Graph Machine LearningGraph Neural NetworksGraph Learning

Simplified Graph Convolution (SGC)

Primary task · Graph Learning

Simplified Graph Convolution (SGC) is a graph neural architecture for learning representations from entities connected by edges. Its interactive page visualises how information travels through a graph, how a node receptive field changes with training depth, and how learned embeddings support node- or graph-level prediction.

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Visual intuition

From data to learned behaviour

A graph model learns by letting connected entities exchange information. Early layers capture immediate neighbours; deeper layers expand the receptive field, allowing a node or whole graph to encode increasingly broader structural context.

Infographic

Conceptual simulation

Watch the learning mechanism form

The structure below is synchronized with the same training state used by the prediction simulation.

Mechanism view

Training control centre

Control both simulations together

Reset regenerates the synthetic data and model state. Train animates to completion. Pause freezes the animation. Train Step advances one learning stage.

Step 0 / 8

Model simulation

Inspect the learned prediction / representation

Synthetic data are generated locally in your browser.

Model description

Understand Simplified Graph Convolution (SGC) after watching it learn

This section connects the animation to the actual statistical or computational idea behind the model.

Deep description

Simplified Graph Convolution (SGC) Simplified Graph Convolution (SGC) is a graph neural architecture for learning representations from entities connected by edges. Its interactive page visualises how information travels through a graph, how a node receptive field changes with training depth, and how learned embeddings support node- or graph-level prediction.

What is learned. During training, the algorithm builds or adjusts the parameters and internal representation used by Simplified Graph Convolution (SGC). The core learning mechanism is: Collapses repeated graph propagation into a fixed linear smoothing operator and removes intermediate nonlinearities, followed by a linear classifier.

How training becomes inference. Start with node features and edges → propagate or attend to neighbour information → update hidden node embeddings → repeat for several layers → apply a node, edge, or graph readout → optimise task loss with back-propagation. Once training stops, the fitted state is reused on unseen inputs rather than being reconstructed from scratch. The resulting output is: Node, edge or graph-level embeddings and predictions produced by graph-aware aggregation.

Why practitioners use it. Very fast and demonstrates how much GCN performance comes from graph smoothing. Typical fits include Fast graph baselines, large node-classification experiments, ablation studies.

What to verify before trusting it. Reduced capacity for complex nonlinear graph relationships. The visual simulation is intentionally simplified, so real use should still validate preprocessing, data independence, hyperparameters, uncertainty and task-appropriate metrics.

Internal statethe parameters and internal representation used by Simplified Graph Convolution (SGC)

Typical outputNode, edge or graph-level embeddings and predictions produced by graph-aware aggregation.

Good fitFast graph baselines, large node-classification experiments, ablation studies.

Main cautionReduced capacity for complex nonlinear graph relationships.

1Training data→

2Learning objective→

3Internal model state→

4Prediction / representation→

5Evaluation

Intuition

What the model is trying to learn

Mathematical lens

Core logic

Most GNNs can be viewed as message passing: compute messages from neighbouring states and edge information, aggregate them with a permutation-invariant operator, then update each node representation. Architectures differ mainly in how messages are weighted, aggregated, propagated, or globally attended.

Training sequence

How learning progresses

Start with node features and edges → propagate or attend to neighbour information → update hidden node embeddings → repeat for several layers → apply a node, edge, or graph readout → optimise task loss with back-propagation.

Original mechanism

Taxonomy description

Collapses repeated graph propagation into a fixed linear smoothing operator and removes intermediate nonlinearities, followed by a linear classifier.

Evaluation guide

How to evaluate this model responsibly

ValidationSplit by nodes, edges or entire graphs according to the deployment unit; avoid connectivity leakage across train/test partitions.

MetricsNode/graph classification: F1/ROC-AUC; regression: MAE/RMSE; link prediction: ROC-AUC/AP.

HPOTune depth, hidden width, dropout, learning rate and propagation/attention settings.

Post-processingCalibrate classification probabilities and assess uncertainty/stability across graph splits.

Hyperparameters

Key parameters

layersTypical: 2–4

Number of message-passing/propagation stages.

hidden_dimTypical: 64

Size of learned node embeddings.

dropoutTypical: 0.0–0.5

Regularisation between graph layers.

Use & trade-offs

Where it fits

Typical applications

Fast graph baselines, large node-classification experiments, ablation studies.

Strengths

Very fast and demonstrates how much GCN performance comes from graph smoothing.

Limitations

Reduced capacity for complex nonlinear graph relationships.

Code example

Minimal Python implementation

import torch

torch.manual_seed(7)
# Five-node undirected graph with self-loops.
A = torch.tensor([
    [1,1,0,0,0],
    [1,1,1,0,0],
    [0,1,1,1,0],
    [0,0,1,1,1],
    [0,0,0,1,1],
], dtype=torch.float32)
X = torch.randn(5, 3)

# SGC removes intermediate nonlinearities and propagates K times first.
deg = A.sum(1)
D = torch.diag(deg.pow(-0.5))
A_norm = D @ A @ D
K = 2
X_sgc = torch.matrix_power(A_norm, K) @ X
classifier = torch.nn.Linear(3, 2)
logits = classifier(X_sgc)

print("STEP 1 · Normalise adjacency and add self-information")
print("STEP 2 · Propagate features", K, "times without hidden nonlinearities")
print("STEP 3 · node logits shape", tuple(logits.shape))

Expected / representative output

STEP 1 · Normalise adjacency and add self-information
STEP 2 · Propagate features 2 times without hidden nonlinearities
STEP 3 · node logits shape (5, 2)