The Impact of Dimensionality on the Stability of Node Embeddings

1. 1
   Define Research Scope

   Articulate clear research questions and hypotheses about how embedding dimension's impact on stability relates to downstream task performance and GNN architecture.
2. 2
   Select GNN Models

   Choose diverse Graph Neural Network (GNN) architectures (e.g., GCN, GraphSAGE, GAT) to ensure generalizability of findings across different models.
3. 3
   Choose Datasets

   Select multiple benchmark graph datasets (e.g., Cora, Citeseer, PPI) with varying characteristics (size, density, domain) to test the robustness of your findings.
4. 4
   Establish Metrics

   Define quantitative metrics for both embedding stability (e.g., cosine similarity variance, Euclidean distance variance, Procrustes analysis) and downstream task performance (e.g., F1-score for classification, AUC-ROC for link prediction).
5. 5
   Design Experiment

   Systematically vary embedding dimensions (e.g., 8, 16, 32, 64, 128, 256) and run each GNN-dataset-dimension configuration 'N' times (e.g., N=10 to 30) using different random seeds. Keep all other hyperparameters constant.
6. 6
   Collect and Analyze Results

   Gather embedding outputs and performance metrics from all runs. Analyze the variance of stability metrics and performance across different dimensions and seeds to identify trends, optimal dimensions, and GNN sensitivities.

import torch
import torch.nn.functional as F
from torch_geometric.nn import GCNConv
from torch_geometric.datasets import Planetoid

class GCN(torch.nn.Module):
    def __init__(self, in_channels, hidden_channels, out_channels):
        super().__init__()
        self.conv1 = GCNConv(in_channels, hidden_channels)
        self.conv2 = GCNConv(hidden_channels, out_channels)

    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = F.relu(x)
        x = F.dropout(x, p=0.5, training=self.training)
        x = self.conv2(x, edge_index)
        return x

# Example of varying embedding dimension (hidden_channels)
dataset = Planetoid(root='/tmp/Cora', name='Cora')
data = dataset[0]

input_dim = dataset.num_node_features
output_dim = dataset.num_classes
random_seeds = [42, 123, 789] # Use more seeds for robust results
embedding_dimensions = [16, 32, 64, 128] # Dimensions to test

results = {}

for emb_dim in embedding_dimensions:
    print(f"Testing with embedding dimension: {emb_dim}")
    dim_results = []
    for seed in random_seeds:
        torch.manual_seed(seed)
        # Initialize model with current embedding dimension
        model = GCN(input_dim, emb_dim, output_dim)
        # Placeholder for training and evaluation
        # In a real scenario, you'd train the model here, collect embeddings,
        # and compute stability/performance metrics.
        print(f"  - Seed {seed}: Model initialized for hidden_channels={emb_dim}")
        # Example: Perform a dummy forward pass to get embeddings
        # embeddings = model(data.x, data.edge_index)
        # dim_results.append(embeddings.detach().cpu().numpy())
        dim_results.append(f"Run {seed} completed for {emb_dim} dim")
    results[emb_dim] = dim_results

print("\nSimulation setup complete. You would now analyze 'results' for stability.")