View source for Graph neural network

{{Short description|Class of artificial neural networks}}
{{Machine learning|Artificial neural network}}

'''Graph neural networks''' ('''GNN''') are specialized [[artificial neural network]]s that are designed for tasks whose inputs are [[Graph (abstract data type)|graphs]].<ref name="wucuipeizhao2022" /><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=2020-02-20 |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>{{Citation |last1=Yang |first1=Kevin |title=Analyzing Learned Molecular Representations for Property Prediction |date=2019-11-20 |arxiv=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}}</ref><ref>{{Cite journal |last=Marchant |first=Jo |date=2020-02-20 |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 }}</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.

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" />

[[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]].]]

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 [[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=2019-01-02 |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" />

[[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=2024-09-12 |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=2024-01-31 |url=https://github.com/FluxML/GeometricFlux.jl |access-date=2024-02-03 |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.

It has been demonstrated that GNNs cannot be more expressive than the [[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]]es can be designed.<ref name=bronstein2021-2 /><ref name=grady2011discrete /><ref name=hajij2022></ref> {{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 its immediate neighbours <math>\mathbf{x}_1</math> to <math>\mathbf{x}_4</math>. Messages are computing via the message function <math>\psi</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}_{uv}</math> be the features of edge <math>(u, v) \in E</math>. An MPNN [[Layer (deep learning)|layer]] can be expressed as follows:<ref name="bronstein2021" />

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

where <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 [[Diameter (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 [[convolutional neural network]]s to graph-structured data.

The formal expression of a GCN layer reads as follows:

:<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>

where <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 graph [[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_{uv} \mathbf{W}^k \mathbf{x}_v\right) </math>

where <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:

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

[[Attention (machine learning)|Attention]] in Machine Learning is a technique that mimics [[Attention|cognitive attention]]. In the context of learning on graphs, the attention coefficient <math>\alpha_{uv}</math> measures ''how important'' is node <math>u \in V</math> to node <math>v \in V</math>.

Normalized attention coefficients are computed as follows:

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

where <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>.

=== Crypto ===
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. We present here several learnable local pooling strategies that have been proposed.<ref name=lui2022 /> For each case, the input is the initial graph is 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

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

where <math>\mathbf{i} = \text{top}_k(\mathbf{y})</math> is the subset of nodes with the top-k highest projection scores, <math>\odot</math> denotes element-wise [[matrix multiplication]], and <math>\text{sigmoid}(\cdot)</math> is the [[sigmoid function]]. In other words, the 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

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

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

where <math>\mathbf{i} = \text{top}_k(\mathbf{y})</math> is the subset of nodes with the top-k highest projection scores, <math>\odot</math> denotes [[Hadamard product (matrices) | element-wise matrix multiplication]].

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">{{Citation |last1=Luan |first1=Sitao |title=The Heterophilic Graph Learning Handbook: Benchmarks, Models, Theoretical Analysis, Applications and Challenges |date=2024-07-12 |url=https://arxiv.org/abs/2407.09618 |access-date=2025-02-02 |arxiv=2407.09618 |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}}</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 journal |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 |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=When Do We Need Graph Neural Networks for Node Classification? |url=https://link.springer.com/chapter/10.1007/978-3-031-53468-3_4 |journal=Complex Networks & Their Applications XII |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=2022-12-06 |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|hypergraphs]]. 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]], [[Clustering|graph clustering]], [[Recommender system|recommender systems]], [[Generative model|generative models]], [[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=2023-12-15 |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 also|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 networks ===
{{See also|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 also|Combinatorial optimization}}
GNNs are used as fundamental building blocks for several combinatorial optimization algorithms.<ref name=cappart2021 /> Examples include computing [[Shortest path problem|shortest paths]] or [[Eulerian path|Eulerian circuits]] for a given graph,<ref name=li2016/> deriving [[Placement (electronic design automation)|chip placements]] superior or competitive to handcrafted human solutions,<ref name=mirhoseini2021 /> and improving expert-designed branching rules in [[branch and bound]].<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 |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 (NDSS) 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 (NDSS)|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 |publisher=AGU|doi=10.1029/2022WR032299 |bibcode=2022WRR....5832299Z |access-date=June 11, 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 |doi=10.1016/j.watres.2023.120264 |pmid=37393807 |bibcode=2023WatRe.24220264Z |access-date=June 11, 2024}}</ref>

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

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