{ "id": "ep92hz02dv", "slug": "message-passing-neural-network", "title": "Message passing neural network", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_computation" }, { "id": "technology.computing.ai.neural_deep", "label": "Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.neural_deep" }, { "id": "technology.computing.ai.nn_architectures", "label": "Neural network architectures", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.nn_architectures" }, { "id": "technology.computing.ai.nn_architectures.graph_networks", "label": "Graph neural network architectures", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.nn_architectures.graph_networks" } ], "geo": [ { "id": "geo.nongeo.t2001.technology.computing.ai.neural_deep", "label": "Non-geographic · 2001 to 2020: Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.neural_deep", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology" }, { "id": "geo.nongeo.t2001.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing" }, { "id": "geo.nongeo.t2001.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai" }, { "id": "geo.nongeo.t2001.technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.ml_computation" }, { "id": "geo.nongeo.t2001.technology.computing.ai.neural_deep", "label": "Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.neural_deep" } ] } ], "excerpt": "A message passing neural network (MPNN) is a graph neural network framework, formalized in 2017, that exchanges messages along edges and excels at molecular property prediction.", "snippet": "A message passing neural network (MPNN) is a graph neural network framework, formalized in 2017, that exchanges messages along edges and excels at molecular property prediction.", "node": "technology.computing.ai.nn_architectures.graph_networks", "markdown": "# Message passing neural network\n\nA message passing neural network (MPNN) is a graph neural network that learns representations of graph-structured data by iteratively exchanging feature vectors, called messages, along edges and updating each node's state from the messages it receives. The framework computes permutation-invariant functions of graphs, and it has proven well suited to molecular property prediction, where molecules are naturally represented as graphs with atoms as nodes and chemical bonds as edges.\n\n| Key fact | Value |\n|---|---|\n| Framework paper | Gilmer, Schoenholz, Riley, Vinyals, and Dahl, *Neural Message Passing for Quantum Chemistry*, 2017 [1](https://doi.org/10.48550/arxiv.1704.01212) |\n| Core equations | \\( m_{v}^{t+1} = \\sum_{w \\in N(v)} M_{t}(h_{v}^{t}, h_{w}^{t}, e_{vw}) \\), then \\( h_{v}^{t+1} = U_{t}(h_{v}^{t}, m_{v}^{t+1}) \\) [2](https://proceedings.mlr.press/v70/gilmer17a.html) |\n| Readout | \\( \\hat{y} = R(\\{h_{v}^{T} \\mid v \\in G\\}) \\), with \\( R \\) invariant to node order [2](https://proceedings.mlr.press/v70/gilmer17a.html) |\n| QM9 result | Chemical accuracy on 11 of 13 targets, state of the art on all 13 (130k molecules, 13 DFT properties) [2](https://proceedings.mlr.press/v70/gilmer17a.html) |\n| Speed vs DFT | Predictions up to 300,000 times faster than DFT simulation [3](https://research.google/blog/predicting-properties-of-molecules-with-machine-learning/) |\n| Typical depth | 3 to 6 message passing steps for most molecular tasks [4](https://engineersofai.com/docs/ml/graph-neural-networks/message-passing-neural-networks) |\n| Expressivity ceiling | At most equal to the 1-Weisfeiler-Leman isomorphism test [5](https://proceedings.neurips.cc/paper/2020/file/a32d7eeaae19821fd9ce317f3ce952a7-Paper.pdf) |\n\n## How it works\n\nEach node \\( v \\) carries a hidden state \\( h_{v}^{t} \\). In the message passing phase, run for \\( T \\) steps, every node computes a message from each neighbor \\( w \\) with a message function \\( M_{t} \\), which may also take the edge features \\( e_{vw} \\) as input, aggregates these messages, and combines the aggregate with its own state through an update function \\( U_{t} \\) [2](https://proceedings.mlr.press/v70/gilmer17a.html):\n\n\\[ m_{v}^{t+1} = \\sum_{w \\in N(v)} M_{t}(h_{v}^{t}, h_{w}^{t}, e_{vw}), \\qquad h_{v}^{t+1} = U_{t}(h_{v}^{t}, m_{v}^{t+1}). \\]\n\nThe aggregation must be a permutation-invariant function such as sum, mean, or max, so the result does not depend on neighbor ordering; PyTorch Geometric expresses the same layer as \\( x_{i}' = \\gamma_{\\Theta}(x_{i}, \\bigoplus_{j \\in N(i)} \\phi_{\\Theta}(x_{i}, x_{j}, e_{j,i})) \\) with \\( \\bigoplus \\) chosen among sum, mean, min, max, or mul.[6](https://pytorch-geometric.readthedocs.io/en/latest/generated/torch_geometric.nn.conv.MessagePassing.html) After \\( T \\) steps, a readout function \\( R \\), itself permutation-invariant, maps the set of node states to an output.[2](https://proceedings.mlr.press/v70/gilmer17a.html)\n\nExpressivity is tied to the [Weisfeiler-Leman test](https://www.edgepedia.org/weisfeiler-leman-test): without special node attributes, an MPNN is at most as good at isomorphism testing as the 1-WL vertex refinement algorithm, even with infinite depth and width, and it cannot determine whether a graph is connected, a node's local clustering coefficient, or whether a cycle is present.[5](https://proceedings.neurips.cc/paper/2020/file/a32d7eeaae19821fd9ce317f3ce952a7-Paper.pdf) Provably reaching the 1-WL bound requires injective aggregation, which motivates the Graph Isomorphism Network.[7](https://arxiv.org/abs/1810.00826)\n\n## How it is done\n\nTraining follows a fixed sequence. For each of \\( T \\) layers, the model computes messages \\( M_{t} \\), aggregates them with a permutation-invariant operator, and applies \\( U_{t} \\); the readout \\( R \\) then produces node, edge, or graph predictions.[2](https://proceedings.mlr.press/v70/gilmer17a.html) In software, a practitioner implements `message()` and `update()` functions and selects the aggregation operator and message flow direction.[6](https://pytorch-geometric.readthedocs.io/en/latest/generated/torch_geometric.nn.conv.MessagePassing.html) For molecular graphs, edges are typically built from a distance cutoff or from \\( K \\)-nearest neighbors; one study found \\( K \\)-nearest-neighbor graphs gave better prediction accuracy than maximum-distance cutoff or Voronoi tessellation graphs.[8](https://export.arxiv.org/pdf/1806.03146)\n\n## Origin\n\nThe MPNN framework was reported by Gilmer and colleagues in *Neural Message Passing for Quantum Chemistry* (2017, arXiv), which recast at least eight existing models, including spectral approaches, neural fingerprints, gated graph networks, molecular graph convolutions, and deep tensor neural networks, into one message-passing form.[1](https://doi.org/10.48550/arxiv.1704.01212) • [2](https://proceedings.mlr.press/v70/gilmer17a.html) A Nature Reviews Methods Primer describes this paper as the first to formalize the idea of message passing.[9](https://www.nature.com/articles/s43586-024-00294-7) It built on earlier work: Scarselli and colleagues' graph neural network model (IEEE Transactions on Neural Networks, 2008) diffused information between nodes until a stable fixed point, computed during each training step, a mechanism MPNN replaced with a fixed, finite number of steps.[10](https://doi.org/10.1109/tnn.2008.2005605) Duvenaud and colleagues' convolutional networks on molecular graphs (2015, arXiv) applied the same local filter to each atom and its neighborhood, followed by global pooling [11](https://doi.org/10.48550/arxiv.1509.09292), and Kipf and Welling's graph convolutional network (2016, arXiv) set off the recent years of development of GNNs.[12](https://doi.org/10.48550/arxiv.1609.02907)\n\n## Variants\n\nVariants differ in the message, aggregation, or update functions [2](https://proceedings.mlr.press/v70/gilmer17a.html):\n\n- **GCN** aggregates with degree normalization: \\( x_{i}^{(k)} = \\sum_{j \\in N(i) \\cup \\{i\\}} \\frac{1}{\\sqrt{\\deg(i)} \\cdot \\sqrt{\\deg(j)}} \\cdot (W^{\\top} \\cdot x_{j}^{(k-1)}) + b \\), implemented with self-loops.[13](https://pytorch-geometric.readthedocs.io/en/latest/tutorial/create_gnn.html) Gilmer et al. write its message as \\( M_{t}(h_{v}^{t}, h_{w}^{t}) = c_{vw} \\cdot h_{w}^{t} \\) with \\( c_{vw} = (\\deg(v) \\cdot \\deg(w))^{-1/2} \\cdot A_{vw} \\).[2](https://proceedings.mlr.press/v70/gilmer17a.html)\n- **GraphSAGE** uses generalized neighborhood aggregation and a concatenation-based update, in which a node's own features and its neighbor aggregate are concatenated before a learned transformation is applied.[14](https://cs.mcgill.ca/~wlh/grl_book/files/GRL_Book-Chapter_5-GNNs.pdf) • [15](https://dl.acm.org/doi/full/10.1145/3816725)\n- **GAT** applies attention, weighting each neighbor's message with data-dependent coefficients computed from node features.[16](https://doi.org/10.17863/cam.48429)\n- **GIN** updates as \\( h_{u}^{(t)} = \\mathrm{MLP}((1+\\epsilon) \\cdot h_{u}^{(t-1)} + \\sum_{v \\in N(u)} h_{v}^{(t-1)}) \\); with injective aggregation it is provably as powerful as the WL test.[7](https://arxiv.org/abs/1810.00826)\n- **Gated graph networks** combine summed, transformed messages with a GRU update.[17](https://www.cs.ox.ac.uk/files/13294/L4.pdf)\n- **Edge-aware models**: among the models Gilmer et al. surveyed, only one had used learned edge features via hidden edge states [2](https://proceedings.mlr.press/v70/gilmer17a.html); a later edge-update network lets messages depend on the receiving atom's state.[8](https://export.arxiv.org/pdf/1806.03146) EdgeConv computes \\( x_{i}^{(k)} = \\max_{j \\in N(i)} h_{\\Theta}(x_{i}^{(k-1)}, x_{j}^{(k-1)} - x_{i}^{(k-1)}) \\) with max aggregation.[13](https://pytorch-geometric.readthedocs.io/en/latest/tutorial/create_gnn.html) The general framework of Battaglia and colleagues jointly updates edge, node, and graph-level embeddings.[14](https://cs.mcgill.ca/~wlh/grl_book/files/GRL_Book-Chapter_5-GNNs.pdf)\n\n## Applications\n\nIn molecular property prediction, molecules are represented with atoms as nodes and chemical bonds as edges, a problem GNNs have quickly proven well suited to.[15](https://dl.acm.org/doi/full/10.1145/3816725) On QM9, the best variant of the original paper, using an edge-network message function, a set2set readout, and virtual graph elements, reached chemical accuracy on 11 of 13 targets and state of the art on all 13.[2](https://proceedings.mlr.press/v70/gilmer17a.html) MPNN variations outperformed all baselines on QM9, with improvements of nearly a factor of four on some targets, and predict 11 properties up to 300,000 times faster than DFT.[3](https://research.google/blog/predicting-properties-of-molecules-with-machine-learning/) An edge-update extension of a continuous-filter MPNN reached a 13.7 meV formation-energy error on the OQMD materials database with \\( K = 24 \\) nearest-neighbor graphs.[8](https://export.arxiv.org/pdf/1806.03146) Beyond property prediction, a GNN screen discovered the antibiotic halicin [18](https://doi.org/10.1016/j.cell.2020.01.021), and applications include link prediction in recommender systems, knowledge graph completion with relation-specific message passing, and materials science.[15](https://dl.acm.org/doi/full/10.1145/3816725) • [17](https://www.cs.ox.ac.uk/files/13294/L4.pdf)\n\n## Limitations and alternatives\n\n**Depth is limited by over-smoothing.** Node representations converge and become indistinguishable as layers stack [15](https://dl.acm.org/doi/full/10.1145/3816725); a survey defines over-smoothing axiomatically as exponential convergence of similarity measures such as Dirichlet energy with depth, observed for GCN, GAT, and [GraphSAGE](https://www.edgepedia.org/graphsage), with Dirichlet energy reaching machine-precision zero within 64 layers on real-world graphs.[19](https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-17.pdf) For most molecular tasks, 3 to 6 steps is optimal, and more steps typically hurt.\n\n**Over-squashing and under-reaching.** Over-squashing arises because the number of nodes in a node's receptive field grows exponentially while the state stays fixed-length; adding a fully connected layer helped in their setting.[17](https://www.cs.ox.ac.uk/files/13294/L4.pdf) Unlike over-smoothing, which is mostly a depth effect, over-squashing is mainly a graph topology effect, occurring when too many signals squeeze through too few edges [15](https://dl.acm.org/doi/full/10.1145/3816725); Topping, Di Giovanni, Chamberlain, Dong, and Bronstein (2021) characterized bottlenecks through negative edge curvature.[20](https://doi.org/10.48550/arxiv.2111.14522) Theory ties required depth to topology: for tasks mixing nodes at high commute time, depth must exceed that commute time, giving impossibility results for bounded-depth MPNNs.[21](https://arxiv.org/pdf/2306.03589) Under-reaching occurs when layers are fewer than the graph diameter, so distant information never arrives.[22](https://proceedings.neurips.cc/paper_files/paper/2025/file/3ee63632d4c26ddc3d350c86ffca381c-Paper-Conference.pdf)\n\n**Expressivity and cost.** MPNNs provably cannot distinguish a 6-cycle from two triangles [23](https://ar5iv.labs.arxiv.org/html/2202.11097), and the 1-WL ceiling applies regardless of size.[5](https://proceedings.neurips.cc/paper/2020/file/a32d7eeaae19821fd9ce317f3ce952a7-Paper.pdf) On cost, a single message passing step on a dense graph requires \\( O(n^{2} \\cdot d^{2}) \\) floating point multiplications; a \"towers\" variant splitting a \\( d \\)-dimensional embedding into \\( k \\) copies of \\( d/k \\) dimensions gave a factor-of-2 inference speedup for \\( k = 8 \\), \\( n = 9 \\), \\( d = 200 \\), though further work is needed for much larger graphs.[2](https://proceedings.mlr.press/v70/gilmer17a.html)\n\nHigher-order GNNs bound expressivity by the \\( k \\)-WL hierarchy; Morris and colleagues' higher-order networks (AAAI 2019) are one line.[24](https://doi.org/10.1609/aaai.v33i01.33014602) Structural message-passing networks maintain a local context matrix per node and are strictly more powerful than MPNNs, simulating any MPNN with the same depth but not vice versa, and recovering graph topology with \\( O(d_{\\max} \\cdot n^{2}) \\) memory where MPNNs would need exponentially growing feature maps.[5](https://proceedings.neurips.cc/paper/2020/file/a32d7eeaae19821fd9ce317f3ce952a7-Paper.pdf) Graph transformers reach all-to-all interaction by operating on a fully connected graph; on benchmarks such as ZINC, QM9, and TMQM, however, encoder-augmented MPNNs are consistently competitive with or better than graph transformers, which mainly help on tasks with long-range dependencies. A position paper argues that higher-order GNNs, graph transformers, rewiring, and subsampling can all be expressed as pairwise message passing over a modified graph, proposing the term \"augmented message passing\" instead of \"beyond message passing\".[23](https://ar5iv.labs.arxiv.org/html/2202.11097)\n\n## References\n\n1. [Gilmer, Justin and colleagues (2017). Neural Message Passing for Quantum Chemistry. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1704.01212)\n2. [Neural Message Passing for Quantum Chemistry (Gilmer et al., ICML 2017, PMLR v70)](https://proceedings.mlr.press/v70/gilmer17a.html)\n3. [Predicting Properties of Molecules with Machine Learning (Google Research blog)](https://research.google/blog/predicting-properties-of-molecules-with-machine-learning/)\n4. [Message Passing Neural Networks, EngineersOfAI](https://engineersofai.com/docs/ml/graph-neural-networks/message-passing-neural-networks)\n5. [Building powerful and equivariant graph neural networks with structural message-passing (SMP, NeurIPS 2020)](https://proceedings.neurips.cc/paper/2020/file/a32d7eeaae19821fd9ce317f3ce952a7-Paper.pdf)\n6. [torch_geometric.nn.conv.MessagePassing, official documentation](https://pytorch-geometric.readthedocs.io/en/latest/generated/torch_geometric.nn.conv.MessagePassing.html)\n7. [How Powerful are Graph Neural Networks? (Xu et al., GIN)](https://arxiv.org/abs/1810.00826)\n8. [Neural Message Passing with Edge Updates for Predicting Properties of Molecules and Materials](https://export.arxiv.org/pdf/1806.03146)\n9. [Graph neural networks | Nature Reviews Methods Primers](https://www.nature.com/articles/s43586-024-00294-7)\n10. [F. Scarselli and colleagues (2008). The Graph Neural Network Model. IEEE Transactions on Neural Networks.](https://doi.org/10.1109/tnn.2008.2005605)\n11. [Duvenaud, David and colleagues (2015). Convolutional Networks on Graphs for Learning Molecular Fingerprints. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1509.09292)\n12. [Kipf, Thomas N., Welling, Max (2016). Semi-Supervised Classification with Graph Convolutional Networks. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1609.02907)\n13. [Creating Message Passing Neural Networks, PyG tutorial](https://pytorch-geometric.readthedocs.io/en/latest/tutorial/create_gnn.html)\n14. [Graph Representation Learning Book, Chapter 5: The Graph Neural Network (Hamilton)](https://cs.mcgill.ca/~wlh/grl_book/files/GRL_Book-Chapter_5-GNNs.pdf)\n15. [Introduction to Graph Neural Networks for Machine Learning Engineers (ACM Computing Surveys)](https://dl.acm.org/doi/full/10.1145/3816725)\n16. [Veličković, P and colleagues (2017). Graph Attention Networks. arXiv (Cornell University).](https://doi.org/10.17863/cam.48429)\n17. [Lecture 4: Message Passing Neural Network Architectures (University of Oxford)](https://www.cs.ox.ac.uk/files/13294/L4.pdf)\n18. [Jonathan M. Stokes and colleagues (2020). A Deep Learning Approach to Antibiotic Discovery. Cell.](https://doi.org/10.1016/j.cell.2020.01.021)\n19. [A Survey on Oversmoothing in Graph Neural Networks (Rusch, Bronstein, Mishra)](https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-17.pdf)\n20. [Topping, Jake and colleagues (2021). Understanding over-squashing and bottlenecks on graphs via curvature. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2111.14522)\n21. [Over-Squashing, Expressivity, and the Misalignment between Task and Topology in Message Passing Neural Networks (Di Giovanni et al.)](https://arxiv.org/pdf/2306.03589)\n22. [What Expressivity Theory Misses: Message Passing Complexity for GNNs (NeurIPS 2025)](https://proceedings.neurips.cc/paper_files/paper/2025/file/3ee63632d4c26ddc3d350c86ffca381c-Paper-Conference.pdf)\n23. [Message passing all the way up (Veličković, 2022)](https://ar5iv.labs.arxiv.org/html/2202.11097)\n24. [Morris, Christopher and colleagues (2019). Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks. AAAI Publications (The Association for the Advancement of Artificial Intelligence (AAAI)).](https://doi.org/10.1609/aaai.v33i01.33014602)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Machine learning and neural computation › Neural networks and deep learning › Neural network architectures › Graph neural network architectures*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/message-passing-neural-network", "markdown_url": "https://www.edgepedia.org/message-passing-neural-network.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Message passing neural network\", Edgepedia (EdgeChat), https://www.edgepedia.org/message-passing-neural-network. Edgepedia Community License 1.0.", "credit_md": "\"[Message passing neural network](https://www.edgepedia.org/message-passing-neural-network)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/message-passing-neural-network](https://www.edgepedia.org/message-passing-neural-network). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Message passing neural network\", Edgepedia (EdgeChat), https://www.edgepedia.org/message-passing-neural-network. Edgepedia Community License 1.0.", "speakable": "A message passing neural network is a graph neural network framework, formalized in 2017, that exchanges messages along edges and excels at molecular property prediction." }