Demystifying Graph Neural Network Explanations

Abstract

Graph neural networks (GNNs) are quickly becoming the standard approach for learning on graph structured data across several domains, but they lack transparency in their decision-making. Several perturbation-based approaches have been developed to provide insights into the decision making process of GNNs. As this is an early research area, the methods and data used to evaluate the generated explanations lack maturity. We explore these existing approaches and identify common pitfalls in three main areas: (1) synthetic data generation process, (2) evaluation metrics, and (3) the final presentation of the explanation. For this purpose, we perform an empirical study to explore these pitfalls along with their unintended consequences and propose remedies to mitigate their effects.

A Background on GNNs and Perturbation-Based Explainer Methods

For a GNN, the goal is to learn a function of features on a graph $G=(V, E)$ with edges $E$ and nodes $V$. The input is comprised of a feature vector $x_i$ for every node i, summarized in a feature matrix $X \in \mathbb {R}^{n \times d_{in}}$ and a representative description of the link structure in the form of an adjacency matrix A. The output of the convolutional layer is a node-level latent representation matrix $Z \in \mathbb {R}^{n \times d_{out}}$, where $d_{out}$ is the number of output latent dimensions per node. Therefore, every convolutional layer can be written as a non-linear function:

$$\begin{aligned} H^{(l+1)} = f(H^{(l)}, A), \end{aligned}$$

with $H^{(0)} = X$ and $H^{(L)} = Z$, L being the number of stacked layers. The vanilla GNN model employed here, uses the propagation rule [3]:

$$\begin{aligned} f(H^{(l)}, A) = \sigma (\hat{D}^{-\frac{1}{2}}\hat{A}\hat{D}^{-\frac{1}{2}}H^{(l)}W^{(l)}), \end{aligned}$$

with $\hat{A} = A + I$, I being the identity matrix. $\hat{D}$ is the diagonal node degree matrix of $\hat{A}$, $W^{(l)}$ is a weight matrix for the ${l}{-}{th}$ neural network layer and $\sigma$ is a non-linear activation function. Taking the latent node representations Z of the last layer we define the logits of node $v_i$ for a node classification task as follows:

$$ \hat{y}_i = \text {softmax}(z_i W^{\top }_{c}), $$

where $W_c \in \mathbb {R}^{d_{out} \times k}$ projects the node representations into the k dimensional classification space.

GNNExplainer: The GNNExplainer takes a trained GNN and its prediction(s), and it returns an explanation in the form of a small subgraph of the input graph together with a small subset of node features that are most influential for the prediction. For their selection, the mutual information between the GNN prediction and the distribution of possible subgraph structures is maximized through optimizing the conditional entropy.

CF-GNNExplainer: The CF-GNNEXPLAINER works by perturbing input data at the instance-level. The instances are nodes in the graph since it is focused on node classification. The method iteratively removes edges from the original adjacency matrix based on matrix sparsification techniques, keeping track of the perturbations that lead to a change in prediction, and returning the perturbation with the smallest change w.r.t. the number of edges, after adding different edges to the subgraph.

ZORRO: ZORRO employs discrete masks to identify important input nodes and node features through a greedy algorithm, where nodes or node features are selected step by step. The goodness of the explanation is measured by the expected deviation from the prediction of the underlying model. A subgraph of the node’s computational graph and its set of features are relevant for a classification decision if the expected classifier score remains nearly the same when randomizing the remaining features.

PGExplainer: The PGExplainer learns approximated discrete masks for edges to explain the predictions. Given an input graph, it first obtains the embeddings for each edge by concatenating node embeddings. Then the predictor uses the edge embeddings to predict the probability of each edge being selected, similarly to an importance score. The approximated discrete masks are then sampled via the reparameterization trick. Finally, the objective function maximises the mutual information between the original predictions and new predictions.