the number of words in sentence \(s_{i}\), and \(y_{j}^{t}\) is the ground-truth event label of words; \(I(O)\) equals 1 if the event type of the word is 'O', otherwise 0; \(\alpha\) is the bias weight large than 1.