Graph neural network: Difference between revisions

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{{Short description|Class of artificial neural networks}}

{{Machine learning|Artificial neural network}}

{{Use dmy dates|date=July 2025}}

A '''Graph neural networknetworks''' ('''GNN''') isare a class ofspecialized [[artificial neural network]]s forthat processingare datadesigned thatfor cantasks bewhose representedinputs asare [[Graph (abstract data type)|graphs]].<ref name="scarselli2008wucuipeizhao2022" /><ref name="scarselli2009" /><ref name="micheli2009" /><ref name="sanchez2021" /><ref name="daigavane2021" />.

One prominent example is molecular drug design.<ref>{{Cite journal |last1=Stokes |first1=Jonathan M. |last2=Yang |first2=Kevin |last3=Swanson |first3=Kyle |last4=Jin |first4=Wengong |last5=Cubillos-Ruiz |first5=Andres |last6=Donghia |first6=Nina M. |last7=MacNair |first7=Craig R. |last8=French |first8=Shawn |last9=Carfrae |first9=Lindsey A. |last10=Bloom-Ackermann |first10=Zohar |last11=Tran |first11=Victoria M. |last12=Chiappino-Pepe |first12=Anush |last13=Badran |first13=Ahmed H. |last14=Andrews |first14=Ian W. |last15=Chory |first15=Emma J. |date=20 February 2020 |title=A Deep Learning Approach to Antibiotic Discovery |journal=Cell |volume=180 |issue=4 |pages=688–702.e13 |doi=10.1016/j.cell.2020.01.021 |issn=1097-4172 |pmc=8349178 |pmid=32084340}}</ref><ref>{{cite arXiv|last1=Yang |first1=Kevin |title=Analyzing Learned Molecular Representations for Property Prediction |date=20 November 2019 |eprint=1904.01561 |last2=Swanson |first2=Kyle |last3=Jin |first3=Wengong |last4=Coley |first4=Connor |last5=Eiden |first5=Philipp |last6=Gao |first6=Hua |last7=Guzman-Perez |first7=Angel |last8=Hopper |first8=Timothy |last9=Kelley |first9=Brian|class=cs.LG }}</ref><ref>{{Cite journal |last=Marchant |first=Jo |date=20 February 2020 |title=Powerful antibiotics discovered using AI |url=https://www.nature.com/articles/d41586-020-00018-3 |journal=Nature |language=en |doi=10.1038/d41586-020-00018-3|pmid=33603175 |url-access=subscription }}</ref> Each input sample is a graph representation of a molecule, where atoms form the nodes and chemical bonds between atoms form the edges. In addition to the graph representation, the input also includes known chemical properties for each of the atoms. Dataset samples may thus differ in length, reflecting the varying numbers of atoms in molecules, and the varying number of bonds between them. The task is to predict the efficacy of a given molecule for a specific medical application, like eliminating [[Escherichia coli|''E. coli'']] bacteria.

[[File:GNN building blocks.png|thumb|Basic building blocks of a Graph neural network (GNN). <math>(1)</math> Permutation equivariant layer. <math>(2)</math> Local pooling layer. <math>(3)</math> Global pooling (or readout) layer. Colors indicate [[Feature (machine learning)|features]].]]

The key design element of GNNs is the use of ''pairwise message passing'', such that graph nodes iteratively update their representations by exchanging information with their neighbors. Several GNN architectures have been proposed,<ref name="scarselli2009" /><ref name="micheli2009" /><ref name="kipf2016" /><ref name="hamilton2017" /><ref name="velickovic2018" /> which implement different flavors of message passing,<ref name="bronstein2021" /><ref name="hajij2022" /> started by recursive<ref name="scarselli2009" /> or convolutional constructive<ref name="micheli2009" /> approaches. {{As of|2022}}, it is an open question whether it is possible to define GNN architectures "going beyond" message passing, or instead every GNN can be built on message passing over suitably defined graphs.<ref name="velickovic2022" />

In the more general subject of "Geometric [[Deep Learning]]", existing neural network architectures can be interpreted as GNNs operating on suitably defined graphs<ref name=bronstein2021 />. [[Convolutional neural network]]s, in the context of [[computer vision]], can be seen as a GNN applied to graphs structured as grids of [[pixel]]s. [[Transformer (machine learning model) | Transformers]], in the context of [[natural language processing]], can be seen as GNNs applied to [[complete graph]]s whose nodes are [[words]] in a [[Sentence (linguistics) | sentence]].

[[File:GNN building blocks.png|thumb|Basic building blocks of a graph neural network (GNN). <math>(1)</math> Permutation equivariant layer. <math>(2)</math> Local pooling layer. <math>(3)</math> Global pooling (or readout) layer. Colors indicate [[Feature (machine learning)|features]].]]

The key design element of GNNs is the use of ''pairwise message passing'', such that graph nodes iteratively update their representations by exchanging information with their neighbors. Since their inception, several different GNN architectures have been proposed<ref name="scarselli2008" /><ref name="kipf2016" /><ref name="hamilton2017" /><ref name="velickovic2018" />, which implement different flavors of message passing<ref name=bronstein2021 />. {{As of|2022}}, whether is it possible to define GNN architectures "going beyond" message passing, or if every GNN can be built on message passing over suitably defined graphs, is an open research question<ref name=velickovic2022 />.

In the more general subject of "geometric [[deep learning]]", certain existing neural network architectures can be interpreted as GNNs operating on suitably defined graphs.<ref name=bronstein2021 /> A [[convolutional neural network]] layer, in the context of [[computer vision]], can be considered a GNN applied to graphs whose nodes are [[pixel]]s and only adjacent pixels are connected by edges in the graph. A [[Transformer (machine learning model)|transformer]] layer, in [[natural language processing]], can be considered a GNN applied to [[complete graph]]s whose nodes are [[words]] or tokens in a passage of [[natural language]] text.

Relevant application domains for GNNs include [[social network]]s<ref name="ying2018" />,

[[Citation graph|citation networks]]<ref name="stanforddata" />,

[[molecular biology]]<ref name="gilmer2017" />,

[[physics]]<ref name=qasim2019 /> and

[[NP-hard]] [[combinatorial optimization]] problems<ref name="li2018" />.

Relevant application domains for GNNs include [[Natural Language Processing|natural language processing]],<ref name="wuchen2023" /> [[social networks]],<ref name="ying2018" /> [[Citation graph|citation networks]],<ref name="stanforddata" /> [[molecular biology]],<ref>{{cite journal |last1=Zhang |first1=Weihang |last2=Cui |first2=Yang |last3=Liu |first3=Bowen |last4=Loza |first4=Martin |last5=Park |first5=Sung-Joon |last6=Nakai |first6=Kenta |date=5 April 2024 |title=HyGAnno: Hybrid graph neural network-based cell type annotation for single-cell ATAC sequencing data |url=https://academic.oup.com/bib/article/25/3/bbae152/7641197 |journal=Briefings in Bioinformatics |volume=25 |issue=3 |pages=bbae152 |doi=10.1093/bib/bbae152|pmid=38581422 |pmc=10998639 }}</ref> chemistry,<ref name="gilmer2017" /><ref>{{Cite journal |last1=Coley |first1=Connor W. |last2=Jin |first2=Wengong |last3=Rogers |first3=Luke |last4=Jamison |first4=Timothy F. |last5=Jaakkola |first5=Tommi S. |last6=Green |first6=William H. |last7=Barzilay |first7=Regina |last8=Jensen |first8=Klavs F. |date=2 January 2019 |title=A graph-convolutional neural network model for the prediction of chemical reactivity |journal=Chemical Science |language=en |volume=10 |issue=2 |pages=370–377 |doi=10.1039/C8SC04228D |pmid=30746086 |pmc=6335848 |issn=2041-6539|doi-access=free }}</ref> [[physics]]<ref name=qasim2019 /> and [[NP-hard]] [[combinatorial optimization]] problems.<ref name="li2018" />

Several [[open source]] [[Library (computing)|libraries]] implementing Graph neural network are available, such as PyTorch Geometric <ref name=fey2019 /> ([[PyTorch]]), TensorFlow GNN<ref name=tfgnn2022 /> ([[TensorFlow]]), and jGraph<ref name=jraph2022/> ([[Google JAX]]).

[[Open source]] [[Library (computing)|libraries]] implementing GNNs include PyTorch Geometric<ref name=fey2019 /> ([[PyTorch]]), TensorFlow GNN<ref name=tfgnn2022 /> ([[TensorFlow]]), Deep Graph Library<ref>{{Cite web |last= |title=Deep Graph Library (DGL) |url=https://www.dgl.ai/ |access-date=12 September 2024 |website=}}</ref> (framework agnostic), jraph<ref name=jraph2022/> ([[Google JAX]]), and GraphNeuralNetworks.jl<ref name=Lucibello2021GNN/>/GeometricFlux.jl<ref>{{Citation |title=FluxML/GeometricFlux.jl |date=31 January 2024 |url=https://github.com/FluxML/GeometricFlux.jl |access-date=3 February 2024 |publisher=FluxML}}</ref> ([[Julia (programming language)|Julia]], [[Flux (machine-learning framework)|Flux]]).

== Architecture ==

The architecture of a generic GNN implements the following fundamental [[Layer (deep learning)|layers]]:<ref name=bronstein2021 />:

*# <em>Permutation equivariant</em>: a permutation equivariant layer [[Map (mathematics)|maps]] a representation of a graph into an updated representation of the same graph. In the literature, permutation equivariant layers are implemented via pairwise message passing between graph nodes.<ref name="bronstein2021" /><ref name="velickovic2022" />. Intuitively, in a message passing layer, nodes <em>update</em> their representations by <em>aggregating</em> the <em>messages</em> received from their immediate neighbours. As such, each message passing layer increases the receptive field of the GNN by one hop.

# <em>Local pooling</em>: a local [[pooling layer]] coarsens the graph via [[Downsampling (signal processing)|downsampling]]. Local pooling is used to increase the receptive field of a GNN, in a similar fashion to pooling layers in [[convolutional neural network]]s. Examples include [[Nearest neighbor graph|k-nearest neighbours pooling]], top-k pooling,<ref name="gao2019" /> and self-attention pooling.<ref name="lee2019" />

# <em>Global pooling</em>: a global pooling layer, also known as ''readout'' layer, provides fixed-size representation of the whole graph. The global pooling layer must be permutation invariant, such that permutations in the ordering of graph nodes and edges do not alter the final output.<ref name="lui2022" /> Examples include element-wise sum, mean or maximum.

* <em>Local pooling</em>: a local pooling layer coarsens the graph via downsampling. Local pooling is used to increase the receptive field of a GNN, in a similar fashion to pooling layers in [[Convolutional neural network]]s. Examples include [[Nearest neighbor graph|k-nearest neighbours pooling]], top-k pooling<ref name=gao2019 />, and self-attention pooling<ref name=lee2019 />.

* <em>Global pooling</em>: a global pooling layer, also known as ''readout'' layer, provides fixed-size representation of the whole graph. The global pooling layer must be permutation invariant, such that permutations in the ordering of graph nodes and edges do not alter the final output<ref name=lui2022 />. Examples include element-wise sum, mean or maximum.

It has been demonstrated that GNNs cannot be more expressive than the Weisfeiler–Lehman[[Weisfeiler Leman graph isomorphism test|Weisfeiler–Leman Graph Isomorphism Test]].<ref name="douglas2011" /><ref name="xu2019" />. In practice, this means that there exist different graph structures (e.g., [[molecules]] with the same [[atoms]] but different [[Chemical bond|bonds]]) that cannot be distinguished by GNNs. More powerful GNNs operating on higher-dimension geometries such as [[Simplicial complex|simplicial complexescomplex]]es can be designed.<ref name=bronstein2021-2 />.<ref name=grady2011discrete /><ref name=hajij2022 /> {{As of|2022}}, whether or not future architectures will overcome the message passing primitive is an open research question.<ref name=velickovic2022 />.

[[File:GNN representational limits.png|thumb|[[Graph isomorphism|Non-isomorphic]] graphs that cannot be distinguished by a GNN due to the limitations of the Weisfeiler-Lehman Graph Isomorphism Test. Colors indicate node [[Feature (machine learning)|features]].]]

== Message passing layers ==

[[File:Message Passing Neural Network.png|thumb|Node representation update in a Message Passing Neural Network (MPNN) layer. Node <math>\mathbf{x}_0</math> receives messages sent by all of hisits immediate neighbours <math>\mathbf{x}_1</math> to <math>\mathbf{x}_4</math>. Messages are computing via the message function <math>\phipsi</math>, which accounts for the features of both senders and receiver.]]

Message passing layers are permutation-equivariant layers mapping a graph into an updated representation of the same graph. Formally, they can be expressed as message passing neural networks (MPNNs).<ref name="bronstein2021" />.

Let <math>G = (V,E)</math> be a [[Graph (discrete mathematics)|graph]], where <math>V</math> is the node set and <math>E</math> is the edge set. Let <math>N_u</math> be the [[Neighbourhood (graph theory)|neighbourhood]] of some node <math>u \in V</math>. Additionally, let <math>\mathbf{x}_u</math> be the [[Feature (machine learning)|features]] of node <math>u \in V</math>, and <math>\mathbf{e}_{u, vuv}</math> be the features of edge <math>(u, v) \in E</math>. An MPNN [[Layer_Layer (deep learning)|layer]] can be expressed as follows:<ref name="bronstein2021" />:

:<math>\mathbf{h}_u = \phi \left( \mathbf{x_ux}_u, \bigoplus_{v \in N_u} \psi(\mathbf{x}_u, \mathbf{x}_v, \mathbf{e}_{uv}) \right)</math>

Wherewhere <math>\phi</math> and <math>\psi</math> are [[differentiable functions]] (e.g., [[artificial neural network]]s), and <math>\bigoplus</math> is a [[permutation]] [[Invariant (mathematics)|invariant]] [[aggregation operator]] that can accept an arbitrary number of inputs (e.g., element-wise sum, mean, or max). In particular, <math>\phi</math> and <math>\psi</math> are referred to as ''update'' and ''message'' functions, respectively. Intuitively, in an MPNN computational block, graph nodes ''update'' their representations by ''aggregating'' the ''messages'' received from their neighbours.

The outputs of one or more MPNN layers are node representations <math>\mathbf{h}_u</math> for each node <math>u \in V</math> in the graph. Node representations can be employed for any downstream task, such as node/graph [[Statistical classification|classification]] or edge prediction.

Graph nodes in an MPNN update their representation aggregating information from their immediate neighbours. As such, stacking <math>n</math> MPNN layers means that one node will be able to communicate with nodes that are at most <math>n</math> "hops" away. In principle, to ensure that every node receives information from every other node, one would need to stack a number of MPNN layers equal to the graph [[DistanceDiameter (graph theory) | diameter]]. However, stacking many MPNN layers may cause issues such as oversmoothing<ref name=chen2021 /> and oversquashing.<ref name=alon2021 />. Oversmoothing refers to the issue of node representations becoming indistinguishable. Oversquashing refers to the bottleneck that is created by squeezing long-range dependencies into fixed-size representations. Countermeasures such as skip connections<ref name=hamilton2017 /><ref name=xu2021 /> (as in [[residual neural network]]s), gated update rules<ref name=li2016 /> and jumping knowledge<ref name=xu2018 /> can mitigate oversmoothing. Modifying the final layer to be a fully-adjacent layer, i.e., by considering the graph as a [[complete graph]], can mitigate oversquashing in problems where long-range dependencies are required.<ref name=alon2021 />

Other "flavours" of MPNN have been developed in the literature,<ref name=bronstein2021 />, such as graph convolutional networks<ref name=kipf2016 /> and graph attention networks,<ref name=velickovic2018 />, whose definitions can be expressed in terms of the MPNN formalism.

=== Graph convolutional network ===

The graph convolutional network (GCN) was first introduced by [[Thomas Kipf]] and [[Max Welling]] in 2017.<ref name=kipf2016 />.

A GCN layer defines a [[Order of approximation|first-order approximation]] of a localized spectral [[Filter (signal processing)|filter]] on graphs. GCNs can be understood as a generalization of [[Convolutionalconvolutional Neuralneural Networknetwork]]s to graph-structured data.

The formal expression of a GCN layer reads as follows:

Line 56 ⟶ 54:

:<math>\mathbf{H} = \sigma\left(\tilde{\mathbf{D}}^{-\frac{1}{2}} \tilde{\mathbf{A}} \tilde{\mathbf{D}}^{-\frac{1}{2}} \mathbf{X} \mathbf{\Theta}\right)</math>

Wherewhere <math>\mathbf{H}</math> is the matrix of node representations <math>\mathbf{h}_u</math>, <math>\mathbf{X}</math> is the matrix of node features <math>\mathbf{x}_u</math>, <math>\sigma(\cdot)</math> is an [[activation function]] (e.g., [[ReLU]]), <math>\tilde{\mathbf{A}}</math> is the graph [[adjacency matrix]] with the addition of self-loops, <math>\tilde{\mathbf{D}}</math> is the normalizedgraph [[degree matrix]] with the addition of self-loops, and <math>\mathbf{\Theta}</math> is a matrix of trainable parameters.

In particular, let <math>\mathbf{A}</math> be the graph adjacency matrix: then, one can define <math>\tilde{\mathbf{A}} = \mathbf{A} + \mathbf{I}</math> and <math>\tilde{\mathbf{D}}_{ii}} = \sum_{j \in V} \tilde{A}_{ij}</math>, where <math>\mathbf{I}</math> denotes the [[identity matrix]]. This normalization ensures that the [[eigenvalue]]s of <math>\tilde{\mathbf{D}}^{-\frac{1}{2}} \tilde{\mathbf{A}} \tilde{\mathbf{D}}^{-\frac{1}{2}}</math> are bounded in the range <math>[0, 1]</math>, avoiding [[Numerical stability|numerical instabilities]] and [[Vanishing gradient problem|exploding/vanishing gradients]].

A limitation of GCNs is that they do not allow multidimensional edge features <math>\mathbf{e}_{uv}</math>.<ref name=kipf2016 />. It is however possible to associate scalar weights <math>w_{uv}</math> to each edge by imposing <math>A_{uv} = w_{uv}</math>, i.e., by setting each nonzero entry in the adjacency matrix equal to the weight of the corresponding edge.

=== Graph attention network ===

The graph attention network (GAT) was introduced by [[Petar Veličković]] et al. in 2018.<ref name=velickovic2018 />.

Graph attention network is a combination of a GNN and an attention layer.

The implementation of attention layer in graphical neural networks helps provide attention or focus to the important information from the data instead of focusing on the whole data.

A multi-head GAT layer can be expressed as follows:

:<math> \mathbf{h}_u = \overset{K}{\underset{k=1}{\Big\Vert}} \sigma \left(\sum_{v \in N_u} \alpha_{ijuv} \mathbf{W}^k \mathbf{x}_v\right) </math>

Wherewhere <math>K</math> is the number of [[Attention (machine learning)|attention]] heads, <math>\Big\Vert</math> denotes [[Concatenation| vector concatenation]], <math>\sigma(\cdot)</math> is an [[activation function]] (e.g., [[ReLU]]), <math>\alpha_{ij}</math> are attention coefficients, and <math>W^k</math> is a matrix of trainable parameters for the <math>k</math>-th attention head.

For the final GAT layer, the outputs from each attention head are averaged before the application of the activation function. Formally, the final GAT layer can be written as:

Line 79 ⟶ 80:

Normalized attention coefficients are computed as follows:

:<math>\alpha_{uv} = \frac{\exp(\text{LeakyReLU}\left(\mathbf{a}^T [\mathbf{W} \mathbf{hx}_u \Vert \mathbf{W} \mathbf{hx}_v \Vert \mathbf{e}_{uv}]\right))}{\sum_{z \in N_u}\exp(\text{LeakyReLU}\left(\mathbf{a}^T [\mathbf{W} \mathbf{hx}_u \Vert \mathbf{W} \mathbf{hx}_z \Vert \mathbf{e}_{uz}]\right))}</math>

Wherewhere <math>\mathbf{a}</math> is a vector of learnable weights, <math>\cdot^T</math> indicates [[Transpose | transposition]], <math>\mathbf{e}_{uv}</math> are the edge features (if present), and <math>\text{LeakyReLU}</math> is a [[Rectifier (neural networks)|modified ReLU]] activation function. Attention coefficients are normalized to make them easily comparable across different nodes.<ref name=velickovic2018 />

A GCN can be seen as a special case of a GAT where attention coefficients are not learnable, but fixed and equal to the edge weights <math>w_{uv}</math>.

=== Gated graph sequence neural network ===

The gated graph sequence neural network (GGS-NN) was introduced by [[Yujia Li]] et al. in 2015.<ref name=li2016 /> The GGS-NN extends the GNN formulation by Scarselli et al.<ref name=scarselli2009 /> to output sequences. The message passing framework is implemented as an update rule to a [[gated recurrent unit]] (GRU) cell.

A GGS-NN can be expressed as follows:

:<math>\mathbf{h}_u^{(0)} = \mathbf{x}_u \, \Vert \, \mathbf{0}</math>

:<math>\mathbf{m}_u^{(l+1)} = \sum_{v \in N_u} \mathbf{\Theta} \mathbf{h}_v</math>

:<math>\mathbf{h}_u^{(l+1)} = \text{GRU}(\mathbf{m}_u^{(l+1)}, \mathbf{h}_u^{(l)})</math>

where <math>\Vert</math> denotes [[Concatenation|vector concatenation]], <math>\mathbf{0}</math> is a vector of zeros, <math>\mathbf{\Theta}</math> is a matrix of learnable parameters, <math>\text{GRU}</math> is a GRU cell, and <math>l</math> denotes the sequence index. In a GGS-NN, the node representations are regarded as the hidden states of a GRU cell. The initial node features <math>\mathbf{x}_u^{(0)}</math> are [[Data structure alignment|zero-padded]] up to the hidden state dimension of the GRU cell. The same GRU cell is used for updating representations for each node.

== Local pooling layers ==

Local pooling layers coarsen the graph via downsampling. In literatureSubsequently, several learnable local pooling strategies that have been proposed are presented.<ref name=lui2022 /> For each case, the input is the initial graph represented by a matrix <math>\mathbf{X}</math> of node features, and the graph adjacency matrix <math>\mathbf{A}</math>. The output is the new matrix <math>\mathbf{X}'</math>of node features, and the new graph adjacency matrix <math>\mathbf{A}'</math>.

=== Top-k pooling ===

We first set

The top-k pooling layer <ref name=gao2019 /> can be formalised as follows:

:<math>\mathbf{y} = \frac{\mathbf{X}\mathbf{p}}{\Vert\mathbf{p}\Vert}</math>

:<math>\mathbf{iy} = \textfrac{top\mathbf{X}_k(\mathbf{yp}}{\Vert\mathbf{p}\Vert})</math>

where <math>\mathbf{p}</math> is a learnable [[Projection (mathematics)|projection]] vector. The projection vector <math>\mathbf{p}</math> computes a scalar projection value for each graph node.

The top-k pooling layer <ref name="gao2019" /> can then be formalised as follows:

:<math>\mathbf{X}' = (\mathbf{X} \odot \text{sigmoid}(\mathbf{y}))_{\mathbf{i}}</math>

:<math>\mathbf{A}' = \mathbf{A}_{\mathbf{i}, \mathbf{Ii}}</math>

Wherewhere <math>\mathbf{Xi} = \text{top}_k(\mathbf{y})</math> is the matrixsubset of nodenodes features,with <math>\mathbf{A}</math>the istop-k thehighest graphprojection adjacency matrixscores, <math>\odot</math> denotes element-wise [[matrix multiplication]], and <math>\mathbftext{psigmoid}(\cdot)</math> is a learnablethe [[Projectionsigmoid (mathematics)|projectionfunction]] vector. TheIn projectionother vectorwords, <math>\mathbf{p}</math> computes a scalar projection value for each graph node. Thethe nodes with the top-k highest projection scores are retained in the new adjacency matrix <math>\mathbf{A}'</math>. The <math>\text{sigmoid}(\cdot)</math> operation makes the projection vector <math>\mathbf{p}</math> trainable by [[backpropagation]], which otherwise would produce discrete outputs.<ref name="gao2019" />.

=== Self-attention pooling ===

We first set

The Self-attention pooling layer <ref name=lee2019 /> can be formalised as follows:

:<math>\mathbf{y} = \text{GNN}(\mathbf{X}, \mathbf{A})</math>

where <math>\text{GNN}</math> is a generic permutation equivariant GNN layer (e.g., GCN, GAT, MPNN).

The Self-attention pooling layer<ref name="lee2019" /> can then be formalised as follows:

:<math>\mathbf{i} = \text{top}_k(\mathbf{y})</math>

:<math>\mathbf{X}' = (\mathbf{X} \odot \mathbf{y})_{\mathbf{i}}</math>

Line 111 ⟶ 126:

:<math>\mathbf{A}' = \mathbf{A}_{\mathbf{i}, \mathbf{i}}</math>

Wherewhere <math>\mathbf{Xi} = \text{top}_k(\mathbf{y})</math> is the matrixsubset of nodenodes features,with <math>\mathbf{A}</math>the istop-k thehighest graphprojection adjacency matrixscores, <math>\odot</math> denotes [[Hadamard product (matrices)|element-wise [[matrix multiplication]], and <math>\text{GNN}</math> is a generic permutation equivariant GNN layer (e.g., GCN, GAT, MPNN).

The self-attention pooling layer can be seen as an extension of the top-k pooling layer. Differently from top-k pooling, the self-attention scores computed in self-attention pooling account both for the graph features and the graph topology.

== Heterophilic Graph Learning ==

[[Homophily]] principle, i.e., nodes with the same labels or similar attributes are more likely to be connected, has been commonly believed to be the main reason for the superiority of Graph Neural Networks (GNNs) over traditional Neural Networks (NNs) on graph-structured data, especially on node-level tasks.<ref name=":0">{{cite arXiv |eprint=2407.09618 |last1=Luan |first1=Sitao |last2=Hua |first2=Chenqing |last3=Lu |first3=Qincheng |last4=Ma |first4=Liheng |last5=Wu |first5=Lirong |last6=Wang |first6=Xinyu |last7=Xu |first7=Minkai |last8=Chang |first8=Xiao-Wen |last9=Precup |first9=Doina |last10=Ying |first10=Rex |last11=Li |first11=Stan Z. |last12=Tang |first12=Jian |last13=Wolf |first13=Guy |last14=Jegelka |first14=Stefanie |title=The Heterophilic Graph Learning Handbook: Benchmarks, Models, Theoretical Analysis, Applications and Challenges |date=2024 |class=cs.LG }}</ref> However, recent work has identified a non-trivial set of datasets where GNN’s performance compared to the NN’s is not satisfactory.<ref>{{Cite book |last1=Luan |first1=Sitao |last2=Hua |first2=Chenqing |last3=Lu |first3=Qincheng |last4=Zhu |first4=Jiaqi |last5=Chang |first5=Xiao-Wen |last6=Precup |first6=Doina |chapter=When do We Need Graph Neural Networks for Node Classification? |date=2024 |editor-last=Cherifi |editor-first=Hocine |editor2-last=Rocha |editor2-first=Luis M. |editor3-last=Cherifi |editor3-first=Chantal |editor4-last=Donduran |editor4-first=Murat |title=Complex Networks & Their Applications XII |chapter-url=https://link.springer.com/chapter/10.1007/978-3-031-53468-3_4 |series=Studies in Computational Intelligence |volume=1141 |language=en |___location=Cham |publisher=Springer Nature Switzerland |pages=37–48|doi=10.1007/978-3-031-53468-3_4 |isbn=978-3-031-53467-6 }}</ref> [[Heterophily]], i.e., low homophily, has been considered the main cause of this empirical observation.<ref name=":1">{{Cite journal |last1=Luan |first1=Sitao |last2=Hua |first2=Chenqing |last3=Lu |first3=Qincheng |last4=Zhu |first4=Jiaqi |last5=Zhao |first5=Mingde |last6=Zhang |first6=Shuyuan |last7=Chang |first7=Xiao-Wen |last8=Precup |first8=Doina |date=6 December 2022 |title=Revisiting Heterophily For Graph Neural Networks |url=https://proceedings.neurips.cc/paper_files/paper/2022/hash/092359ce5cf60a80e882378944bf1be4-Abstract-Conference.html |journal=Advances in Neural Information Processing Systems |language=en |volume=35 |pages=1362–1375|arxiv=2210.07606 }}</ref> People have begun to revisit and re-evaluate most existing graph models in the heterophily scenario across various kinds of graphs, e.g., [[Heterogeneous network|heterogeneous graphs]], [[Temporal network|temporal graphs]] and [[hypergraph]]s. Moreover, numerous graph-related applications are found to be closely related to the heterophily problem, e.g. [[Fraud detection|graph fraud/anomaly detection]], [[Adversarial attack|graph adversarial attacks and robustness]], privacy, [[federated learning]] and [[Point cloud|point cloud segmentation]], [[cluster analysis|graph clustering]], [[recommender system]]s, [[generative model]]s, [[link prediction]], [[Graph isomorphism|graph classification]] and [[Graph coloring|coloring]], etc. In the past few years, considerable effort has been devoted to studying and addressing the heterophily issue in graph learning.<ref name=":0" /><ref name=":1" /><ref>{{Cite journal |last1=Luan |first1=Sitao |last2=Hua |first2=Chenqing |last3=Xu |first3=Minkai |last4=Lu |first4=Qincheng |last5=Zhu |first5=Jiaqi |last6=Chang |first6=Xiao-Wen |last7=Fu |first7=Jie |last8=Leskovec |first8=Jure |last9=Precup |first9=Doina |date=15 December 2023 |title=When Do Graph Neural Networks Help with Node Classification? Investigating the Homophily Principle on Node Distinguishability |url=https://proceedings.neurips.cc/paper_files/paper/2023/hash/5ba11de4c74548071899cf41dec078bf-Abstract-Conference.html |journal=Advances in Neural Information Processing Systems |language=en |volume=36 |pages=28748–28760}}</ref>

== Applications ==

=== Protein folding ===

{{See Alsoalso|AlphaFold}}

Graph neural networks are one of the main building blocks of [[AlphaFold]], an artificial intelligence program developed by [[Google]]'s [[DeepMind]] for solving the [[protein folding]] problem in [[biology]]. AlphaFold achieved first place in several [[CASP]] competitions.<ref name=guardian2018 /><ref name=mit2020 />.<ref name=xu2018 />

=== Social Networksnetworks ===

{{See Alsoalso|Recommender system}}

[[Social networks]] are a major application ___domain for GNNs due to their natural representation as [[social graph]]s. GNNs are used to develop recommender systems based on both [[social relations]] and item relations.<ref name=fan2019 /><ref name=ying2018/>.

=== Combinatorial optimization ===

{{See Alsoalso|Combinatorial Optimizationoptimization}}

GNNs are used as fundamental building blocks for several combinatorial optimization algorithm algorithms.<ref name=cappart2021 />. Examples include computing superior[[Shortest path problem|shortest paths]] or competitive[[Eulerian path|Eulerian circuits]] for a given graph,<ref name=li2016/> deriving [[Placement (electronic design automation)|chip placements]] superior or competitive withto handcrafted human solutions,<ref name=mirhoseini2021 />, and improving expert-designed bracingbranching rules in [[Branchbranch and Boundbound]].<ref name=gasse2019 />.

=== Cyber security ===

{{See also|Intrusion detection system}}

When viewed as a graph, a network of computers can be analyzed with GNNs for anomaly detection. Anomalies within provenance graphs often correlate to malicious activity within the network. GNNs have been used to identify these anomalies on individual nodes<ref>{{Cite journal |last1=Wang |first1=Su |last2=Wang |first2=Zhiliang |last3=Zhou |first3=Tao |last4=Sun |first4=Hongbin |last5=Yin |first5=Xia |last6=Han |first6=Dongqi |last7=Zhang |first7=Han |last8=Shi |first8=Xingang |last9=Yang |first9=Jiahai |date=2022 |title=Threatrace: Detecting and Tracing Host-Based Threats in Node Level Through Provenance Graph Learning |url=https://ieeexplore.ieee.org/document/9899459/;jsessionid=NzAXdLahhjEX-xmrFzOROk4qxoaz40aJFvKcZRgjck8-zCOucJi7!380715771 |journal=IEEE Transactions on Information Forensics and Security |volume=17 |pages=3972–3987 |doi=10.1109/TIFS.2022.3208815 |issn=1556-6021|arxiv=2111.04333 |bibcode=2022ITIF...17.3972W |s2cid=243847506 }}</ref> and within paths<ref>{{Cite journal |last1=Wang |first1=Qi |last2=Hassan |first2=Wajih Ul |last3=Li |first3=Ding |last4=Jee |first4=Kangkook |last5=Yu |first5=Xiao |date=2020 |title=You Are What You Do: Hunting Stealthy Malware via Data Provenance Analysis. |journal=Network and Distributed Systems Security Symposium|doi=10.14722/ndss.2020.24167 |isbn=978-1-891562-61-7 |s2cid=211267791 |doi-access=free }}</ref> to detect malicious processes, or on the edge level<ref>{{Cite journal |last1=King |first1=Isaiah J. |last2=Huang |first2=H. Howie |date=2022 |title=Euler: Detecting Network Lateral Movement via Scalable Temporal Link Prediction |url=https://www.ndss-symposium.org/wp-content/uploads/2022-107A-paper.pdf |journal=In Proceedings of the 29th Network and Distributed Systems Security Symposium|doi=10.14722/ndss.2022.24107 |s2cid=248221601 }}</ref> to detect [[Network Lateral Movement|lateral movement]].

=== Water distribution networks ===

{{See also|Water distribution system}}

Water distribution systems can be modelled as graphs, being then a straightforward application of GNN. This kind of algorithm has been applied to water demand forecasting,<ref>{{cite journal |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2022WR032299|title=Graph Convolutional Recurrent Neural Networks for Water Demand Forecasting|last=Zanfei|first=Ariele |display-authors=etal |date=2022|journal=Water Resources Research|volume=58 |issue=7 |article-number=e2022WR032299 |publisher=AGU|doi=10.1029/2022WR032299 |bibcode=2022WRR....5832299Z |access-date=11 June 2024}}</ref> interconnecting District Measuring Areas to improve the forecasting capacity. Other application of this algorithm on water distribution modelling is the development of metamodels.<ref>{{cite journal |url=https://www.sciencedirect.com/science/article/abs/pii/S0043135423007005|title=Shall we always use hydraulic models? A graph neural network metamodel for water system calibration and uncertainty assessment|last=Zanfei|first=Ariele |journal=Water Research |display-authors=etal |date=2023|volume=242 |article-number=120264 |doi=10.1016/j.watres.2023.120264 |pmid=37393807 |bibcode=2023WatRe.24220264Z |access-date=11 June 2024|url-access=subscription }}</ref>

=== Computer Vision ===

{{See also|Computer vision}}

To represent an image as a graph structure, the image is first divided into multiple patches, each of which is treated as a node in the graph. Edges are then formed by connecting each node to its nearest neighbors based on spatial or feature similarity. This graph-based representation enables the application of graph learning models to visual tasks. The relational structure helps to enhance feature extraction and improve performance on image understanding.<ref>{{cite arXiv |eprint=2206.00272 |last1=Han |first1=Kai |last2=Wang |first2=Yunhe |last3=Guo |first3=Jianyuan |last4=Tang |first4=Yehui |last5=Wu |first5=Enhua |title=Vision GNN: An Image is Worth Graph of Nodes |date=2022 |class=cs.CV }}</ref>

=== Text and NLP ===

{{See also|Natural language processing}}

Graph-based representation of text helps to capture deeper semantic relationships between words. Many studies have used graph networks to enhance performance in various text processing tasks such as text classification, question answering, Neural Machine Translation (NMT), event extraction, fact verification, etc.<ref>{{Cite journal |last1=Zhou |first1=Jie |last2=Cui |first2=Ganqu |last3=Hu |first3=Shengding |last4=Zhang |first4=Zhengyan |last5=Yang |first5=Cheng |last6=Liu |first6=Zhiyuan |last7=Wang |first7=Lifeng |last8=Li |first8=Changcheng |last9=Sun |first9=Maosong |date=1 January 2020 |title=Graph neural networks: A review of methods and applications |journal=AI Open |volume=1 |pages=57–81 |doi=10.1016/j.aiopen.2021.01.001 |issn=2666-6510|doi-access=free }}</ref>

==References==

{{Reflist|refs=

<ref name="scarselli2008wuchen2023">{{Cite journal|last1=ScarselliWu|first1=FrancoLingfei|last2=GoriChen|first2=MarcoYu|last3=TsoiShen |first3=Ah ChungKai|last4=HagenbuchnerGuo|first4=MarkusXiaojie|last5=MonfardiniGao|first5=GabrieleHanning|datelast6=2009Li|titlefirst6=The Graph Neural Network ModelShucheng|urllast7=https://ieeexplore.ieee.org/document/4700287Pei|journalfirst7=IEEE Transactions on Neural NetworksJian|volumelast8=20Long|issuefirst8=1Bo|pagesdate=61–802023|doititle=10.1109/TNN.2008.2005605|pmid=19068426|s2cid=206756462|issn=1941-0093}}</ref>Graph Neural Networks for Natural Language Processing: A Survey

|url=https://www.nowpublishers.com/article/Details/MAL-096|journal=Foundations and Trends in Machine Learning|volume=16|issue=2|pages=119–328|doi=10.1561/2200000096 |pmid=19068426|s2cid=206756462|issn=1941-0093|arxiv=2106.06090}}</ref>

<ref name="sanchez2021">{{Cite journal|last1=Sanchez-Lengeling|first1=Benjamin|last2=Reif|first2=Emily|last3=Pearce|first3=Adam|last4=Wiltschko|first4=Alex|date=2021-09-02|title=A Gentle Introduction to Graph Neural Networks|url=https://distill.pub/2021/gnn-intro|journal=Distill|language=en|volume=6|issue=9|pages=e33|doi=10.23915/distill.00033|issn=2476-0757|doi-access=free}}</ref>

<ref name="daigavane2021wucuipeizhao2022">{{Cite journal|lastlast1=DaigavaneWu|firstfirst1=AmeyaLingfei|last2=RavindranCui|first2=BalaramanPeng|last3=AggarwalPei |first3=GauravJian|last4=Zhao|first4=Liang|date=2021-09-022022|title=UnderstandingGraph ConvolutionsNeural onNetworks: Foundations, Frontiers, and GraphsApplications|url=https://distillgraph-neural-networks.github.pubio/2021/understanding-gnns|journal=Distill|language=en|volume=6|issue=9Springer Singapore|pages=e32725|doiurl-access=10.23915/distill.00032|issn=2476<!-0757-WP:URLACCESS-->}}</ref>

<ref name="gilmer2017scarselli2009">{{Cite journal|last1=GilmerScarselli|first1=JustinFranco|last2=SchoenholzGori|first2=Samuel S.Marco|last3=RileyTsoi |first3=PatrickAh F.Chung|last4=VinyalsHagenbuchner|first4=OriolMarkus|last5=DahlMonfardini|first5=George E.Gabriele|date=2017-07-172009|title=NeuralThe MessageGraph PassingNeural forNetwork Quantum Chemistry|url=http://proceedings.mlr.press/v70/gilmer17a.htmlModel|journal=InternationalIEEE ConferenceTransactions on MachineNeural LearningNetworks|languagevolume=en20|publisherissue=PMLR1|pages=1263–127261–80|arxivdoi=170410.012121109/TNN.2008.2005605 |pmid=19068426|bibcode=2009ITNN...20...61S |s2cid=206756462|issn=1941-0093}}</ref>

<ref name="kipf2016micheli2009">{{Cite journal|last1=KipfMicheli|first1=Thomas N|last2=Welling|first2=Max|date=2016Alessio|title=Semi-supervisedNeural classificationNetwork withfor graphGraphs: convolutionalA networks|url=https://ieeexplore.ieee.org/document/4700287Contextual Constructive Approach|journal=InternationalIEEE ConferenceTransactions on LearningNeural Networks|year=2009 Representations|volume=520|issue=13|pages=61–80498–511|doi=10.1109/TNN.2008.20056052010350 |pmid=1906842619193509|arxivbibcode=16092009ITNN.02907..20..498M |s2cid=20675646217486263|issn=1045-9227}}</ref>

<ref name="hamilton2017sanchez2021">{{Cite journal|last1=HamiltonSanchez-Lengeling|first1=WilliamBenjamin|last2=YingReif|first2=RexEmily |last3=LeskovecPearce|first3=JureAdam|last4=Wiltschko|first4=Alex|date=20172 September 2021|title=InductiveA RepresentationGentle LearningIntroduction onto LargeGraph GraphsNeural Networks|url=https://csdistill.stanford.edupub/people2021/jure/pubs/graphsagegnn-nips17.pdfintro|journal=Neural Information Processing SystemsDistill|volume=316|arxivissue=17069|pages=e33 |doi=10.0221623915/distill.00033|viaissn=2476-0757|doi-access=Stanfordfree}}</ref>

<ref name="velickovic2018daigavane2021">{{Cite journal|last1=VeličkovićDaigavane|first1=PetarAmeya|last2=CucurullRavindran|first2=GuillemBalaraman |last3=CasanovaAggarwal|first3=ArantxaGaurav|last4date=Romero2 September 2021|first4title=AdrianaUnderstanding Convolutions on Graphs |last5url=Liòhttps://distill.pub/2021/understanding-gnns|first5journal=PietroDistill|last6volume=Bengio6|first6issue=Yoshua9|datepages=2018-02-04|title=Graphe32 Attention Networks|arxivdoi=171010.1090323915/distill.00032|journals2cid=International Conference on Learning Representations239678898|volumeissn=62476-0757|doi-access=free}}</ref>

<ref name="gilmer2017">{{Cite journal|last1=Gilmer|first1=Justin|last2=Schoenholz|first2=Samuel S. |last3=Riley|first3=Patrick F.|last4=Vinyals|first4=Oriol|last5=Dahl|first5=George E.|date=17 July 2017|title=Neural Message Passing for Quantum Chemistry|url=http://proceedings.mlr.press/v70/gilmer17a.html |journal=Proceedings of Machine Learning Research|language=en|pages=1263–1272|arxiv=1704.01212}}</ref>

<ref name=stanforddata>{{Cite web|title=Stanford Large Network Dataset Collection|url=https://snap.stanford.edu/data/|access-date=2021-07-05|website=snap.stanford.edu}}</ref>

<ref name="li2018kipf2016">{{citeCite journal |last1=Li Kipf|first1=ZhuwenThomas N|last2=Chen Welling|first2=Qifeng Max|last3date=Koltun |first3=Vladlen2016 |title=CombinatorialSemi-supervised optimizationclassification with graph convolutional networks and guided tree search |journal=NeuralIEEE InformationTransactions Processingon SystemsNeural |date=2018Networks |volume=31 5|pageissue=537--5461|pages=61–80 |urldoi=https:/10.1109/TNN.2008.2005605|pmid=19068426|arxiv=1609.org/abs/181002907|bibcode=2009ITNN.10659..20...61S |s2cid=206756462}}</ref>

<ref name="bronstein2021hamilton2017">{{citeCite journal |last1=Bronstein Hamilton|first1=Michael M. William|last2=Bruna Ying|first2=JoanRex |last3=Cohen Leskovec|first3=Taco Jure|last4date=Veličković |first4=Petar 2017|title=GeometricInductive DeepRepresentation Learning: Grids,on Groups,Large Graphs Geodesics and Gauges |journalurl=arXiv https://cs.stanford.edu/people/jure/pubs/graphsage-nips17.pdf|datejournal=MayNeural 4,Information 2021Processing Systems|urlvolume=https://31|arxiv=1706.org/abs/2104.1347802216|via=Stanford}}</ref>.

<ref name="velickovic2018">{{Cite arXiv|last1=Veličković|first1=Petar|last2=Cucurull|first2=Guillem |last3=Casanova|first3=Arantxa|last4=Romero|first4=Adriana|last5=Liò|first5=Pietro|last6=Bengio |first6=Yoshua|date=4 February 2018 |title=Graph Attention Networks|eprint=1710.10903 |class=stat.ML}}</ref>

<ref name=douglas2011>{{cite arXiv|last=Douglas|first=B. L.|date=2011-01-27|title=The Weisfeiler–Lehman Method and Graph Isomorphism Testing|class=math.CO|eprint=1101.5211}}</ref>.

<ref name=stanforddata>{{Cite web|title=Stanford Large Network Dataset Collection |url=https://snap.stanford.edu/data/|access-date=5 July 2021|website=snap.stanford.edu}}</ref>

<ref name=xu2019>{{Cite journal|last1=Xu|first1=Keyulu|last2=Hu|first2=Weihua|last3=Leskovec|first3=Jure|last4=Jegelka|first4=Stefanie|date=2019-02-22|title=How Powerful are Graph Neural Networks?|arxiv=1810.00826|journal=International Conference on Learning Representations|volume=7}}</ref>

<ref name="li2018">{{cite book |last1=Li |first1=Zhuwen |last2=Chen |first2=Qifeng |last3=Koltun |first3=Vladlen |title=Neural Information Processing |chapter=Text Simplification with Self-Attention-Based Pointer-Generator Networks |series=Lecture Notes in Computer Science |date=2018 |volume=31 |pages=537–546 |doi=10.1007/978-3-030-04221-9_48 |arxiv=1810.10659 |isbn=978-3-030-04220-2 }}</ref>

<ref name=velickovic2022>{{cite journal |last1=Veličković |first1=Petar |title=Message passing all the way up |journal=arXiv |doi=10.48550/ARXIV.2202.11097 |url=https://arxiv.org/abs/2202.11097}}</ref>

<ref name=qasim2019"bronstein2021">{{cite journalarXiv |last1=QasimBronstein |first1=ShahMichael RukhM. |last2=KieselerBruna |first2=JanJoan |last3=IiyamaCohen |first3=YutaroTaco |last4=PieriniVeličković |first4=Maurizio PieriniPetar |title=LearningGeometric representationsDeep ofLearning: irregularGrids, particle-detectorGroups, geometryGraphs withGeodesics distance-weightedand graph networksGauges |journaldate=The4 EuropeanMay Physical Journal C2021 |dateclass=2019 |volume=79 |issue=7cs.LG |doieprint=102104.1140/epjc/s10052-019-7113-913478}}</ref>

<ref name=douglas2011>{{cite arXiv|last=Douglas|first=B. L.|date=27 January 2011|title=The Weisfeiler–Lehman Method and Graph Isomorphism Testing|class=math.CO|eprint=1101.5211}}</ref>

<ref name=lui2022>{{cite journal |last1=Liu |first1=Chuang |last2=Zhan |first2=Yibing |last3=Li |first3=Chang |last4=Du |first4=Bo |last5=Wu |first5=Jia |last6=Hu |first6=Wenbin |last7=Liu |first7=Tongliang |last8=Tao |first8=Dacheng |title=Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities |journal=arXiv |date=2022 |doi=10.48550/arXiv.2204.07321 |url=https://arxiv.org/abs/2204.07321}}</ref>

<ref name=bronstein2021-2xu2019>{{cite journalCite arXiv|last1=Bodnar Xu|first1=Christian Keyulu|last2=Frasca Hu|first2=Fabrizio Weihua|last3=Guang Wang Leskovec|first3=YuJure |last4=Otter Jegelka|first4=Nina Stefanie|last5author4-link=MontúfarStefanie Jegelka|first5date=Guido22 |last6=Liò |first6=Pietro |last7=Bronstein |first7=MichaelFebruary 2019|title=WeisfeilerHow andPowerful Lehmanare GoGraph Topological: Message Passing SimplicialNeural Networks? |journaleprint=arXiv |date=2021 |doi=101810.48550/arXiv.2103.0321200826 |urlclass=https://arxiv.org/abs/2103cs.03212LG}}</ref>

<ref name=gao2019velickovic2022>{{cite journalarXiv |last1=GaoVeličković |first1=Hongyang |last2=Ji |first2=Shuiwang JiPetar |title=GraphMessage U-Netspassing |journal=Internationalall Conferencethe onway Machine Learningup |dateyear=20192022 |doiclass=10cs.48550/arXiv.1905.05178LG |urleprint=https://arxiv.org/abs/19052202.0517811097}}</ref>

<ref name=lee2019qasim2019>{{cite journal |last1=LeeQasim |first1=JunhyunShah Rukh |last2=LeeKieseler |first2=InyeopJan |last3=KangIiyama |first3=JaewooYutaro |last4=Pierini |first4=Maurizio Pierini |title=SelfLearning representations of irregular particle-Attentiondetector Graphgeometry Poolingwith distance-weighted graph networks |journal=InternationalThe ConferenceEuropean ofPhysical MachineJournal LearningC |date=2019 |volume=79 |issue=7 |page=608 |doi=10.485501140/arXiv.1904.08082epjc/s10052-019-7113-9|s2cid=88518244 |urldoi-access=https://free |arxiv=1902.org/abs/190407987 |bibcode=2019EPJC.08082..79..608Q }}</ref>

<ref name=alon2021lui2022>{{cite journalarXiv |last1=AlonLiu |first1=UriChuang |last2=YahavZhan |first2=EranYibing |titlelast3=OnLi the|first3=Chang Bottleneck|last4=Du of|first4=Bo Graph|last5=Wu Neural|first5=Jia Networks|last6=Hu and|first6=Wenbin its|last7=Liu Practical|first7=Tongliang Implications|last8=Tao |journalfirst8=InternationalDacheng Conference|title=Graph onPooling Learningfor Graph Neural Networks: Progress, Challenges, and RepresentationsOpportunities |date=20212022 |doiclass=10cs.48550/arXiv.2006.05205LG |urleprint=https://arxiv.org/abs/20062204.0520507321}}</ref>

<ref name=chen2021hajij2022>{{cite journalarXiv |last1=ChenHajij |first1=DeliM. |last2=LinZamzmi |first2=YankaiG. |last3=LiPapamarkou |first3=WeiT. |last4=LiMiolane |first4=PengN. |last5=ZhouGuzmán-Sáenz |first5=JieA. |last6=SunRamamurthy |first6=XuK. N. |titlelast7=MeasuringSchaub and|first7=M. Relieving the Over-smoothing Problem for Graph Neural Networks from theT. |title=Topological Viewdeep |journal=AAAIlearning: ConferenceGoing onbeyond Artificialgraph Intelligencedata |date=20202022 |doiclass=10cs.48550/arXiv.1909.03211LG |urleprint=https://arxiv.org/abs/19092206.0321100606}}</ref>

<ref name=bronstein2021-2>{{cite arXiv |last1=Bodnar |first1=Christian |last2=Frasca |first2=Fabrizio |last3=Guang Wang |first3=Yu |last4=Otter |first4=Nina |last5=Montúfar |first5=Guido |last6=Liò |first6=Pietro |last7=Bronstein |first7=Michael |title=Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks |date=2021 |class=cs.LG |eprint=2103.03212}}</ref>

<ref name=gao2019>{{cite arXiv |last1=Gao |first1=Hongyang |last2=Ji |first2=Shuiwang Ji |title=Graph U-Nets |date=2019 |class=cs.LG |eprint=1905.05178}}</ref>

<ref name=lee2019>{{cite arXiv |last1=Lee |first1=Junhyun |last2=Lee |first2=Inyeop |last3=Kang |first3=Jaewoo |title=Self-Attention Graph Pooling |date=2019 |class=cs.LG |eprint=1904.08082}}</ref>

<ref name=alon2021>{{cite arXiv |last1=Alon |first1=Uri |last2=Yahav |first2=Eran |title=On the Bottleneck of Graph Neural Networks and its Practical Implications |date=2021 |class=cs.LG |eprint=2006.05205}}</ref>

<ref name=chen2021>{{cite journal |last1=Chen |first1=Deli |last2=Lin |first2=Yankai |last3=Li |first3=Wei |last4=Li |first4=Peng |last5=Zhou |first5=Jie |last6=Sun |first6=Xu |title=Measuring and Relieving the Over-Smoothing Problem for Graph Neural Networks from the Topological View |journal=Proceedings of the AAAI Conference on Artificial Intelligence |date=2020 |volume=34 |issue=4 |pages=3438–3445 |doi=10.1609/aaai.v34i04.5747 |s2cid=202539008 |arxiv=1909.03211}}</ref>

<ref name=guardian2018>{{cite news|last1=Sample|first1=Ian|date=2 December 2018|title=Google's DeepMind predicts 3D shapes of proteins|work=The Guardian|url=https://www.theguardian.com/science/2018/dec/02/google-deepminds-ai-program-alphafold-predicts-3d-shapes-of-proteins|access-date=30 November 2020}}</ref>

<ref name=mit2020>{{cite web |title=DeepMind's protein-folding AI has solved a 50-year-old grand challenge of biology |url=https://www.technologyreview.com/2020/11/30/1012712/deepmind-protein-folding-ai-solved-biology-science-drugs-disease/ |website=MIT Technology Review |access-date=30 November 2020 |language=en}}</ref>

<ref name=fan2019>{{cite journalbook |last1=Fan |first1=Wenqi |last2=Ma |first2=Yao |last3=Li |first3=Qing |last4=He |first4=Yuan |last5=Zhao |first5=Eric |last6=Tang |first6=Jiliang |last7=Yin |first7=Dawei |title=Graph Neural Networks for Social Recommendation |journaldate=The Web Conference2019 |datepages=2019417–426 |doi=10.485501145/arXiv3308558.1902.072433313488 |urlhdl=https:/10397/81232 |isbn=9781450366748 |s2cid=67769538 |arxiv.org/abs/=1902.07243}}</ref>

<ref name=ying2018>{{cite journalbook |last1=Ying |first1=Rex |last2=He |first2=Ruining |last3=Chen |first3=Kaifeng |last4=Eksombatchai |first4=Pong |last5=Hamilton |first5=William L. |last6=Leskovec |first6=Jure |title=Graph Convolutional Neural Networks for Web-Scale Recommender Systems |journal=ACM SIGKDD Conference on Knowledge Discovery and Data Mining |date=2018 |pages=974–983 |doi=10.485501145/arXiv3219819.1806.019733219890 |urlisbn=https://9781450355520 |s2cid=46949657 |arxiv.org/abs/=1806.01973}}</ref>

<ref name=gasse2019>{{cite journalarXiv |last1=Gasse |first1=Maxime |last2=Chételat |first2=Didier |last3=Ferroni |first3=Nicola |last4=Charlin |first4=Laurent |last5=Lodi |first5=Andrea |title=Exact Combinatorial Optimization with Graph Convolutional Neural Networks |journal=Conference and Workshop on Neural Information Processing Systems |date=2019 |doiclass=10cs.48550/arXiv.1906.01629LG |urleprint=https://arxiv.org/abs/1906.01629}}</ref>

<ref name=cappart2021>{{cite journalarXiv |last1=Cappart |first1=Quentin |last2=Chételat |first2=Didier |last3=Khalil |first3=Elias |last4=Lodi |first4=Andrea |last5=Morris |first5=Christopher |last6=Veličković |first6=Petar |title=Combinatorial optimization and reasoning with graph neural networks |journalyear=arXiv2021 |doiclass=10cs.48550/arXiv.2102.09544LG |urleprint=https://arxiv.org/abs/2102.09544}}</ref>

<ref name=mirhoseini2021>{{cite journal |last1=Mirhoseini |first1=Azalia |last2=Goldie |first2=Anna |last3=Yazgan |first3=Mustafa |last4=Jiang |first4=Joe Wenjie |last5=Songhori |first5=Ebrahim |last6=Wang |first6=Shen |last7=Lee |first7=Young-Joon |last8=Johnson |first8=Eric |last9=Pathak |first9=Omkar |last10=Nazi |first10=Azade |last11=Pak |first11=Jiwoo |last12=Tong |first12=Andy |last13=Srinivasa |first13=Kavya |last14=Hang |first14=William |last15=Tuncer |first15=Emre |last16=Le |first16=Quoc V. |last17=Laudon |first17=James |last18=Ho |first18=Richard |last19=Carpenter |first19=Roger |last20=Dean |first20=Jeff |title=A graph placement methodology for fast chip design |journal=Nature |date=2021 |volume=594 |issue=7862 |pages=207-212207–212 |doi=10.1038/s41586-021-03544-w|pmid=34108699 |bibcode=2021Natur.594..207M |s2cid=235395490 }}</ref>

<ref name=fey2019>{{cite journalarXiv |last1=Matthias |first1=Fey |last2=Lenssen |first2=Jan E. |title=Fast Graph Representation Learning with PyTorch Geometric |journal=ICLR Workshop on Representation Learning on Graphs and Manifolds |date=2019 |doiclass=10cs.48550/arXiv.1903.02428LG |urleprint=https://arxiv.org/abs/1903.02428}}</ref>

<ref name=tfgnn2022>{{cite web |title=Tensorflow GNN |website=[[GitHub]] |url=https://github.com/tensorflow/gnn |access-date=30 June 2022}}</ref>

<ref name=jraph2022>{{cite web |title=jraph |website=[[GitHub]] |url=https://github.com/deepmind/jraph |access-date=30 June 2022}}</ref>

<ref name=xu2021>{{cite arXiv |last1=Xu |first1=Keyulu |last2=Zhang |first2=Mozhi |last3=Jegelka |first3=Stephanie|author3-link=Stefanie Jegelka |last4=Kawaguchi |first4=Kenji |title=Optimization of Graph Neural Networks: Implicit Acceleration by Skip Connections and More Depth |date=2021 |class=cs.LG |eprint=2105.04550}}</ref>

<ref name=li2016>{{cite arXiv |last1=Li |first1=Yujia |last2=Tarlow |first2=Daniel |last3=Brockschmidt |first3=Mark |last4=Zemel |first4=Richard |title=Gated Graph Sequence Neural Networks |date=2016 |class=cs.LG |eprint=1511.05493 }}</ref>

<ref name=grady2011discrete>{{cite book |last1=Grady |first1=Leo |last2=Polimeni |first2=Jonathan |title=Discrete Calculus: Applied Analysis on Graphs for Computational Science |url=http://leogrady.net/wp-content/uploads/2017/01/grady2010discrete.pdf |date=2011 |publisher=Springer }}</ref>

<ref name=xu2018>{{cite arXiv |last1=Xu |first1=Keyulu |last2=Li |first2=Chengtao |last3=Tian |first3=Yonglong |last4=Sonobe |first4=Tomohiro |last5=Kawarabayashi |first5=Ken-ichi |last6=Jegelka |first6=Stefanie|author6-link=Stefanie Jegelka |title=Representation Learning on Graphs with Jumping Knowledge Networks |date=2018 |class=cs.LG |eprint=1806.03536}}</ref>

<ref name=Lucibello2021GNN>{{cite web |last=Lucibello |first=Carlo |title=GraphNeuralNetworks.jl |website=[[GitHub]] |url=https://github.com/CarloLucibello/GraphNeuralNetworks.jl |year=2021 |access-date=21 September 2023}}</ref>

}}

== External links ==

[[Category:Machine learning]]

* [https://distill.pub/2021/gnn-intro/ A Gentle Introduction to Graph Neural Networks]

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