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# <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 either on higher-dimension geometries such as [[simplicial complex]]es, can orbe ondesigned.<ref sparsename=bronstein2021-2 convolutions/><ref name=grady2011discrete /><ref name="Giraldo-SGNNs-1"hajij2022></ref> {{citeAs journalof|title=Higher-Order2022}}, Sparsewhether Convolutionsor Innot Graphfuture Neuralarchitectures Networks|journal=IEEEwill Internationalovercome Conferencethe onmessage Acoustics,passing primitive Speechis andan Signalopen Processing|date=2023|url=https://ieeexploreresearch question.ieee.org/document/10096494|last1=Giraldo|first1=H.|last2=Javed|first2=S.|last3=Mahmood|first3=A.|last4=Malliaros|first4=F.|last5=Bouwmans|first5=T.}}</ref> or on Sobolev normname=velickovic2022 />
<ref name="Giraldo-SGNNs-2">{{cite journal|title=Higher-Order GNNs Meet Efficiency: Sparse Sobolev Graph Neural Networks|journal=IEEE Transactions on Signal and Information Processing Over Networks|date=2024|url=https://ieeexplore.ieee.org/document/10758782|last1=Giraldo|first1=H.|last2=Torovic|first2=A.|last3=Einizade|first3=A.|last4=Castro-Correa|first4=J.|last5=Badiey|first5=M.|last6=Malliaros|first6=F.|last7=Bouwmans|first7=T.}}</ref> can be designed <ref name=bronstein2021-2 /><ref name=grady2011discrete /><ref name=hajij2022></ref>. Thus, Graph Neural Networks (GNNs) show great promise in modeling relationships between nodes in a graph, but capturing higher-order relationships remains a challenge for large-scale networks. {{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]].]]
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>
=== Traffic Forecasting ===
GNNs are well suited to traffic. For example, a road network is naturally a graph, with road intersections as the nodes and road connections as the edges. With graphs as the input, several GNN-based models have demonstrated superio.r performance to conventional approaches on tasks including road traffic flow and speed forecasting problems.
=== Computer Vision ===
Moving objects detection in video surveillance is a challenging task in computer vision. Graph neural networks allow to separate background and foreground objects
<ref name="Prummel">{{cite journal|title=Graph Neural Networks for Moving Object Detection in Videos|journal=Pattern Recognition and Computer Vision in the New AI Era, Edited by C.H. Chen, World Scientific Publishing|date=April 2025|last1=Prummel|first1=W.|last2=Giraldo|first2=H.|last3=Zakharova|first3=A.|last4=Bouwmans|first4=T.}}</ref> . Transductive <ref name="Giraldo">{{cite journal|title=Graph CNN for Moving Object Detection in Complex Environments from Unseen Videos|journal=Fourth Workshop on Robust Subspace Learning and Computer Vision, ICCV 2021|date=October 2021|url=https://ieeexplore.ieee.org/document/9607835|last1=Giraldo|first1=H.|last2=Javed|first2=S.|last3=Werghi|first3=N.|last4=Bouwmans|first4=T.}}</ref> and inductive GNNs <ref name="Prummel1">{{cite journal|title=Inductive Graph Neural Networks for Moving Object Segmentation|journal=IEEE International Conference on Image Processing, ICIP 2023|date=October 2023|url=https://arxiv.org/abs/2305.09585|last1=Prummel|first1=W.|last2=Giraldo|first2=H.|last3=Zakharova|first3=A.|last4=Bouwmans|first4=T.}}</ref>
have shown interesting performance to detect moving objects in urban environments <ref name="Prummel2">{{cite journal|title=Transductive and Inductive GNNs for Physical Moving Objects Detection in Surface Scenes for Digital Twins|journal=Chapter 10, Handbook on “DIGITAL TWINS: Concept, Applications and Challenges”, Edited by L. Sharma and P. Garg, CRC Press, Taylor and Francis Group|date=July 2025|last1=Prummel|first1=W.|last2=Giraldo|first2=H.|last3=Subudhi|first3=B.|last4=Zakharova|first4=A.|last5=Bouwmans|first5=T.}}</ref> and natural environments <ref name="Kapoor">{{cite journal|title=Graph-based Moving Object Segmentation for Underwater Videos using Semi-supervised Learning|journal= Computer Vision and Image Understanding|date=2025|url= https://www.sciencedirect.com/science/article/pii/S107731422500013X|
last1=Kapoor|first1=M.|last2=Prummel|first2=W.|last3=Giraldo|first3=H.|last4=Subudhi|first3=B.|last5=Zakharova|first5=A.|last6=Bouwmans|first6=T.|last6=Bansal|first6=A.}}</ref>.
==References==
== External links ==
* [https://distill.pub/2021/gnn-intro/ A Gentle Introduction to Graph Neural Networks]
* [https://sites.google.com/view/gsp-website/graph-neural-networks A List of References on Graph Neural Networks]
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