Explainability in Graph Neural Networks¶
Author: Filippo Maria Bianchi.
Adapted from the original tutorial of Simone Scardapane.
Colab notebook here.
Libraries:
Download the data
mutag = pyg.datasets.TUDataset(root='.', name='MUTAG')
Print some statistics about the dataset
print(f"graph samples: {len(mutag)}")
print(f"classes: {mutag.num_classes}") # Binary (graph-level) classification
print(f"node features: {mutag.num_features}") # One-hot encoding for each node type (atom)
print(f"edge features: {mutag.num_edge_features}") # One-hot encoding for the bond type (we will ignore this)
graph samples: 188 classes: 2 node features: 7 edge features: 4
Each graph in the dataset is represented as an instance of the generic Data object
mutag_0 = mutag[0]
print(type(mutag_0))
<class 'torch_geometric.data.data.Data'>
# x contains the node features
mutag_0.x.shape
torch.Size([17, 7])
# y contains the corresponding class
mutag_0.y
tensor([1])
The Edges are stored in a COO format, with a 2xE list (edge_index[:, i] are the source and target nodes of the $i$-th edge)
mutag_0.edge_index.shape
torch.Size([2, 38])
# We print the first four edges in the list
mutag_0.edge_index[:, 0:4]
tensor([[0, 0, 1, 1],
[1, 5, 0, 2]])
Inside pyg.utils there are a number of useful tools.
E.g., we can check that the graph is undirected (the adjacency matrix is symmetric)
pyg.utils.is_undirected(mutag_0.edge_index)
True
We define a simple function for plotting the graph using the tools from networkx
colors = list(mcolors.TABLEAU_COLORS)
def draw_graph(g: pyg.data.Data, ax=None):
# Get a different color for each atom type
node_color = [colors[i.item()] for i in g.x.argmax(dim=1)]
# Convert to networkx
g = pyg.utils.to_networkx(g, to_undirected=True)
# Draw on screen
pos = nx.planar_layout(g)
pos = nx.spring_layout(g, pos=pos)
nx.draw_networkx(g, node_color=node_color, with_labels=False,
node_size=150, ax=ax)
draw_graph(mutag_0)
We can also use a more advanced drawing script (taken from here) that also shows the atom type
draw_molecule(to_molecule(mutag_0))
As an experiment, we load the graph with the adjacency in a SparseTensor format instead of the COO list
mutag_adj = pyg.datasets.TUDataset(root='.', name='MUTAG',
transform=pyg.transforms.ToSparseTensor())
We now have the adjacency matrix available as an additional propertyty)
mutag_adj[0].adj_t # This gives us useful information (size, nnz, density)
SparseTensor(row=tensor([ 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7,
8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14,
15, 16]),
col=tensor([ 1, 5, 0, 2, 1, 3, 2, 4, 9, 3, 5, 6, 0, 4, 4, 7, 6, 8,
7, 9, 13, 3, 8, 10, 9, 11, 10, 12, 11, 13, 14, 8, 12, 12, 15, 16,
14, 14]),
size=(17, 17), nnz=38, density=13.15%)
The SparseTensor format supports several useful methods.
For example, we can propagate the features on the graph (cornerstone of graph-based methods) efficiently with a sparse-dense matrix multiplication
print(torch_sparse.matmul(mutag_adj[0].adj_t, mutag_adj[0].x).shape)
torch.Size([17, 7])
3. Data loaders¶
Data loaders are a nice utility to automatically build mini-batches from the dataset.
A batch can either be a set of graphs, or subgraphs extracted from a single graph.
First, we split the original dataset into a training and a test part with a split stratified on the class
train_idx, test_idx = train_test_split(range(len(mutag)), stratify=[m.y[0].item() for m in mutag], test_size=0.25)
Then, we build the two loaders
train_loader = pyg.loader.DataLoader(mutag[train_idx], batch_size=32, shuffle=True)
test_loader = pyg.loader.DataLoader(mutag[test_idx], batch_size=32)
Let us load the first batch of data
batch = next(iter(train_loader))
The batch is built by considering all the subgraphs as a single giant graph with unconnected components
print(f"x shape: {batch.x.shape}") # All the nodes of the 32 graphs are put together
print(f"y shape: {batch.y.shape}") # A single label for each graph
print(f"edges shape: {batch.edge_index.shape}") # Edge list of all 32 graphs
x shape: torch.Size([566, 7]) y shape: torch.Size([32]) edges shape: torch.Size([2, 1242])
There is an additional property in batch that links each node to its corresponding graph index
print(batch.batch.shape)
print(batch.batch[0:30]) # print the first 30 elements
torch.Size([566])
tensor([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1])
We can perform graph-level operations with torch_scatter (more details here)
# Sum all the nodes in the same graph
print(torch_scatter.scatter_sum(batch.x, batch.batch, dim=0).shape)
torch.Size([32, 7])
# Average all the nodes in the same graph
print(torch_scatter.scatter_mean(batch.x, batch.batch, dim=0).shape)
torch.Size([32, 7])
# Alternatively, PyG has this implemented as a functional layer
pyg.nn.global_mean_pool(batch.x, batch.batch).shape
torch.Size([32, 7])
4. Building the GNN¶
Layers in PyG are very similar to standard PyTorch layers.
This is a standard graph convolutional layer.
gcn = pyg.nn.GCNConv(7, 12)
gcn(batch.x, batch.edge_index).shape
torch.Size([566, 12])
Different layers have different properties
See this cheatsheet from more info.
For example, GCNConv accepts an additional edge_weight parameter to weight each edge.
We will use this later on, to mask the corresponding edges for the prediction.
4.1 Pytorch lightning¶
PyTorch Lightning is a lightweight wrapper that helps organize PyTorch code, making it more readable and maintainable.
It simplifies the complex engineering tasks involved in PyTorch models removing a lot of boilerplate code.
If you are not famliar with PyTorch Lightning, see this 5-minutes intro.
4.1.1 The Lightning Module¶
class MUTAGClassifier(ptlight.LightningModule):
def __init__(self):
# The model is just GCNConv --> GCNConv --> global pooling --> Dropout --> Linear
super().__init__()
self.gc1 = pyg.nn.GCNConv(7, 64)
self.gc2 = pyg.nn.GCNConv(64, 256)
self.linear = torch.nn.Linear(256, 1)
def forward(self, x, edge_index=None, batch=None, edge_weight=None):
# Note: "edge_weight" is not used for training, but only for the explainability part
if edge_index == None:
x, edge_index, batch = x.x, x.edge_index, x.batch
x = F.relu(self.gc1(x, edge_index, edge_weight))
x = F.relu(self.gc2(x, edge_index, edge_weight))
x = pyg.nn.global_mean_pool(x, batch)
x = F.dropout(x)
x = self.linear(x)
return x
def configure_optimizers(self):
optimizer = torch.optim.Adam(self.parameters(), lr=1e-3)
return optimizer
def training_step(self, batch, _):
y_hat = self.forward(batch.x, batch.edge_index, batch.batch)
loss = F.binary_cross_entropy_with_logits(y_hat, batch.y.unsqueeze(1).float())
self.log("train_loss", loss)
self.log("train_accuracy", accuracy(y_hat, batch.y.unsqueeze(1), task='binary'), prog_bar=True, batch_size=32)
return loss
def validation_step(self, batch, _):
x, edge_index, batch_idx = batch.x, batch.edge_index, batch.batch
y_hat = self.forward(x, edge_index, batch_idx)
self.log("val_accuracy", accuracy(y_hat, batch.y.unsqueeze(1), task='binary'), prog_bar=True, batch_size=32)
We instantiate the classifier
gnn = MUTAGClassifier() # PL module
Sanity check: we run a single mini-batch and check the output shape
gnn(batch.x, batch.edge_index, batch.batch).shape
torch.Size([32, 1])
4.1.2 The training loop¶
Save weights of the model during training
checkpoint_callback = ptlight.callbacks.ModelCheckpoint(
dirpath='./checkpoints/',
filename='gnn-{epoch:02d}',
every_n_epochs=50,
save_top_k=-1)
Create the trainer
trainer = ptlight.Trainer(max_epochs=150, callbacks=[checkpoint_callback])
GPU available: True (cuda), used: True TPU available: False, using: 0 TPU cores IPU available: False, using: 0 IPUs HPU available: False, using: 0 HPUs /home/filippo/anaconda3/envs/pyg/lib/python3.10/site-packages/pytorch_lightning/trainer/connectors/logger_connector/logger_connector.py:67: Starting from v1.9.0, `tensorboardX` has been removed as a dependency of the `pytorch_lightning` package, due to potential conflicts with other packages in the ML ecosystem. For this reason, `logger=True` will use `CSVLogger` as the default logger, unless the `tensorboard` or `tensorboardX` packages are found. Please `pip install lightning[extra]` or one of them to enable TensorBoard support by default
Start the training loop
# This is not a particularly well-designed model, we expect approximately 70-80% val accuracy
trainer.fit(gnn, train_loader, test_loader)
You are using a CUDA device ('NVIDIA GeForce RTX 3090') that has Tensor Cores. To properly utilize them, you should set `torch.set_float32_matmul_precision('medium' | 'high')` which will trade-off precision for performance. For more details, read https://pytorch.org/docs/stable/generated/torch.set_float32_matmul_precision.html#torch.set_float32_matmul_precision
LOCAL_RANK: 0 - CUDA_VISIBLE_DEVICES: [0]
| Name | Type | Params
-----------------------------------
0 | gc1 | GCNConv | 512
1 | gc2 | GCNConv | 16.6 K
2 | linear | Linear | 257
-----------------------------------
17.4 K Trainable params
0 Non-trainable params
17.4 K Total params
0.070 Total estimated model params size (MB)
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/home/filippo/anaconda3/envs/pyg/lib/python3.10/site-packages/pytorch_lightning/trainer/connectors/data_connector.py:441: The 'val_dataloader' does not have many workers which may be a bottleneck. Consider increasing the value of the `num_workers` argument` to `num_workers=23` in the `DataLoader` to improve performance. /home/filippo/anaconda3/envs/pyg/lib/python3.10/site-packages/pytorch_lightning/trainer/connectors/data_connector.py:441: The 'train_dataloader' does not have many workers which may be a bottleneck. Consider increasing the value of the `num_workers` argument` to `num_workers=23` in the `DataLoader` to improve performance. /home/filippo/anaconda3/envs/pyg/lib/python3.10/site-packages/pytorch_lightning/loops/fit_loop.py:293: The number of training batches (5) is smaller than the logging interval Trainer(log_every_n_steps=50). Set a lower value for log_every_n_steps if you want to see logs for the training epoch.
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IOPub message rate exceeded. The Jupyter server will temporarily stop sending output to the client in order to avoid crashing it. To change this limit, set the config variable `--ServerApp.iopub_msg_rate_limit`. Current values: ServerApp.iopub_msg_rate_limit=1000.0 (msgs/sec) ServerApp.rate_limit_window=3.0 (secs)
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`Trainer.fit` stopped: `max_epochs=150` reached.
5. Explainability¶
5.1 GNNExplainer¶
GNNExplainer is a simple method to search for instance-level explanations on a trained GNN.
It optimizes for the smallest feature and edge masks that do not vary the prediction of the trained model.
We take a single graph from the test dataset
idx = 0
test_graph = mutag[test_idx[idx]]
draw_graph(test_graph)
# Explanations are mostly interesting for mutagenic graphs
print(test_graph.y)
tensor([1])
PyG has an implementation of GNNExplainer that we will use.
It provides a mask that can remove individual features for each node.
# Instantiate the explainer
explainer = Explainer(
model=gnn,
algorithm=GNNExplainer(epochs=300),
explanation_type='model',
node_mask_type='attributes',
edge_mask_type='object',
model_config=dict(
mode='binary_classification',
task_level='graph',
return_type='raw',
),
)
# Compute the explanations
explanation = explainer(test_graph.x, test_graph.edge_index)
These are the resulting masks.
print(explanation)
Explanation(node_mask=[22, 7], edge_mask=[50], prediction=[1, 1], target=[1], x=[22, 7], edge_index=[2, 50])
We mostly care about the edges here.
explanation.edge_mask
tensor([0.0322, 0.0369, 0.0410, 0.0401, 0.0363, 0.0415, 0.0340, 0.0341, 0.0368,
0.0443, 0.0419, 0.0393, 0.0369, 0.0333, 0.0417, 0.0404, 0.0408, 0.0363,
0.0389, 0.0335, 0.0252, 0.0400, 0.0465, 0.0368, 0.0397, 0.0549, 0.0455,
0.0473, 0.0420, 0.0431, 0.0352, 0.0485, 0.0410, 0.0450, 0.0552, 0.0453,
0.0584, 0.0482, 0.0626, 0.0737, 0.0592, 0.0607, 0.9391, 0.9240, 0.9459,
0.0922, 0.0538, 0.0559, 0.9325, 0.9292])
Get the relevant edges (highest values inside the edge mask)
relevant_edges = explanation.edge_mask > 0.90
We build the subgraph corresponding to the explanation
explanation_subgraph = pyg.data.Data(test_graph.x, test_graph.edge_index[:, relevant_edges])
A good explanation should contain $CI_2O$ groups, which are known to be mutagenic
draw_graph(explanation_subgraph)
We can also use the other visualizer
draw_molecule(to_molecule(explanation_subgraph))
5.2 Integrated Gradients¶
Instead of using a custom-made explainer, we can also try using a "standard" one.
Here we use the implementation of Integrated Gradients provided by Captum.
How IG works?¶
- Baseline Selection: Choose a baseline representing a non-informative input (e.g., a black image).
- Path Construction: Construct a path that gradually changes the baseline into the actual input.
Gradient Calculation: At each step along this path, the gradient of the model output with respect to the input is calculated.
This gradient signifies how much each input feature contributes to the output.
- Integration: The gradients across all the steps are summed up to quantify the importance of each input feature in determining the model's output.
In our case, we will integrate a "saliency vector" moving from a graph with no connectivity to our actual graph.
We define a custom function whose input is the edge mask.
batch_idx = torch.zeros(test_graph.x.shape[0], dtype=torch.int64) # All nodes belong to the same graph
gnn.eval() # disable dropout, turn off gradients computation
def model_forward(edge_mask, graph):
out = gnn(graph.x, graph.edge_index, batch_idx, edge_weight=edge_mask[0])
return out
We initialize the edge mask with all 1s
edge_mask = torch.ones(test_graph.edge_index.shape[1]).unsqueeze(0).requires_grad_(True)
print(edge_mask.shape)
torch.Size([1, 50])
model_forward(edge_mask, test_graph)
tensor([[1.6257]], grad_fn=<AddmmBackward0>)
We compute IG using the Captum IntegratedGradients class.
# Instantiate IG with the custom function we just defined
ig = IntegratedGradients(model_forward)
# Compute the mask
ig_mask = ig.attribute(edge_mask, additional_forward_args=test_graph, internal_batch_size=1)
This is the same as before, we just use the new mask
explanation = pyg.data.Data(test_graph.x, test_graph.edge_index[:, ig_mask[0] > 0.1])
explanation = pyg.transforms.RemoveIsolatedNodes()(pyg.transforms.ToUndirected()(explanation))
draw_graph(explanation)
5.3 TracIn¶
TracIn (Traceable Inputs) measures the impact of each training example on a specific prediction (e.g., the class of test_graph).
It helps to identify which training data points are most influential for a particular prediction.
TracIn computes the gradients of the loss for a particular data point (e.g., test_graph) and for all training samples.
By comparing these gradients, TracIn estimates the cumulative influence of each training example on the classification of the particular data point.
First, we manually select (some of) the checkpoints.
!ls checkpoints
'gnn-epoch=149.ckpt' 'gnn-epoch=49.ckpt' 'gnn-epoch=99.ckpt'
ckpt = ['checkpoints/gnn-epoch=49.ckpt',
'checkpoints/gnn-epoch=99.ckpt',
'checkpoints/gnn-epoch=149.ckpt'
]
We will compute the gradient only with respect to the last linear layer of the GNN.
Remember that our GNN model is GCNConv $\rightarrow$ GCNConv $\rightarrow$ global pooling $\rightarrow$ Dropout $\rightarrow$ Linear
params = gnn.linear.weight.shape[1]
print(params)
256
We will store here the gradients for the test graph on all checkpoints
test_gradients = torch.zeros((len(ckpt), params))
for k in range(len(ckpt)):
# Manually load the checkpoint
checkpoint = torch.load(ckpt[k])
gnn.load_state_dict(checkpoint['state_dict'])
gnn.eval()
# Compute the gradient on the test graph
y = gnn(test_graph.x, test_graph.edge_index)
test_gradients[k] = torch.autograd.grad(y, gnn.linear.weight)[0]
We do the same also for all training points
mutag_train = mutag[train_idx] # Training set
train_gradients = torch.zeros((len(mutag_train), len(ckpt), params))
for k in range(len(ckpt)):
# Manually load the checkpoint
checkpoint = torch.load(ckpt[k])
gnn.load_state_dict(checkpoint['state_dict'])
gnn.eval()
# Same as before, but we loop over all training graphs
for i in range(len(mutag_train)):
y = gnn(mutag_train[i].x, mutag_train[i].edge_index)
train_gradients[i, k] = torch.autograd.grad(y, gnn.linear.weight)[0]
# test_gradients.shape = ( 3, 256)
# train_gradients.shape = (141, 3, 256)
# This computes the relevance score for all graphs
relevance = (test_gradients * train_gradients).sum(dim=[1, 2])
relevance.shape
torch.Size([141])
# Take the top-k highest values (proponents), and top-k lowest values (opponents)
_, proponents = torch.topk(relevance, k=3, largest=True)
_, opponents = torch.topk(relevance, k=3, largest=False)
# Let's draw once again the test_graph
draw_graph(test_graph)
fig, axs = plt.subplots(1, 3, layout='tight', figsize=(10, 4))
for c, i in enumerate(proponents):
draw_graph(mutag_train[i], ax=axs[c])
sign_i = gnn(mutag_train[i].x, mutag_train[i].edge_index).sign()
axs[c].set_title(f"Class: {mutag_train[i].y.numpy()[0]}")
fig.suptitle('Proponents', fontsize=16)
plt.show()
fig, axs = plt.subplots(1, 3, layout='tight', figsize=(10, 4))
for c, i in enumerate(opponents):
draw_graph(mutag_train[i], ax=axs[c])
sign_i = gnn(mutag_train[i].x, mutag_train[i].edge_index).sign()
axs[c].set_title(f"Class: {mutag_train[i].y.numpy()[0]}")
fig.suptitle('Opponents', fontsize=16)
plt.show()
The end¶
- Questions?
- Contact: filippo.m.bianchi@uit.no