[32] Goyal P and Ferrara E 2018 Graph embedding techniques, applications, and performance: A survey *Knowl.-Based Syst.* **151** 78–94

[33] Chen F, Wang Y C, Wang B and Kuo C C J 2020 *APSIPA Trans. Signal Inf. Process.* **9** e15

[34] Li M M, Huang K and Zitnik M 2022 Graph representation learning in biomedicine and healthcare *Nat. Biomed. Eng.* **6** 1353–69

[35] Yang C, Xiao Y, Zhang Y, Sun Y and Han J 2020 *IEEE Trans. Knowl. Data Eng.* **34** 1843–55

[36] Zhou D, Huang J and Schölkopf B 2006 Learning with hypergraphs: clustering, classification and embedding *Proc. 19th Int. Conf. on Neural Information Processing Systems (NIPS '06)* (MIT Press) pp 1601–8

[37] Gui H, Liu J, Tao F, Jiang M, Norick B and Han J 2016 Large-scale embedding learning in heterogeneous event data 2016 IEEE 16th Int. Conf. on Data Mining (ICDM) pp 907–12

[38] Feng Y, You H, Zhang Z, Ji R and Gao Y 2019 Hypergraph neural networks *Proc. AAAI Conf. on Artificial Intelligence* vol 33 pp 3558–65

[39] Wold S, Esbensen K and Geladi P 1987 Principal component analysis *Chemometr. Intell. Lab. Syst.* **2** 37–52

[40] Samko O, Marshall A and Rosin P 2006 Selection of the optimal parameter value for the Isomap algorithm *Pattern Recognit. Lett.* **27** 968–79

[41] Roweis S T and Saul L K 2000 Nonlinear Dimensionality Reduction by Locally Linear Embedding *Science* **290** 2323–6

[42] van der Maaten L and Hinton G 2008 *J. Mach. Learn. Res.* **9** 2579–605

[43] McInnes L, Healy J, Saul N and Großberger L 2018 UMAP: Uniform Manifold Approximation and Projection *J. Open Source Softw.* **3** 861

[44] Belkin M and Niyogi P 2003 Laplacian Eigenmaps for Dimensionality Reduction and Data Representation *Neural Comput.* **15** 1373–96

[45] Ghojogh B, Karray F and Crowley M 2019 Eigenvalue and generalized eigenvalue problems: tutorial (arXiv:1903.11240)

[46] Luo D, Ding C H Q, Nie F and Huang H 2011 Cauchy graph embedding *Int. Conf. on Machine Learning* pp 553–60 (available at: https://icml.cc/2011/papers/353_icmlpaper.pdf)

[47] Ahmed A, Shervashidze N, Narayanamurthy S, Josifovski V and Smola A J 2013 Distributed large-scale natural graph factorization *Proc. 22nd Int. Conf. on World Wide Web (WWW '13)* (Association for Computing Machinery) pp 37–48

[48] Cao S, Lu W and Xu Q 2015 GraRep: learning graph representations with global structural information *Proc. 24th ACM Int. on Conf. on Information and Knowledge Management (CIKM '15)* (Association for Computing Machinery) pp 891–900

[49] Mikolov T, Chen K, Corrado G and Dean J 2013 Efficient estimation of word representations in vector space 1st Int. Conf. on Learning Representations (ICLR 2013) (Scottsdale, Arizona, USA, 2–4 May 2013) (Workshop Track Proc.) (arXiv:1301.3781)

[50] Gutmann M and Hyvärinen A 2010 Noise-contrastive estimation: a new estimation principle for unnormalized statistical models *Proc. 13th Int. Conf. on Artificial Intelligence and Statistics (Proc. Machine Learning Research)* vol 9) ed Y W Teh and M Titterington (PMLR) pp 297–304 (available at: https://proceedings.mlr.press/v9/gutmann10a.html)

[51] Ou M, Cui P, Pei J, Zhang Z and Zhu W 2016 Asymmetric transitivity preserving graph embedding *Proc. 22nd ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '16)* (Association for Computing Machinery) pp 1105–14

[52] Wang X, Cui P, Wang J, Pei J, Zhu W and Yang S 2017 Community preserving network embedding *Proc. 31st AAAI Conf. on Artificial Intelligence (AAAI '17)* (AAAI Press) pp 203–9

[53] Yang C, Liu Z, Zhao D, Sun M and Chang E Y 2015 Network representation learning with rich text information *Proc. 24th Int. Conf. on Artificial Intelligence (IJCAI '15)* (AAAI Press) pp 2111–7

[54] Yu H F, Jain P, Kar P and Dhillon I 2014 Large-scale multi-label learning with missing labels *Proc. 31st Int. Conf. on Machine Learning (Proc. Machine Learning Research)* vol 32) ed E P Xing and T Jebara (PMLR) pp 593–601 (available at: https://proceedings.mlr.press/v32/yu14.html)

[55] He X and Niyogi P 2004 *Advances in Neural Information Processing Systems* vol 16 pp 153–60

[56] Shaw B and Jebara T 2009 Structure preserving embedding *Proc. 26th Annual Int. Conf. on Machine Learning (ICML '09)* (Association for Computing Machinery) pp 937–44

[57] Lin Y Y, Liu T L and Chen H T 2005 Semantic manifold learning for image retrieval *Proc. 13th Annual ACM Int. Conf. on Multimedia (MULTIMEDIA '05)* (Association for Computing Machinery) pp 249–58

[58] Huang X, Li J and Hu X 2017 Label informed attributed network embedding *Proc. 10th ACM Int. Conf. on Web Search and Data Mining (WSDM '17)* (Association for Computing Machinery) pp 731–9

[59] Zhang D, Yin J, Zhu X and Zhang C 2016 Homophily, structure and content augmented network representation learning 2016 IEEE 16th Int. Conf. on Data Mining (ICDM) pp 609–18

[60] Tu C, Zhang W, Liu Z and Sun M 2016 Max-margin deepwalk: discriminative learning of network representation *Proc. 25th Int. Joint Conf. on Artificial Intelligence (IJCAI '16)* (AAAI Press) pp 3889–95

[61] Zhang D, Yin J, Zhu X and Zhang C 2016 Collective classification via discriminative matrix factorization on sparsely labeled networks *Proc. 25th ACM Int. on Conf. on Information and Knowledge Management (CIKM '16)* (Association for Computing Machinery) pp 1563–72

[62] Nickel M, Tresp V and Kriegel H P 2012 Factorizing YAGO: scalable machine learning for linked data *Proc. 21st Int. Conf. on World Wide Web (WWW '12)* (Association for Computing Machinery) pp 271–80

[63] Yang B, tau Yih W, He X, Gao J and Deng L 2015 *CoRR* (arXiv:1412.6575)

[64] Trouillon T, Welbl J, Riedel S, Gaussier E and Bouchard G 2016 Complex embeddings for simple link prediction *Proc. 33rd Int. Conf. on Int. Conf. on Machine Learning (ICML '16)* vol 48 (JMLR.org) pp 2071–80

[65] Schaub M T, Delvenne J C, Lambiotte R and Barahona M 2019 Multiscale dynamical embeddings of complex networks *Phys. Rev. E* **99** 062308

[66] Perozzi B, Al-Rfou R and Skiena S 2014 DeepWalk: online learning of social representations *Proc. 20th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '14)* (Association for Computing Machinery) pp 701–10

[67] Mikolov T, Sutskever I, Chen K, Corrado G S and Dean J 2013 Distributed representations of words and phrases and their compositionality *Advances in Neural Information Processing Systems* vol 26, ed C J C Burges, L Bottou, M Welling, Z Ghahramani and W (Curran Associates, Inc.) (available at: https://proceedings.neurips.cc/paper/2013/file/9aa42b31882ec039965f3c4923ce901b-paper.pdf)

[68] Mnih A and Hinton G E 2008 A scalable hierarchical distributed language model *Advances in Neural Information Processing Systems* vol 21

[69] Qiu J, Dong Y, Ma H, Li J, Wang K and Tang J 2018 Network embedding as matrix factorization: unifying DeepWalk, LINE, PTE and Node2vec *Proc. 11th ACM Int. Conf. on Web Search and Data Mining (WSDM '18)* (Association for Computing Machinery) pp 459–67
[70] Cen Y, Zou X, Zhang J, Yang H, Zhou J and Tang J 2019 Representation learning for attributed multiplex heterogeneous network Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '19) (Association for Computing Machinery) pp 1358–68

[71] Dursun C, Smith JR, Hayman G, Kwitek A E and Bozdag S 2020 NECO: a node embedding algorithm for multiplex heterogeneous networks 2020 IEEE Int. Conf. on Bioinformatics and Biomedicine (BIBM) (IEEE Computer Society) pp 146–9

[72] Grover A and Leskovec J 2016 node2vec: scalable feature learning for networks (arXiv:1607.00653)

[73] Wilson J D, Baybay M, Sankar R and Stillman P E 2018 arXiv:1809.06437

[74] Liu W, Chen P Y, Yeung S, Suzumura T and Chen L 2017 Principled multilayer network embedding 2017 IEEE Int. Conf. on Data Mining Workshops (ICDMW) pp 134–41

[75] Chen H, Perozzi B, Hu Y and Skiena S 2018 HARP: hierarchical representation learning for networks 32nd AAAI Conf. on Artificial Intelligence

[76] Li J, Zhu J and Zhang B 2016 Discriminative deep random walk for network classification Proc. 54th Annual Meeting of the Association for Computational Linguistics

[77] Perozzi B, Kulkarni V, Chen H and Skiena S 2017 Don't walk, skip! Online learning of multi-scale network embeddings Proc. 2017 IEEE/ACM Int. Conf. on Advances in Social Networks Analysis and Mining 2017 (ASONAM '17) (Association for Computing Machinery) pp 258–65

[78] Yang C and Liu Z 2015 arXiv:1501.00358

[79] Ribeiro L F, Saverese P H and Figueiredo D R 2017 struc2vec: learning node representations from structural identity Proc. 23rd ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '17) (Association for Computing Machinery) pp 385–94

[80] Salvador S and Chan P 2007 Toward accurate dynamic time warping in linear time and space Intell. Data Anal. 11 561–80

[81] Li C, Li Z, Wang S, Yang Y, Zhang X and Zhou J 2017 Semi-supervised network embedding Database Systems for Advanced Applications ed S Candan, L Chen, T B Pedersen, L Chang and W Hua (Springer International Publishing) pp 131–47

[82] Chen J, Zhang Q and Huang X 2016 Incorporate group information to enhance network embedding Proc. 25th ACM Int. on Conf. on Information and Knowledge Management (CIKM '16) (Association for Computing Machinery) pp 1901–4

[83] Pan S, Wu J, Zhu X, Zhang C and Wang Y 2016 Tri-party deep network representation Proc. 25th Int. Joint Conf. on Artificial Intelligence (IJCAI' 16) (AAAI Press) pp 1895–901

[84] Lyu T, Zhang Y and Zhang Y 2017 Enhancing the network embedding quality with structural similarity Proc. 2017 ACM on Conf. on Information and Knowledge Management (CIKM '17) (Association for Computing Machinery) pp 147–56

[85] Wu F, Lu X, Song J, Yan S, Zhang Z M, Rui Y and Zhuang Y 2016 Learning of Multimodal Representations With Random Walks on the Click Graph IEEE Trans. Image Process. 25 630–42

[86] Zhang C, Swami A and Chawla N V 2019 SHNE: representation learning for semantic-associated heterogeneous networks Proc. 12th ACM Int. Conf. on Web Search and Data Mining (WSDM '19) (Association for Computing Machinery) pp 690–8

[87] Wang X, Zhang Y and Shi C 2019 Heterogeneous graph attention network Proc. AAAI Conf. on Artificial Intelligence vol 33 pp 5337–44

[88] Chen T and Sun Y 2017 Task-guided and path-augmented heterogeneous network embedding for author identification Proc. 10th ACM Int. Conf. on Web Search and Data Mining (WSDM '17) (Association for Computing Machinery) pp 295–304

[89] Hussein R, Yang D and Cudré-Mauroux P 2018 Are meta-paths necessary? Revisiting heterogeneous graph embeddings Proc. 27th ACM Int. Conf. on Information and Knowledge Management (CIKM '18) (Association for Computing Machinery) pp 437–46

[90] He Y, Song Y, Li J, Ji C, Peng J and Peng H 2019 HeteSpaceyWalk: a heterogeneous spacey random walk for heterogeneous information network embedding Proc. 28th ACM Int. Conf. on Information and Knowledge Management (CIKM '19) (Association for Computing Machinery) pp 639–48

[91] Park C, Kim D, Zhu Q, Han J and Yu H 2019 Task-guided pair embedding in heterogeneous network Proc. 28th ACM Int. Conf. on Information and Knowledge Management (CIKM '19) (Association for Computing Machinery) pp 489–98

[92] Dong Y, Chawla N V and Swami A 2017 metapath2vec: scalable representation learning for heterogeneous networks Proc. 23rd ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '17) (Association for Computing Machinery) pp 135–44

[93] Fu T Y, Lee W C and Lei Z 2017 HIN2Vec: explore meta-paths in heterogeneous information networks for representation learning Proc. 2017 ACM on Conf. on Information and Knowledge Management (CIKM '17) (Association for Computing Machinery) pp 1797–806

[94] Huang Z and Mamoulis N 2017 arXiv:1701.05291

[95] Shi C, Hu B, Zhao W X and Yu P S 2019 Heterogeneous Information Network Embedding for Recommendation IEEE Trans. Knowl. Data Eng. 31 357–70

[96] Tsitsulin A, Mottin D, Karras P and Müller E 2018 VERSE: versatile graph embeddings from similarity measures Proc. 2018 World Wide Web Conf. (WWW '18) (International World Wide Web Conferences Steering Committee) pp 539–48

[97] Jeh G and Widom J 2002 SimRank: a measure of structural-context similarity Proc. 8th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '02) (Association for Computing Machinery) pp 538–43

[98] Mnih A and Teh Y W 2012 A fast and simple algorithm for training neural probabilistic language models Proc. 29th Int. Coference on Int. Conf. on Machine Learning (ICML '12) (Omnipress) pp 419–26

[99] Pio-Lopez L, Valdeolivas A and Tichit L 2020 Multiverse: a multiplex and multiplex-heterogeneous network embedding approach Sci. Rep. 11 8794

[100] Tang J, Qu M, Wang M, Zhang M, Yan J and Mei Q 2015 LINE: large-scale information network embedding Proc. 24th Int. Conf. on World Wide Web (WWW '15) (International World Wide Web Conferences Steering Committee) pp 1067–77

[101] Zhang X, Chen W and Yan H 2016 TLINE: scalable transductive network embedding Information Retrieval Technology ed S Ma, J R Wen, Y Liu, Z Dou, M Zhang, Y Chang and X Zhao (Springer International Publishing) pp 98–110

[102] Tang J, Qu M and Mei Q 2015 PTE: predictive text embedding through large-scale heterogeneous text networks Proc. 21th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '15) (Association for Computing Machinery) pp 1165–74

[103] Zhou C, Liu Y, Liu X, Liu Z and Gao J 2017 Scalable graph embedding for asymmetric proximity Proc. AAAI Conf. on Artificial Intelligence vol 31

[104] Zitnik M and Leskovec J 2017 Predicting multicellular function through multi-layer tissue networks Bioinformatics 33 i190–8

[105] Xu L, Wei X, Cao J and Yu P S 2017 Embedding of embedding (EOE): joint embedding for coupled heterogeneous networks Proc. 10th ACM Int. Conf. on Web Search and Data Mining (WSDM '17) (Association for Computing Machinery) pp 741–9

[106] Zhang H, Qiu L, Yi L and Song Y 2018 Scalable multiplex network embedding Int. Joint Conf. on Artificial Intelligence (IJCAI)

[107] Bagavathi A and Krishnan S 2018 Multi-Net: a scalable multiplex network embedding framework Complex Networks and Their Applications VII (Springer International Publishing) pp 119–31
[70] Cen Y, Zou X, Zhang J, Yang H, Zhou J and Tang J 2019 Representation learning for attributed multiplex heterogeneous network Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '19) (Association for Computing Machinery) pp 1358–68

[71] Dursun C, Smith JR, Hayman G, Kwitek A E and Bozdag S 2020 NECO: a node embedding algorithm for multiplex heterogeneous networks 2020 IEEE Int. Conf. on Bioinformatics and Biomedicine (BIBM) (IEEE Computer Society) pp 146–9

[72] Grover A and Leskovec J 2016 node2vec: scalable feature learning for networks (arXiv:1607.00653)

[73] Wilson J D, Baybay M, Sankar R and Stillman P E 2018 arXiv:1809.06437

[74] Liu W, Chen P Y, Yeung S, Suzumura T and Chen L 2017 Principled multilayer network embedding 2017 IEEE Int. Conf. on Data Mining Workshops (ICDMW) pp 134–41

[75] Chen H, Perozzi B, Hu Y and Skiena S 2018 HARP: hierarchical representation learning for networks 32nd AAAI Conf. on Artificial Intelligence

[76] Li J, Zhu J and Zhang B 2016 Discriminative deep random walk for network classification Proc. 54th Annual Meeting of the Association for Computational Linguistics

[77] Perozzi B, Kulkarni V, Chen H and Skiena S 2017 Don't walk, skip! Online learning of multi-scale network embeddings Proc. 2017 IEEE/ACM Int. Conf. on Advances in Social Networks Analysis and Mining 2017 (ASONAM '17) (Association for Computing Machinery) pp 258–65

[78] Yang C and Liu Z 2015 arXiv:1501.00358

[79] Ribeiro L F, Saverese P H and Figueiredo D R 2017 struc2vec: learning node representations from structural identity Proc. 23rd ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '17) (Association for Computing Machinery) pp 385–94

[80] Salvador S and Chan P 2007 Toward accurate dynamic time warping in linear time and space Intell. Data Anal. 11 561–80

[81] Li C, Li Z, Wang S, Yang Y, Zhang X and Zhou J 2017 Semi-supervised network embedding Database Systems for Advanced Applications ed S Candan, L Chen, T B Pedersen, L Chang and W Hua (Springer International Publishing) pp 131–47

[82] Chen J, Zhang Q and Huang X 2016 Incorporate group information to enhance network embedding Proc. 25th ACM Int. on Conf. on Information and Knowledge Management (CIKM '16) (Association for Computing Machinery) pp 1901–4

[83] Pan S, Wu J, Zhu X, Zhang C and Wang Y 2016 Tri-party deep network representation Proc. 25th Int. Joint Conf. on Artificial Intelligence (IJCAI' 16) (AAAI Press) pp 1895–901

[84] Lyu T, Zhang Y and Zhang Y 2017 Enhancing the network embedding quality with structural similarity Proc. 2017 ACM on Conf. on Information and Knowledge Management (CIKM '17) (Association for Computing Machinery) pp 147–56

[85] Wu F, Lu X, Song J, Yan S, Zhang Z M, Rui Y and Zhuang Y 2016 Learning of Multimodal Representations With Random Walks on the Click Graph IEEE Trans. Image Process. 25 630–42

[86] Zhang C, Swami A and Chawla N V 2019 SHNE: representation learning for semantic-associated heterogeneous networks Proc. 12th ACM Int. Conf. on Web Search and Data Mining (WSDM '19) (Association for Computing Machinery) pp 690–8

[87] Wang X, Zhang Y and Shi C 2019 Heterogeneous graph attention network Proc. AAAI Conf. on Artificial Intelligence vol 33 pp 5337–44

[88] Chen T and Sun Y 2017 Task-guided and path-augmented heterogeneous network embedding for author identification Proc. 10th ACM Int. Conf. on Web Search and Data Mining (WSDM '17) (Association for Computing Machinery) pp 295–304

[89] Hussein R, Yang D and Cudré-Mauroux P 2018 Are meta-paths necessary? Revisiting heterogeneous graph embeddings Proc. 27th ACM Int. Conf. on Information and Knowledge Management (CIKM '18) (Association for Computing Machinery) pp 437–46

[90] He Y, Song Y, Li J, Ji C, Peng J and Peng H 2019 HeteSpaceyWalk: a heterogeneous spacey random walk for heterogeneous information network embedding Proc. 28th ACM Int. Conf. on Information and Knowledge Management (CIKM '19) (Association for Computing Machinery) pp 639–48

[91] Park C, Kim D, Zhu Q, Han J and Yu H 2019 Task-guided pair embedding in heterogeneous network Proc. 28th ACM Int. Conf. on Information and Knowledge Management (CIKM '19) (Association for Computing Machinery) pp 489–98

[92] Dong Y, Chawla N V and Swami A 2017 metapath2vec: scalable representation learning for heterogeneous networks Proc. 23rd ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '17) (Association for Computing Machinery) pp 135–44

[93] Fu T Y, Lee W C and Lei Z 2017 HIN2Vec: explore meta-paths in heterogeneous information networks for representation learning Proc. 2017 ACM on Conf. on Information and Knowledge Management (CIKM '17) (Association for Computing Machinery) pp 1797–806

[94] Huang Z and Mamoulis N 2017 arXiv:1701.05291

[95] Shi C, Hu B, Zhao W X and Yu P S 2019 Heterogeneous Information Network Embedding for Recommendation IEEE Trans. Knowl. Data Eng. 31 357–70

[96] Tsitsulin A, Mottin D, Karras P and Müller E 2018 VERSE: versatile graph embeddings from similarity measures Proc. 2018 World Wide Web Conf. (WWW '18) (International World Wide Web Conferences Steering Committee) pp 539–48

[97] Jeh G and Widom J 2002 SimRank: a measure of structural-context similarity Proc. 8th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '02) (Association for Computing Machinery) pp 538–43

[98] Mnih A and Teh Y W 2012 A fast and simple algorithm for training neural probabilistic language models Proc. 29th Int. Coference on Int. Conf. on Machine Learning (ICML '12) (Omnipress) pp 419–26

[99] Pio-Lopez L, Valdeolivas A and Tichit L 2020 Multiverse: a multiplex and multiplex-heterogeneous network embedding approach Sci. Rep. 11 8794

[100] Tang J, Qu M, Wang M, Zhang M, Yan J and Mei Q 2015 LINE: large-scale information network embedding Proc. 24th Int. Conf. on World Wide Web (WWW '15) (International World Wide Web Conferences Steering Committee) pp 1067–77

[101] Zhang X, Chen W and Yan H 2016 TLINE: scalable transductive network embedding Information Retrieval Technology ed S Ma, J R Wen, Y Liu, Z Dou, M Zhang, Y Chang and X Zhao (Springer International Publishing) pp 98–110

[102] Tang J, Qu M and Mei Q 2015 PTE: predictive text embedding through large-scale heterogeneous text networks Proc. 21th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '15) (Association for Computing Machinery) pp 1165–74

[103] Zhou C, Liu Y, Liu X, Liu Z and Gao J 2017 Scalable graph embedding for asymmetric proximity Proc. AAAI Conf. on Artificial Intelligence vol 31

[104] Zitnik M and Leskovec J 2017 Predicting multicellular function through multi-layer tissue networks Bioinformatics 33 i190–8

[105] Xu L, Wei X, Cao J and Yu P S 2017 Embedding of embedding (EOE): joint embedding for coupled heterogeneous networks Proc. 10th ACM Int. Conf. on Web Search and Data Mining (WSDM '17) (Association for Computing Machinery) pp 741–9

[106] Zhang H, Qiu L, Yi L and Song Y 2018 Scalable multiplex network embedding Int. Joint Conf. on Artificial Intelligence (IJCAI)

[107] Bagavathi A and Krishnan S 2018 Multi-Net: a scalable multiplex network embedding framework Complex Networks and Their Applications VII (Springer International Publishing) pp 119–31
[108] An Q and Yu L 2021 Brief. Bioinform. 22 bbab275

[109] Bordes A, Usunier N, Garcia-Durán A, Weston J and Yakhnenko O 2013 Translating embeddings for modeling multi-relational data Proc. 26th Int. Conf. on Neural Information Processing Systems (NIPS '13) vol 2 (Curran Associates, Inc.) pp 2787–95

[110] Wang Z, Zhang J, Feng J and Chen Z 2014 Knowledge graph embedding by translating on hyperplanes Proc. 28th AAAI Conf. on Artificial Intelligence (AAAI '14) (AAAI Press) pp 1112–9

[111] Lin Y, Liu Z, Sun M, Liu Y and Zhu X 2015 Learning entity and relation embeddings for knowledge graph completion Proc. 29th AAAI Conf. on Artificial Intelligence (AAAI' 15) (AAAI Press) pp 2181–7

[112] Ji G, He S, Xu L, Liu K and Zhao J 2015 Knowledge graph embedding via dynamic mapping matrix Proc. 53rd Annual Meeting of the Association for Computational Linguistics and the 7th Int. Joint Conf. on Natural Language Processing (Volume 1: Long Papers) (Association for Computational Linguistics) pp 687–96 (available at: https://aclanthology.org/P15-1067)

[113] Sun Z, Deng Z, Nie J Y and Tang J 2019 arXiv:1902.10197

[114] Chang S, Han W, Tang J, Qi G J, Aggarwal C C and Huang T S 2015 Heterogeneous network embedding via deep architectures Proc. 21th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '15) (Association for Computing Machinery) pp 119–28

[115] Cao S, Lu W and Xu Q 2016 Deep neural networks for learning graph representations Proc. 30th AAAI Conf. on Artificial Intelligence (AAAI '16) (AAAI Press) pp 1145–52

[116] Wang D, Cui P and Zhu W 2016 Structural deep network embedding Proc. 22nd ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '16) (Association for Computing Machinery) pp 1225–34

[117] Kipf T and Welling M 2016 arXiv:1611.07308

[118] Alzubaidi L, Zhang J, Humaidi A J, Al-Dujaili A, Duan Y, Al-Shamma O, Santamaría J, Fadhel M A, Al-Amidie M and Farhan L 2021 J. Big Data 8 53

[119] Xie R, Liu Z, Jia J, Luan H and Sun M 2016 Representation learning of knowledge graphs with entity descriptions Proc. 30th AAAI Conf. on Artificial Intelligence (AAAI '16) (AAAI Press) pp 2659–65

[120] Niepert M, Ahmed M and Kutzkov K 2016 Learning convolutional neural networks for graphs Proc. The 33rd Int. Conf. on Machine Learning (Proc. Machine Learning Research vol 48) ed M F Balcan and K Q Weinberger (PMLR) pp 2014–23 (available at: https://proceedings.mlr.press/v48/niepert16.html)

[121] Gaier A and Ha D 2019 Weight agnostic neural networks Advances in Neural Information Processing Systems vol 32, ed H Wallach, H Larochelle, A Beygelzimer, F d'Alché-Buc, E Fox and R Garnett (Curran Associates, Inc.)

[122] Man T, Shen H, Liu S, Jin X and Cheng X 2016 Predict anchor links across social networks via an embedding approach Proc. 25th Int. Joint Conf. on Artificial Intelligence (IJCAI '16) (AAAI Press) pp 1823–9

[123] Li C, Ma J, Guo X and Mei Q 2017 DeepCas: an end-to-end predictor of information cascades Proc. 26th Int. Conf. on World Wide Web (WWW '17) (International World Wide Web Conferences Steering Committee) pp 577–86

[124] Defferrard M, Bresson X and Vandergheynst P 2016 Convolutional neural networks on graphs with fast localized spectral filtering Proc. 30th Int. Conf. on Neural Information Processing Systems (NIPS '16) (Curran Associates, Inc.) pp 3844–52

[125] Zitnik M, Agrawal M and Leskovec J 2018 Modeling polypharmacy side effects with graph convolutional networks Bioinformatics 34 i457–66

[126] Kipf T and Welling M 2017 arXiv:1609.02907

[127] Hamilton W L, Ying R and Leskovec J 2017 Inductive representation learning on large graphs Proc. 31st Int. Conf. on Neural Information Processing Systems (NIPS '17) (Curran Associates, Inc.) pp 1025–35

[128] Veličković P, Cucurull G, Casanova A, Romero A, Liò P and Bengio Y 2017 6th Int. Conf. on Learning Representations

[129] Xu Q, Wang Q, Xu C and Qu L 2017 Attentive graph-based recursive neural network for collective vertex classification Proc. 2017 ACM on Conf. on Information and Knowledge Management (CIKM '17) (Association for Computing Machinery) pp 2403–6

[130] Abu-El-Haija S, Perozzi B, Al-Rfou R and Alemi A A 2018 Watch your step: learning node embeddings via graph attention Advances in Neural Information Processing Systems vol 31, ed S Bengio, H Wallach, H Larochelle, K Grauman, N Cesa-Bianchi and R Garnett (Curran Associates, Inc.) (available at: https://proceedings.neurips.cc/paper/2018/file/ 8a94ecfa54dcb88a2fa993bfa6388f9e-paper.pdf)

[131] Xu K, Li C, Tian Y, Sonobe T, Kawarabayashi K-I and Jegelka S 2018 Representation learning on graphs with jumping knowledge networks Int. Conf. on Machine Learning

[132] Ying R, You J, Morris C, Ren X, Hamilton W L and Leskovec J 2018 Hierarchical graph representation learning with differentiable pooling Proc. 32nd Int. Conf. on Neural Information Processing Systems (NIPS '18) (Curran Associates, Inc.) pp 4805–15

[133] Zhang C, Song D, Huang C, Swami A and Chawla N V 2019 Heterogeneous graph neural network Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '19) (Association for Computing Machinery) pp 793–803

[134] Zhou J, Cui G, Hu S, Zhang Z, Yang C, Liu Z, Wang L, Li C and Sun M 2020 AI Open 1 57–81

[135] Goodfellow I, Pouget-Abadie J, Mirza M, Xu B, Warde-Farley D, Ozair S, Courville A and Bengio Y 2014 Generative adversarial nets Advances in Neural Information Processing Systems vol 27 ed Z Ghahramani, M Welling, C Cortes, N Lawrence and K Weinberger (Curran Associates, Inc.) (available at: https://proceedings.neurips.cc/paper/2014/file/ 5ca3e9b122f61f8f06494c97b1afccf3-paper.pdf)

[136] Wang H, Wang J, Wang J, Zhao M, Zhang W, Zhang F, Xie X and Guo M 2018 Proc. AAAI Conf. on Artificial Intelligence vol 32 (available at: https://ojs.aaai.org/index.php/AAAI/article/view/11872)

[137] Dai Q, Li Q, Tang J and Wang D 2018 Adversarial network embedding Proc. 32nd AAAI Conf. on Artificial Intelligence and 30th Innovative Applications of Artificial Intelligence Conf. and 8th AAAI Symp. on Educational Advances in Artificial Intelligence (AAAI '18/IAAI '18/EAAI'18) (AAAI Press)

[138] Gao H, Pei J and Huang H 2019 ProGAN: network embedding via proximity generative adversarial network Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '19) (Association for Computing Machinery) pp 1308–16

[139] McClelland J L et al 1987 Parallel Distributed Processing, Volume 2: Explorations in the Microstructure of Cognition: Psychological and Biological Models (MIT Press)

[140] Wang S, Tang J, Morstatter F and Liu H 2016 Paired restricted Boltzmann machine for linked data Proc. 25th ACM Int. on Conf. on Information and Knowledge Management (CIKM '16) (Association for Computing Machinery) pp 1753–62

[141] Bianconi G 2021 Higher-Order Networks (Cambridge University Press)

[142] Battiston F, Cencetti G, Iacopini I, Latora V, Lucas M, Patania A, Young J G and Petri G 2020 Phys. Rep. 874 1–92

[143] Battiston F et al 2021 The physics of higher-order interactions in complex systems Nat. Phys. 17 1093–8

[144] Torres L, Blevins A S, Bassett D and Eliassi-Rad T 2021 The why, how, and when of representations for complex systems SIAM Rev. 63 435–85
[108] An Q and Yu L 2021 Brief. Bioinform. 22 bbab275

[109] Bordes A, Usunier N, Garcia-Durán A, Weston J and Yakhnenko O 2013 Translating embeddings for modeling multi-relational data Proc. 26th Int. Conf. on Neural Information Processing Systems (NIPS '13) vol 2 (Curran Associates, Inc.) pp 2787–95

[110] Wang Z, Zhang J, Feng J and Chen Z 2014 Knowledge graph embedding by translating on hyperplanes Proc. 28th AAAI Conf. on Artificial Intelligence (AAAI '14) (AAAI Press) pp 1112–9

[111] Lin Y, Liu Z, Sun M, Liu Y and Zhu X 2015 Learning entity and relation embeddings for knowledge graph completion Proc. 29th AAAI Conf. on Artificial Intelligence (AAAI' 15) (AAAI Press) pp 2181–7

[112] Ji G, He S, Xu L, Liu K and Zhao J 2015 Knowledge graph embedding via dynamic mapping matrix Proc. 53rd Annual Meeting of the Association for Computational Linguistics and the 7th Int. Joint Conf. on Natural Language Processing (Volume 1: Long Papers) (Association for Computational Linguistics) pp 687–96 (available at: https://aclanthology.org/P15-1067)

[113] Sun Z, Deng Z, Nie J Y and Tang J 2019 arXiv:1902.10197

[114] Chang S, Han W, Tang J, Qi G J, Aggarwal C C and Huang T S 2015 Heterogeneous network embedding via deep architectures Proc. 21th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '15) (Association for Computing Machinery) pp 119–28

[115] Cao S, Lu W and Xu Q 2016 Deep neural networks for learning graph representations Proc. 30th AAAI Conf. on Artificial Intelligence (AAAI '16) (AAAI Press) pp 1145–52

[116] Wang D, Cui P and Zhu W 2016 Structural deep network embedding Proc. 22nd ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '16) (Association for Computing Machinery) pp 1225–34

[117] Kipf T and Welling M 2016 arXiv:1611.07308

[118] Alzubaidi L, Zhang J, Humaidi A J, Al-Dujaili A, Duan Y, Al-Shamma O, Santamaría J, Fadhel M A, Al-Amidie M and Farhan L 2021 J. Big Data 8 53

[119] Xie R, Liu Z, Jia J, Luan H and Sun M 2016 Representation learning of knowledge graphs with entity descriptions Proc. 30th AAAI Conf. on Artificial Intelligence (AAAI '16) (AAAI Press) pp 2659–65

[120] Niepert M, Ahmed M and Kutzkov K 2016 Learning convolutional neural networks for graphs Proc. The 33rd Int. Conf. on Machine Learning (Proc. Machine Learning Research vol 48) ed M F Balcan and K Q Weinberger (PMLR) pp 2014–23 (available at: https://proceedings.mlr.press/v48/niepert16.html)

[121] Gaier A and Ha D 2019 Weight agnostic neural networks Advances in Neural Information Processing Systems vol 32, ed H Wallach, H Larochelle, A Beygelzimer, F d'Alché-Buc, E Fox and R Garnett (Curran Associates, Inc.)

[122] Man T, Shen H, Liu S, Jin X and Cheng X 2016 Predict anchor links across social networks via an embedding approach Proc. 25th Int. Joint Conf. on Artificial Intelligence (IJCAI '16) (AAAI Press) pp 1823–9

[123] Li C, Ma J, Guo X and Mei Q 2017 DeepCas: an end-to-end predictor of information cascades Proc. 26th Int. Conf. on World Wide Web (WWW '17) (International World Wide Web Conferences Steering Committee) pp 577–86

[124] Defferrard M, Bresson X and Vandergheynst P 2016 Convolutional neural networks on graphs with fast localized spectral filtering Proc. 30th Int. Conf. on Neural Information Processing Systems (NIPS '16) (Curran Associates, Inc.) pp 3844–52

[125] Zitnik M, Agrawal M and Leskovec J 2018 Modeling polypharmacy side effects with graph convolutional networks Bioinformatics 34 i457–66

[126] Kipf T and Welling M 2017 arXiv:1609.02907

[127] Hamilton W L, Ying R and Leskovec J 2017 Inductive representation learning on large graphs Proc. 31st Int. Conf. on Neural Information Processing Systems (NIPS '17) (Curran Associates, Inc.) pp 1025–35

[128] Veličković P, Cucurull G, Casanova A, Romero A, Liò P and Bengio Y 2017 6th Int. Conf. on Learning Representations

[129] Xu Q, Wang Q, Xu C and Qu L 2017 Attentive graph-based recursive neural network for collective vertex classification Proc. 2017 ACM on Conf. on Information and Knowledge Management (CIKM '17) (Association for Computing Machinery) pp 2403–6

[130] Abu-El-Haija S, Perozzi B, Al-Rfou R and Alemi A A 2018 Watch your step: learning node embeddings via graph attention Advances in Neural Information Processing Systems vol 31, ed S Bengio, H Wallach, H Larochelle, K Grauman, N Cesa-Bianchi and R Garnett (Curran Associates, Inc.) (available at: https://proceedings.neurips.cc/paper/2018/file/ 8a94ecfa54dcb88a2fa993bfa6388f9e-paper.pdf)

[131] Xu K, Li C, Tian Y, Sonobe T, Kawarabayashi K-I and Jegelka S 2018 Representation learning on graphs with jumping knowledge networks Int. Conf. on Machine Learning

[132] Ying R, You J, Morris C, Ren X, Hamilton W L and Leskovec J 2018 Hierarchical graph representation learning with differentiable pooling Proc. 32nd Int. Conf. on Neural Information Processing Systems (NIPS '18) (Curran Associates, Inc.) pp 4805–15

[133] Zhang C, Song D, Huang C, Swami A and Chawla N V 2019 Heterogeneous graph neural network Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '19) (Association for Computing Machinery) pp 793–803

[134] Zhou J, Cui G, Hu S, Zhang Z, Yang C, Liu Z, Wang L, Li C and Sun M 2020 AI Open 1 57–81

[135] Goodfellow I, Pouget-Abadie J, Mirza M, Xu B, Warde-Farley D, Ozair S, Courville A and Bengio Y 2014 Generative adversarial nets Advances in Neural Information Processing Systems vol 27 ed Z Ghahramani, M Welling, C Cortes, N Lawrence and K Weinberger (Curran Associates, Inc.) (available at: https://proceedings.neurips.cc/paper/2014/file/ 5ca3e9b122f61f8f06494c97b1afccf3-paper.pdf)

[136] Wang H, Wang J, Wang J, Zhao M, Zhang W, Zhang F, Xie X and Guo M 2018 Proc. AAAI Conf. on Artificial Intelligence vol 32 (available at: https://ojs.aaai.org/index.php/AAAI/article/view/11872)

[137] Dai Q, Li Q, Tang J and Wang D 2018 Adversarial network embedding Proc. 32nd AAAI Conf. on Artificial Intelligence and 30th Innovative Applications of Artificial Intelligence Conf. and 8th AAAI Symp. on Educational Advances in Artificial Intelligence (AAAI '18/IAAI '18/EAAI'18) (AAAI Press)

[138] Gao H, Pei J and Huang H 2019 ProGAN: network embedding via proximity generative adversarial network Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '19) (Association for Computing Machinery) pp 1308–16

[139] McClelland J L et al 1987 Parallel Distributed Processing, Volume 2: Explorations in the Microstructure of Cognition: Psychological and Biological Models (MIT Press)

[140] Wang S, Tang J, Morstatter F and Liu H 2016 Paired restricted Boltzmann machine for linked data Proc. 25th ACM Int. on Conf. on Information and Knowledge Management (CIKM '16) (Association for Computing Machinery) pp 1753–62

[141] Bianconi G 2021 Higher-Order Networks (Cambridge University Press)

[142] Battiston F, Cencetti G, Iacopini I, Latora V, Lucas M, Patania A, Young J G and Petri G 2020 Phys. Rep. 874 1–92

[143] Battiston F et al 2021 The physics of higher-order interactions in complex systems Nat. Phys. 17 1093–8

[144] Torres L, Blevins A S, Bassett D and Eliassi-Rad T 2021 The why, how, and when of representations for complex systems SIAM Rev. 63 435–85
[145] Salnikov V, Cassese D and Lambiotte R 2018 Simplicial complexes and complex systems Eur. J. Phys. 40 014001

[146] Zomorodian A 2012 Topological data analysis Advances in Applied and Computational Topology vol 70 (American Mathematical Society) pp 1–39

[147] Huang S, Elhoseiny M, Elgammal A and Yang D 2015 Learning hypergraph-regularized attribute predictors 2015 IEEE Conf. on Computer Vision and Pattern Recognition (CVPR) pp 409–17

[148] Yang D, Qu B, Yang J and Cudre-Mauroux P 2020 IEEE Trans. Knowl. Data Eng. 34 1843–55

[149] Gui H, Liu J, Tao F, Jiang M, Norick B, Kaplan L and Han J 2017 Embedding Learning with Events in Heterogeneous Information Networks IEEE Trans. Knowl. Data Eng. 29 2428–841

[150] Tu K, Cui P, Wang X, Wang F and Zhu W 2018 Structural deep embedding for hyper-networks Proc. 32nd AAAI Conf. on Artificial Intelligence and 30th Innovative Applications of Artificial Intelligence Conf. and 8th AAAI Symp. on Educational Advances in Artificial Intelligence (AAAI ’18/IAAI ’18/EAAI ’18) (AAAI Press)

[151] Bai S, Zhang F and Torr P H 2021 Hypergraph convolution and hypergraph attention Pattern Recognit. 110 107637

[152] Papillon M, Sanborn S, Hajij M and Miolane N 2023 arXiv:2304.10031

[153] Bodnar C, Frasca F, Wang Y, Otter N, Montufar G F, Lio P and Bronstein M 2021 Weisfeiler and Lehman go topological: message passing simplicial networks Int. Conf. on Machine Learning (PMLR) pp 1026–37

[154] Bodnar C, Frasca F, Otter N, Wang Y, Lio P, Montufar G F and Bronstein M 2021 Advances in Neural Information Processing Systems vol 34 pp 2625–40

[155] Ebli S, Defferrard M and Spreemann G 2020 arXiv:2010.03633

[156] Chen Y, Gel Y R and Poor H V 2022 BScNets: block simplicial complex neural networks Proc. AAAI Conf. on Artificial Intelligence vol 36 pp 6333–41

[157] Hajij M, Istvan K and Zamzmi G 2020 arXiv:2010.00743

[158] Schaub M T, Zhu Y, Seby J B, Roddenberry T M and Segarra S 2021 Signal Process. 187 108149

[159] Giusti L, Battiloro C, Testa L, Di Lorenzo P, Sardellitti S and Barbarossa S 2022 arXiv:2209.08179

[160] Torres L, Chan K S and Eliassi-Rad T 2020 GLEE: Geometric Laplacian Eigenmap Embedding J. Complex Netw. 8 cnaa007

[161] Meng C, Yang J, Ribeiro B and Neville J 2019 HATS: a hierarchical sequence-attention framework for inductive set-of-sets embeddings Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery & Data Mining pp 783–92

[162] Boguna M, Bonamassa I, De Domenico M, Havlin S, Krioukov D and Serrano M A 2021 Network geometry Nat. Rev. Phys. 3 114–35

[163] Aste T, Di Matteo T and Hyde S 2005 Complex networks on hyperbolic surfaces Physica A 346 20–26

[164] Kleinberg R 2007 Geographic routing using hyperbolic space IEEE INFOCOM 2007—26th IEEE Int. Conf. on Computer Communications (IEEE) pp 1902–9

[165] Boguna M, Krioukov D and Claffy K C 2009 Navigability of complex networks Nat. Phys. 5 74–80

[166] Zhou Y, Smith B H and Sharpee T O 2018 Hyperbolic geometry of the olfactory space Sci. Adv. 4 eeaq1458

[167] Sharpee T O 2019 An argument for hyperbolic geometry in neural circuits Curr. Opin. Neurobiol. 58 101–4

[168] Krioukov D, Papadopoulos F, Kitsak M, Vahdat A and Boguná M 2010 Hyperbolic geometry of complex networks Phys. Rev. E 82 036106

[169] Papadopoulos F, Kitsak M, Serrano M A, Boguná M and Krioukov D 2012 Popularity versus similarity in growing networks Nature 489 537–40

[170] Boguná M, Papadopoulos F and Krioukov D 2010 Sustaining the Internet with hyperbolic mapping Nat. Commun. 1 62

[171] Kitsak M, Voitalov I and Krioukov D 2020 Link prediction with hyperbolic geometry Phys. Rev. Res. 2 043113

[172] Faqeeh A, Osat S and Radicchi F 2018 Characterizing the Analogy Between Hyperbolic Embedding and Community Structure of Complex Networks Phys. Rev. Lett. 121 098301

[173] Zuev K, Boguná M, Bianconi G and Krioukov D 2015 Emergence of Soft Communities from Geometric Preferential Attachment Sci. Rep. 5 9421

[174] Nickel M and Kiela D 2017 Advances in Neural Information Processing Systems vol 30

[175] García-Pérez G, Allard A, Serrano M A and Boguñá M 2019 Mercator: uncovering faithful hyperbolic embeddings of complex networks New J. Phys. 21 123033

[176] Muscoloni A, Thomas J M, Ciucci S, Bianconi G and Cannistraci C V 2017 Machine learning meets complex networks via coalescent embedding in the hyperbolic space Nat. Commun. 8 1615

[177] Kovács B and Palla G 2021 Optimisation of the coalescent hyperbolic embedding of complex networks Sci. Rep. 11 8350

[178] Patania A, Allard A and Young J G 2023 arXiv:2301.10403

[179] Kitsak M, Ganin A, Elmokashfi A, Cui H, Eisenberg D A, Alderson D L, Korkin D and Linkov I 2023 Finding shortest and nearly shortest path nodes in large substantially incomplete networks by hyperbolic mapping Nat. Commun. 14 186

[180] Kovács B and Palla G 2023 Model-independent embedding of directed networks into Euclidean and hyperbolic spaces Commun. Phys. 6 28

[181] Sarkar R 2012 Low distortion delaunay embedding of trees in hyperbolic plane Graph Drawing: 19th Int. Symp., (GD 2011) (Eindhoven, The Netherlands, 21–23 September 2011) (Revised Selected Papers vol 19) (Springer) pp 355–66

[182] Sala F, De Sa C, Gu A and Ré C 2018 Representation tradeoffs for hyperbolic embeddings Int. Conf. on Machine Learning (PMLR) pp 4460–9

[183] Chami I, Wolf A, Juan D C, Sala F, Ravi S and Ré C 2020 arXiv:2005.00545

[184] Chami I, Ying Z, Ré C and Leskovec J 2019 Advances in Neural Information Processing Systems vol 32

[185] Fernández-Gracia J and Onnela J P 2019 Flexible model of network embedding Sci. Rep. 9 11710

[186] Massara G P, Di Matteo T and Aste T 2016 J. Complex Netw. 5 161–78

[187] Bianconi G and Rahmede C 2016 Network geometry with flavor: From complexity to quantum geometry Phys. Rev. E 93 032315

[188] Bianconi G and Rahmede C 2017 Sci. Rep. 7 1–9

[189] Tumminello M, Aste T, Di Matteo T and Mantegna R N 2005 A tool for filtering information in complex systems Proc. Natl Acad. Sci. 102 10421–6

[190] Clough J R and Evans T S 2017 Embedding graphs in Lorentzian spacetime PLoS One 12 e0187301

[191] Fanuel M, Alaíz C M, Fernández A and Suykens J A 2018 Magnetic eigenmaps for the visualization of directed networks Appl. Comput. Harmon. Anal. 44 189–99

[192] Gong X, Higham D J and Zygalakis K 2021 Directed network Laplacians and random graph models R. Soc. Open Sci. 8 211144

[193] Gong X, Higham D J and Zygalakis K 2023 Generative hypergraph models and spectral embedding Sci. Rep. 13 540
[145] Salnikov V, Cassese D and Lambiotte R 2018 Simplicial complexes and complex systems Eur. J. Phys. 40 014001

[146] Zomorodian A 2012 Topological data analysis Advances in Applied and Computational Topology vol 70 (American Mathematical Society) pp 1–39

[147] Huang S, Elhoseiny M, Elgammal A and Yang D 2015 Learning hypergraph-regularized attribute predictors 2015 IEEE Conf. on Computer Vision and Pattern Recognition (CVPR) pp 409–17

[148] Yang D, Qu B, Yang J and Cudre-Mauroux P 2020 IEEE Trans. Knowl. Data Eng. 34 1843–55

[149] Gui H, Liu J, Tao F, Jiang M, Norick B, Kaplan L and Han J 2017 Embedding Learning with Events in Heterogeneous Information Networks IEEE Trans. Knowl. Data Eng. 29 2428–841

[150] Tu K, Cui P, Wang X, Wang F and Zhu W 2018 Structural deep embedding for hyper-networks Proc. 32nd AAAI Conf. on Artificial Intelligence and 30th Innovative Applications of Artificial Intelligence Conf. and 8th AAAI Symp. on Educational Advances in Artificial Intelligence (AAAI ’18/IAAI ’18/EAAI ’18) (AAAI Press)

[151] Bai S, Zhang F and Torr P H 2021 Hypergraph convolution and hypergraph attention Pattern Recognit. 110 107637

[152] Papillon M, Sanborn S, Hajij M and Miolane N 2023 arXiv:2304.10031

[153] Bodnar C, Frasca F, Wang Y, Otter N, Montufar G F, Lio P and Bronstein M 2021 Weisfeiler and Lehman go topological: message passing simplicial networks Int. Conf. on Machine Learning (PMLR) pp 1026–37

[154] Bodnar C, Frasca F, Otter N, Wang Y, Lio P, Montufar G F and Bronstein M 2021 Advances in Neural Information Processing Systems vol 34 pp 2625–40

[155] Ebli S, Defferrard M and Spreemann G 2020 arXiv:2010.03633

[156] Chen Y, Gel Y R and Poor H V 2022 BScNets: block simplicial complex neural networks Proc. AAAI Conf. on Artificial Intelligence vol 36 pp 6333–41

[157] Hajij M, Istvan K and Zamzmi G 2020 arXiv:2010.00743

[158] Schaub M T, Zhu Y, Seby J B, Roddenberry T M and Segarra S 2021 Signal Process. 187 108149

[159] Giusti L, Battiloro C, Testa L, Di Lorenzo P, Sardellitti S and Barbarossa S 2022 arXiv:2209.08179

[160] Torres L, Chan K S and Eliassi-Rad T 2020 GLEE: Geometric Laplacian Eigenmap Embedding J. Complex Netw. 8 cnaa007

[161] Meng C, Yang J, Ribeiro B and Neville J 2019 HATS: a hierarchical sequence-attention framework for inductive set-of-sets embeddings Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery & Data Mining pp 783–92

[162] Boguna M, Bonamassa I, De Domenico M, Havlin S, Krioukov D and Serrano M A 2021 Network geometry Nat. Rev. Phys. 3 114–35

[163] Aste T, Di Matteo T and Hyde S 2005 Complex networks on hyperbolic surfaces Physica A 346 20–26

[164] Kleinberg R 2007 Geographic routing using hyperbolic space IEEE INFOCOM 2007—26th IEEE Int. Conf. on Computer Communications (IEEE) pp 1902–9

[165] Boguna M, Krioukov D and Claffy K C 2009 Navigability of complex networks Nat. Phys. 5 74–80

[166] Zhou Y, Smith B H and Sharpee T O 2018 Hyperbolic geometry of the olfactory space Sci. Adv. 4 eeaq1458

[167] Sharpee T O 2019 An argument for hyperbolic geometry in neural circuits Curr. Opin. Neurobiol. 58 101–4

[168] Krioukov D, Papadopoulos F, Kitsak M, Vahdat A and Boguná M 2010 Hyperbolic geometry of complex networks Phys. Rev. E 82 036106

[169] Papadopoulos F, Kitsak M, Serrano M A, Boguná M and Krioukov D 2012 Popularity versus similarity in growing networks Nature 489 537–40

[170] Boguná M, Papadopoulos F and Krioukov D 2010 Sustaining the Internet with hyperbolic mapping Nat. Commun. 1 62

[171] Kitsak M, Voitalov I and Krioukov D 2020 Link prediction with hyperbolic geometry Phys. Rev. Res. 2 043113

[172] Faqeeh A, Osat S and Radicchi F 2018 Characterizing the Analogy Between Hyperbolic Embedding and Community Structure of Complex Networks Phys. Rev. Lett. 121 098301

[173] Zuev K, Boguná M, Bianconi G and Krioukov D 2015 Emergence of Soft Communities from Geometric Preferential Attachment Sci. Rep. 5 9421

[174] Nickel M and Kiela D 2017 Advances in Neural Information Processing Systems vol 30

[175] García-Pérez G, Allard A, Serrano M A and Boguñá M 2019 Mercator: uncovering faithful hyperbolic embeddings of complex networks New J. Phys. 21 123033

[176] Muscoloni A, Thomas J M, Ciucci S, Bianconi G and Cannistraci C V 2017 Machine learning meets complex networks via coalescent embedding in the hyperbolic space Nat. Commun. 8 1615

[177] Kovács B and Palla G 2021 Optimisation of the coalescent hyperbolic embedding of complex networks Sci. Rep. 11 8350

[178] Patania A, Allard A and Young J G 2023 arXiv:2301.10403

[179] Kitsak M, Ganin A, Elmokashfi A, Cui H, Eisenberg D A, Alderson D L, Korkin D and Linkov I 2023 Finding shortest and nearly shortest path nodes in large substantially incomplete networks by hyperbolic mapping Nat. Commun. 14 186

[180] Kovács B and Palla G 2023 Model-independent embedding of directed networks into Euclidean and hyperbolic spaces Commun. Phys. 6 28

[181] Sarkar R 2012 Low distortion delaunay embedding of trees in hyperbolic plane Graph Drawing: 19th Int. Symp., (GD 2011) (Eindhoven, The Netherlands, 21–23 September 2011) (Revised Selected Papers vol 19) (Springer) pp 355–66

[182] Sala F, De Sa C, Gu A and Ré C 2018 Representation tradeoffs for hyperbolic embeddings Int. Conf. on Machine Learning (PMLR) pp 4460–9

[183] Chami I, Wolf A, Juan D C, Sala F, Ravi S and Ré C 2020 arXiv:2005.00545

[184] Chami I, Ying Z, Ré C and Leskovec J 2019 Advances in Neural Information Processing Systems vol 32

[185] Fernández-Gracia J and Onnela J P 2019 Flexible model of network embedding Sci. Rep. 9 11710

[186] Massara G P, Di Matteo T and Aste T 2016 J. Complex Netw. 5 161–78

[187] Bianconi G and Rahmede C 2016 Network geometry with flavor: From complexity to quantum geometry Phys. Rev. E 93 032315

[188] Bianconi G and Rahmede C 2017 Sci. Rep. 7 1–9

[189] Tumminello M, Aste T, Di Matteo T and Mantegna R N 2005 A tool for filtering information in complex systems Proc. Natl Acad. Sci. 102 10421–6

[190] Clough J R and Evans T S 2017 Embedding graphs in Lorentzian spacetime PLoS One 12 e0187301

[191] Fanuel M, Alaíz C M, Fernández A and Suykens J A 2018 Magnetic eigenmaps for the visualization of directed networks Appl. Comput. Harmon. Anal. 44 189–99

[192] Gong X, Higham D J and Zygalakis K 2021 Directed network Laplacians and random graph models R. Soc. Open Sci. 8 211144

[193] Gong X, Higham D J and Zygalakis K 2023 Generative hypergraph models and spectral embedding Sci. Rep. 13 540
[194] Zhang X, He Y, Brugnone N, Perlmutter M and Hirn M 2021 *Advances in Neural Information Processing Systems* vol 34 pp 27003–15

[195] Böttcher L and Porter M A 2022 arXiv:2212.06257

[196] Singer A and Wu H T 2012 Vector diffusion maps and the connection Laplacian *Commun. Pure Appl. Math.* 65 1067–144

[197] Bodnar C, Di Giovanni F, Chamberlain B P, Liò P and Bronstein M M 2022 arXiv:2202.04579

[198] Barbero F, Bodnar C, de Ocáriz Borde H S, Bronstein M, Veličković P and Liò P 2022 Sheaf neural networks with connection Laplacians *Topological, Algebraic and Geometric Learning Workshops 2022* (PMLR) pp 28–36

[199] Krzakala F, Moore C, Mossel E, Neeman J, Sly A, Zdeborová L and Zhang P 2013 Spectral redemption in clustering sparse networks *Proc. Natl Acad. Sci.* 110 20935–40

[200] Torres L, Suárez-Serrato P and Eliassi-Rad T 2019 Non-backtracking cycles: length spectrum theory and graph mining *Appl. Netw. Sci.* 4 1–35

[201] Gu W, Tandon A, Ahn Y Y and Radicchi F 2021 Principled approach to the selection of the embedding dimension of networks *Nat. Commun.* 12 3772

[202] Zhang Y J, Yang K C and Radicchi F 2021 Systematic comparison of graph embedding methods in practical tasks *Phys. Rev. E* 104 044315

[203] Tandon A, Albeshri A, Thyananthan V, Alhalabi W, Radicchi F and Fortunato S 2021 Community detection in networks using graph embeddings *Phys. Rev. E* 103 022316

[204] Srinivasan B and Ribeiro B 2019 arXiv:1910.00452

[205] Macqueen J 1967 Some methods for classification and analysis of multivariate observations 5th Berkeley Symp. on Mathematical Statistics and Probability pp 281–97

[206] Kojaku S, Radicchi F, Ahn Y Y and Fortunato S 2023 Network community detection via neural embeddings (arXiv:2306.13400)

[207] Chami I, Gu A, Nguyen D P and Re C 2021 HoroPCA: hyperbolic dimensionality reduction via horospherical projections *Proc. 38th Int. Conf. on Machine Learning (Proc. Machine Learning Research* vol 139) ed M Meila and T Zhang (PMLR) pp 1419–29 (available at: https://proceedings.mlr.press/v139/chami21a.html)

[208] Rissaki A, Scarone B, Liu D, Pandey A, Klein B, Eliassi-Rad T and Borkin M A 2022 BiaScope: visual unfairness diagnosis for graph embeddings 2022 IEEE Visualization in Data Science (VDS) pp 27–36

[209] MacArthur B D, Sánchez-García R J and Anderson J W 2008 Symmetry in complex networks *Discrete Appl. Math.* 156 3525–31

[210] MacArthur B D and Sánchez-García R J 2009 Spectral characteristics of network redundancy *Phys. Rev. E* 80 026117

[211] Sánchez-García R J 2020 Exploiting symmetry in network analysis *Commun. Phys.* 3 87

[212] Wang J, Huang Y, Wu F X and Pan Y 2012 Symmetry compression method for discovering network motifs *IEEE/ACM Trans. Comput. Biol. Bioinform.* 9 1776–89

[213] Dai H, Dai B and Song L 2016 Discriminative embeddings of latent variable models for structured data *Proc. 33rd Int. Conf. on Int. Conf. on Machine Learning (ICML '16)* vol 48 (JMLR.org) pp 2702–11

[214] Duvenaud D K, Maclaurin D, Iparraguirre J, Bombarell R, Hirzel T, Aspuru-Guzik A and Adams R P 2015 Convolutional networks on graphs for learning molecular fingerprints *Advances in Neural Information Processing Systems* vol 28, ed C Cortes, N Lawrence, D Lee, M Sugiyama and R Garnett (Curran Associates, Inc.) (available at: https://proceedings.neurips.cc/paper/ 2015/file/f9be311e65d81a9ad8150a60844bb94c-paper.pdf)

[215] Kearnes S, McCloskey K, Berndl M, Pande V and Riley P 2016 Molecular graph convolutions: moving beyond fingerprints *J. Comput.-Aided Mol. Des.* 30 595–608

[216] Bruna J, Zaremba W, Szlam A D and LeCun Y 2014 CoRR (arXiv:1312.6203)

[217] Feng J, Huang M, Yang Y and Zhu X 2016 GAKE: graph aware knowledge embedding *Proc. COLING 2016, the 26th Int. Conf. on Computational Linguistics: Technical Papers* (The COLING 2016 Organizing Committee) pp 641–51 (available at: https://aclanthology.org/C16-1062)

[218] Fan J, Cannistra A, Fried I, Lim T, Schaffner T, Crovella M, Hescott B and Leiserson M D 2018 bioRxiv Preprint (22 March 2023)

[219] Heimann M, Shen H, Safavi T and Koutra D 2018 REGAL: representation learning-based graph alignment *Proc. 27th ACM Int. Conf. on Information and Knowledge Management (CIKM '18)* (Association for Computing Machinery) pp 117–26

[220] Fahimipour A K and Gross T 2020 Mapping the bacterial metabolic niche space *Nat. Commun.* 11 4887

[221] Diaz-Papkovich A, Anderson-Trocmé L, Ben-Eghan C and Gravel S 2019 UMAP reveals cryptic population structure and phenotype heterogeneity in large genomic cohorts *PLoS Genet.* 15 e1008432

[222] Han P, Yang P, Zhao P, Shang S, Liu Y, Zhou J, Gao X and Kalnis P 2019 GCN-MF: disease-gene association identification by graph convolutional networks and matrix factorization *Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '19)* (Association for Computing Machinery) pp 705–13

[223] Rotmensch M, Halpern Y, Tlimat A, Horng S and Sontag D 2017 Learning a Health Knowledge Graph from Electronic Medical Records *Sci. Rep.* 7 5994

[224] Wu T, Wang Y, Wang Y, Zhao E, Yuan Y and Yang Z 2019 arXiv:1910.02574

[225] Graham R L and Winkler P M 1985 On isometric embeddings of graphs *Trans. Am. Math. Soc.* 288 527–36

[226] Linial N, London E and Rabinovich Y 1995 The geometry of graphs and some of its algorithmic applications *Combinatorica* 15 215–45

[227] Linial N 2003 arXiv:math/0304466

[228] Indyk P, Matoušek J and Sidiropoulos A 2017 Low-distortion embeddings of finite metric spaces *Handbook of Discrete and Computational Geometry* (Chapman and Hall/CRC) pp 211–31

[229] Agrawal A, Ali A and Boyd S 2021 Minimum-Distortion Embedding *Found. Trends Mach. Learn.* 14 211–378
[194] Zhang X, He Y, Brugnone N, Perlmutter M and Hirn M 2021 *Advances in Neural Information Processing Systems* vol 34 pp 27003–15

[195] Böttcher L and Porter M A 2022 arXiv:2212.06257

[196] Singer A and Wu H T 2012 Vector diffusion maps and the connection Laplacian *Commun. Pure Appl. Math.* 65 1067–144

[197] Bodnar C, Di Giovanni F, Chamberlain B P, Liò P and Bronstein M M 2022 arXiv:2202.04579

[198] Barbero F, Bodnar C, de Ocáriz Borde H S, Bronstein M, Veličković P and Liò P 2022 Sheaf neural networks with connection Laplacians *Topological, Algebraic and Geometric Learning Workshops 2022* (PMLR) pp 28–36

[199] Krzakala F, Moore C, Mossel E, Neeman J, Sly A, Zdeborová L and Zhang P 2013 Spectral redemption in clustering sparse networks *Proc. Natl Acad. Sci.* 110 20935–40

[200] Torres L, Suárez-Serrato P and Eliassi-Rad T 2019 Non-backtracking cycles: length spectrum theory and graph mining *Appl. Netw. Sci.* 4 1–35

[201] Gu W, Tandon A, Ahn Y Y and Radicchi F 2021 Principled approach to the selection of the embedding dimension of networks *Nat. Commun.* 12 3772

[202] Zhang Y J, Yang K C and Radicchi F 2021 Systematic comparison of graph embedding methods in practical tasks *Phys. Rev. E* 104 044315

[203] Tandon A, Albeshri A, Thyananthan V, Alhalabi W, Radicchi F and Fortunato S 2021 Community detection in networks using graph embeddings *Phys. Rev. E* 103 022316

[204] Srinivasan B and Ribeiro B 2019 arXiv:1910.00452

[205] Macqueen J 1967 Some methods for classification and analysis of multivariate observations 5th Berkeley Symp. on Mathematical Statistics and Probability pp 281–97

[206] Kojaku S, Radicchi F, Ahn Y Y and Fortunato S 2023 Network community detection via neural embeddings (arXiv:2306.13400)

[207] Chami I, Gu A, Nguyen D P and Re C 2021 HoroPCA: hyperbolic dimensionality reduction via horospherical projections *Proc. 38th Int. Conf. on Machine Learning (Proc. Machine Learning Research* vol 139) ed M Meila and T Zhang (PMLR) pp 1419–29 (available at: https://proceedings.mlr.press/v139/chami21a.html)

[208] Rissaki A, Scarone B, Liu D, Pandey A, Klein B, Eliassi-Rad T and Borkin M A 2022 BiaScope: visual unfairness diagnosis for graph embeddings 2022 IEEE Visualization in Data Science (VDS) pp 27–36

[209] MacArthur B D, Sánchez-García R J and Anderson J W 2008 Symmetry in complex networks *Discrete Appl. Math.* 156 3525–31

[210] MacArthur B D and Sánchez-García R J 2009 Spectral characteristics of network redundancy *Phys. Rev. E* 80 026117

[211] Sánchez-García R J 2020 Exploiting symmetry in network analysis *Commun. Phys.* 3 87

[212] Wang J, Huang Y, Wu F X and Pan Y 2012 Symmetry compression method for discovering network motifs *IEEE/ACM Trans. Comput. Biol. Bioinform.* 9 1776–89

[213] Dai H, Dai B and Song L 2016 Discriminative embeddings of latent variable models for structured data *Proc. 33rd Int. Conf. on Int. Conf. on Machine Learning (ICML '16)* vol 48 (JMLR.org) pp 2702–11

[214] Duvenaud D K, Maclaurin D, Iparraguirre J, Bombarell R, Hirzel T, Aspuru-Guzik A and Adams R P 2015 Convolutional networks on graphs for learning molecular fingerprints *Advances in Neural Information Processing Systems* vol 28, ed C Cortes, N Lawrence, D Lee, M Sugiyama and R Garnett (Curran Associates, Inc.) (available at: https://proceedings.neurips.cc/paper/ 2015/file/f9be311e65d81a9ad8150a60844bb94c-paper.pdf)

[215] Kearnes S, McCloskey K, Berndl M, Pande V and Riley P 2016 Molecular graph convolutions: moving beyond fingerprints *J. Comput.-Aided Mol. Des.* 30 595–608

[216] Bruna J, Zaremba W, Szlam A D and LeCun Y 2014 CoRR (arXiv:1312.6203)

[217] Feng J, Huang M, Yang Y and Zhu X 2016 GAKE: graph aware knowledge embedding *Proc. COLING 2016, the 26th Int. Conf. on Computational Linguistics: Technical Papers* (The COLING 2016 Organizing Committee) pp 641–51 (available at: https://aclanthology.org/C16-1062)

[218] Fan J, Cannistra A, Fried I, Lim T, Schaffner T, Crovella M, Hescott B and Leiserson M D 2018 bioRxiv Preprint (22 March 2023)

[219] Heimann M, Shen H, Safavi T and Koutra D 2018 REGAL: representation learning-based graph alignment *Proc. 27th ACM Int. Conf. on Information and Knowledge Management (CIKM '18)* (Association for Computing Machinery) pp 117–26

[220] Fahimipour A K and Gross T 2020 Mapping the bacterial metabolic niche space *Nat. Commun.* 11 4887

[221] Diaz-Papkovich A, Anderson-Trocmé L, Ben-Eghan C and Gravel S 2019 UMAP reveals cryptic population structure and phenotype heterogeneity in large genomic cohorts *PLoS Genet.* 15 e1008432

[222] Han P, Yang P, Zhao P, Shang S, Liu Y, Zhou J, Gao X and Kalnis P 2019 GCN-MF: disease-gene association identification by graph convolutional networks and matrix factorization *Proc. 25th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (KDD '19)* (Association for Computing Machinery) pp 705–13

[223] Rotmensch M, Halpern Y, Tlimat A, Horng S and Sontag D 2017 Learning a Health Knowledge Graph from Electronic Medical Records *Sci. Rep.* 7 5994

[224] Wu T, Wang Y, Wang Y, Zhao E, Yuan Y and Yang Z 2019 arXiv:1910.02574

[225] Graham R L and Winkler P M 1985 On isometric embeddings of graphs *Trans. Am. Math. Soc.* 288 527–36

[226] Linial N, London E and Rabinovich Y 1995 The geometry of graphs and some of its algorithmic applications *Combinatorica* 15 215–45

[227] Linial N 2003 arXiv:math/0304466

[228] Indyk P, Matoušek J and Sidiropoulos A 2017 Low-distortion embeddings of finite metric spaces *Handbook of Discrete and Computational Geometry* (Chapman and Hall/CRC) pp 211–31

[229] Agrawal A, Ali A and Boyd S 2021 Minimum-Distortion Embedding *Found. Trends Mach. Learn.* 14 211–378