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This is Lecture 1 of Stanford CS224W: Machine Learning with Graphs, Fall 2025, dated 2025-09-23. This reconstruction uses the course schedule and the lecture slides; the slides credit Jure Leskovec and the course team.
Materials and gaps
Public materials include 01-intro.pdf and the readings listed on the schedule. Canvas video, live Q&A, board work, and Ed discussions are unavailable, so this article does not reconstruct them. The public 2021 videos are not evidence for a 2025 lecture.
Complete lecture agenda
1. Course map and tools
The slides first expose the entire course: node embeddings, GNNs, graph transformers, knowledge graphs, graph generation, relational deep learning, and GNN+LLM. The unifying question is whether relationships between data points carry predictive signal.
2. A common language for graph data
A graph consists of nodes and edges. It may be directed or undirected, weighted or unweighted, and may attach features to both nodes and edges. Social systems, molecules, citation networks, and relational databases can all be expressed in this vocabulary.
3. Hand-designed features and representation learning
A traditional pipeline designs features such as degree and clustering coefficient before fitting a predictor. Representation learning instead learns vectors from neighborhoods and structure. That shift is the spine of the course.
4. Node-, edge-, and graph-level tasks
The task scale determines the output: node classification, link prediction, and graph classification require different readouts. Writing down the prediction unit before choosing a model prevents basic splitting and evaluation mistakes.
5. A route through the lecture
For self-study, choose one tiny graph and write each node's degree, neighborhood, and one prediction target. Reuse it as a sanity check for embeddings and message passing in later lectures.
Slide-by-slide expansion
A graph is a modeling decision
The deck connects social, information, technological, and biological networks. Nodes may be people, pages, proteins, or drugs; edges may encode interaction, hyperlinks, or physical binding. The useful question is whether relationships change the prediction. Write a graph-construction contract before selecting a model: node meaning, edge existence, direction, weight, and what is observable at prediction time.
Graph variants and representation
Directed and undirected edges have different semantics; weighted edges may encode frequency, distance, or trust; multirelational graphs cannot safely collapse relation types; dynamic graphs require timestamps to prevent future leakage. Adjacency matrices expose linear algebra, while edge lists preserve sparsity. Large real graphs should not be represented as dense n-by-n arrays.
Task scale determines labels and splits
Node, edge, and graph tasks require more than different heads. Node prediction may use masks, link prediction must remove held-out edges from training structure, and graph classification splits whole graphs. Define the prediction unit before constructing a split.
Hand-designed features remain baselines
Degree, clustering coefficients, centrality, motifs, and graphlets remain informative. Representation learning changes how features are obtained, not the need for a credible baseline. A GNN that fails to beat a degree-based predictor has not demonstrated complex relational reasoning.
Encoder, decoder, and objective
Read the course as three blocks: an encoder maps graph objects to vectors, a decoder turns vectors into scores, and an objective defines success. Changing negative sampling or loss can change the learned geometry even when the encoder architecture is unchanged.
Permutation is a hard constraint
Renumbering nodes does not change a graph. Graph outputs should be invariant, while node outputs should be equivariant. Sum, mean, max, and graph readouts are designed around this requirement.
Where the tools fit
The slides list PyTorch Geometric, GraphGym, SNAP.PY, and NetworkX. Use a tiny hand-computed graph before trusting a library. Differences usually expose self-loop, direction, normalization, or batching assumptions.
A repeatable minimal experiment
Build a four-node path with constant features. Predict degree, endpoint status, and graph connectivity; then permute node IDs. Node outputs should permute and the graph output should remain unchanged. Reuse this test throughout the series.
Case studies and a complete modeling audit
1. Social-network adoption
For adoption prediction, follow edges are directed, while interaction edges carry time and frequency. Collapsing them into one undirected binary edge discards all three signals. Cut the graph at prediction time, compare attributes and degree first, then add neighbor behavior.
2. Molecular graphs
Atoms are nodes and bonds are typed edges; readout must ignore atom ordering, arbitrary edge augmentation can violate valence, and random splits over related scaffolds can make testing unrealistically easy.
3. Knowledge graphs
A person—born_in—city triple is directed and typed, while an unrecorded triple is not necessarily false. TransE, R-GCN, and inductive KG models build on this directed, typed, incomplete setting.
4. The dependency chain
Embeddings define geometry; GNNs replace lookup with shared encoders; later lectures expose components, limits, transformers, heterogeneous graphs, RDL, LLMs, agents, and generation.
5. Evidence boundary
Slides establish formulas, examples, and agenda, not live questions or Canvas demonstrations. Worked examples here are explicit reconstructions.
Final acceptance exercise
Write a graph problem specification with examples, schema, feature availability, prediction unit, split, baseline, and metric. Run edge ablation and node-permutation tests.
Where this lecture leads
The concepts from Lecture 1 are composed in later lectures. Keep one small graph, a notation sheet, and a baseline. For every new model, identify whether it changes the data, message, aggregation, update, objective, or evaluation.
Expected results for the four-node graph
Continue the invented path example with undirected edges A—B—C—D, no self-loops, and initial feature 1 at every node. Degrees are 1, 2, 2, 1: A and D are endpoints; B and C are interior nodes.
One aggregation that sums neighbors’ initial features also returns 1, 2, 2, 1. No training is needed to compute this. If code differs, check edge direction and self-loops first. Reorder input nodes to C, A, D, B: outputs must follow those nodes as 2, 1, 1, 2, rather than keeping the original array fixed.
Check: What if aggregation uses mean instead of sum? All four outputs are 1, because every neighbor feature is 1. This shows that aggregation affects which signal is retained; it does not establish that mean is worse on every task.
Course video sources
Checked live: the archived Fall 2025 official page lists no public recording link (its Canvas entry is empty), and the current Fall 2026 official page says lecture videos are on Canvas for enrolled students only; the Canvas entry redirects to the Stanford sign-in page. The public Stanford Online YouTube playlists currently cover only the 2021 and 2023 offerings, whose lectures differ from the Fall 2025 offering this article follows, so they are not embedded or treated as Fall 2025 recordings.
Course and recording entries:
- Official course / lecture source
- CS224W Fall 2025 official archive
- Stanford Online CS224W public playlist (2021 offering, not Fall 2025)
- Stanford Online CS224W public playlist (2021 / 2023 offerings, not Fall 2025)
Checked: 2026-10-10.
Update Log
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2026-10-11: Added an invented teaching example, worked reasoning, and a comprehension check with an answer.
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2026-10-10: Added explicit video status and checked recording sources and access notes.
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2026-10-10: Rechecked video status. Confirmed the official pages list no public Fall 2025 recording, Canvas requires sign-in, and public playlists cover only 2021/2023; status unchanged and check date added.
References
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