{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T14:40:44Z","timestamp":1773758444334,"version":"3.50.1"},"reference-count":51,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2020,3,7]],"date-time":"2020-03-07T00:00:00Z","timestamp":1583539200000},"content-version":"vor","delay-in-days":1,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS-1741197"],"award-info":[{"award-number":["IIS-1741197"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Graph embedding seeks to build a low-dimensional representation of a graph $G$. This low-dimensional representation is then used for various downstream tasks. One popular approach is Laplacian Eigenmaps (LE), which constructs a graph embedding based on the spectral properties of the Laplacian matrix of $G$. The intuition behind it, and many other embedding techniques, is that the embedding of a graph must respect node similarity: similar nodes must have embeddings that are close to one another. Here, we dispose of this distance-minimization assumption. Instead, we use the Laplacian matrix to find an embedding with geometric properties instead of spectral ones, by leveraging the so-called simplex geometry of $G$. We introduce a new approach, Geometric Laplacian Eigenmap Embedding, and demonstrate that it outperforms various other techniques (including LE) in the tasks of graph reconstruction and link prediction.<\/jats:p>","DOI":"10.1093\/comnet\/cnaa007","type":"journal-article","created":{"date-parts":[[2020,1,22]],"date-time":"2020-01-22T12:15:03Z","timestamp":1579695303000},"source":"Crossref","is-referenced-by-count":39,"title":["GLEE: Geometric Laplacian Eigenmap Embedding"],"prefix":"10.1093","volume":"8","author":[{"given":"Leo","family":"Torres","sequence":"first","affiliation":[{"name":"Network Science Institute, Northeastern University, Boston, MA 02115, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kevin S","family":"Chan","sequence":"additional","affiliation":[{"name":"U.S. CCDC Army Research Laboratory, Adelphi, MD 20783, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tina","family":"Eliassi-Rad","sequence":"additional","affiliation":[{"name":"Network Science Institute and Khoury College of Computer Sciences, Northeastern University, Boston, MA 02115, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2020,3,6]]},"reference":[{"key":"2021031102343501800_B1","article-title":"A tutorial on network embeddings","author":"Chen,","year":"2018","journal-title":"Preprint"},{"key":"2021031102343501800_B2","article-title":"Fast, warped graph embedding: unifying framework and one-click algorithm","author":"Chen,","year":"2017","journal-title":"CoRR"},{"key":"2021031102343501800_B3","article-title":"On the equivalence between node embeddings and structural graph representations","author":"Srinivasan,","year":"2019","journal-title":"CoRR"},{"key":"2021031102343501800_B4","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1145\/3292500.3330992","author":"Jin,","year":"2019","journal-title":"Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD 2019"},{"key":"2021031102343501800_B5","doi-asserted-by":"crossref","DOI":"10.1093\/comnet\/cny036","article-title":"The simplex geometry of graphs","author":"Devriendt,","year":"2019","journal-title":"J. Compl. Netw."},{"key":"2021031102343501800_B6","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511973611","author":"Fiedler,","year":"2011","journal-title":"Matrices and Graphs in Geometry, vol. 139 of Encyclopedia of Mathematics and its Applications"},{"key":"2021031102343501800_B7","first-page":"585","author":"Belkin,","year":"2002","journal-title":"Advances in Neural Information Processing Systems 14, NIPS 2014"},{"key":"2021031102343501800_B8","doi-asserted-by":"crossref","first-page":"1373","DOI":"10.1162\/089976603321780317","article-title":"Laplacian eigenmaps for dimensionality reduction and data representation","volume":"15","author":"Belkin,","year":"2003","journal-title":"Neural Comput."},{"key":"2021031102343501800_B9","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1086\/jar.33.4.3629752","article-title":"An information flow model for conflict and fission in small groups","volume":"33","author":"Zachary,","year":"1977","journal-title":"J. Anthropol. Res."},{"key":"2021031102343501800_B10","first-page":"2722","article-title":"Theoretical justification of popular link prediction heuristics","author":"Sarkar,","year":"2011","journal-title":"Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011"},{"key":"2021031102343501800_B11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41467-019-09177-y","article-title":"Network-based prediction of protein interactions","volume":"10","author":"Kov\u00e1cs,","year":"2019","journal-title":"Nat. Commun."},{"key":"2021031102343501800_B12","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1137\/090771806","article-title":"Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions","volume":"53","author":"Halko,","year":"2011","journal-title":"SIAM Rev."},{"key":"2021031102343501800_B13","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719574","volume":"50","author":"Trefethen,","year":"1997","journal-title":"Numerical Linear Algebra"},{"key":"2021031102343501800_B14","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198805090.001.0001","author":"Newman,","year":"2018","journal-title":"Networks"},{"key":"2021031102343501800_B15","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1090\/bull\/1557","article-title":"Graphs, vectors, and matrices","volume":"54","author":"Spielman,","year":"2017","journal-title":"Bull. Am. Math. Soc."},{"key":"2021031102343501800_B16","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511921681","author":"Van Mieghem,","year":"2010","journal-title":"Graph Spectra for Complex Networks"},{"key":"2021031102343501800_B17","doi-asserted-by":"crossref","first-page":"1913","DOI":"10.1137\/080734029","article-title":"Graph sparsification by effective resistances","volume":"40","author":"Spielman,","year":"2011","journal-title":"SIAM J. Comput."},{"key":"2021031102343501800_B18","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s11222-007-9033-z","article-title":"A tutorial on spectral clustering","volume":"17","author":"Von Luxburg,","year":"2007","journal-title":"Stat. Comput."},{"key":"2021031102343501800_B19","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s10115-013-0671-5","article-title":"Efficiently spotting the starting points of an epidemic in a large graph","volume":"38","author":"Prakash,","year":"2014","journal-title":"Knowl. Inform. Syst."},{"key":"2021031102343501800_B20","article-title":"An upper bound for the epidemic threshold in exact Markovian SIR and SIS epidemics on networks","author":"Van Mieghem,","year":"2014","journal-title":"53rd IEEE Conference on Decision and Control, CDC 2014"},{"key":"2021031102343501800_B21","article-title":"On the robustness of complex networks by using the algebraic connectivity","author":"Jamakovic,","year":"2008","journal-title":"NETWORKING 2008 Ad Hoc and Sensor Networks, Wireless Networks, Next Generation Internet: 7th International IFIP-TC6 Networking Conference"},{"key":"2021031102343501800_B22","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.ifacol.2015.10.339","article-title":"Robustness and algebraic connectivity of random interdependent networks","volume":"48","author":"Shahrivar,","year":"2015","journal-title":"IFAC-PapersOnLine"},{"key":"2021031102343501800_B23","doi-asserted-by":"crossref","first-page":"1867","DOI":"10.1016\/j.camwa.2004.08.015","article-title":"Drawing graphs by eigenvectors: theory and practice","volume":"49","author":"Koren,","year":"2005","journal-title":"Comput. Math. Appl."},{"key":"2021031102343501800_B24","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1021\/ci9900938","article-title":"Characterizing graph drawing with eigenvectors","volume":"40","author":"Pisanski,","year":"2000","journal-title":"J. Chem. Inform. Comput. Sci."},{"key":"2021031102343501800_B25","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.knosys.2018.03.022","article-title":"Graph embedding techniques, applications, and performance: a survey","volume":"151","author":"Goyal,","year":"2018","journal-title":"Knowl.-Based Syst."},{"key":"2021031102343501800_B26","first-page":"52","article-title":"Representation learning on graphs: methods and applications","volume":"40","author":"Hamilton,","year":"2017","journal-title":"IEEE Data Eng. Bull."},{"key":"2021031102343501800_B27","article-title":"Incrementally updated spectral embeddings","author":"Charisopoulos,","year":"2019","journal-title":"CoRR"},{"key":"2021031102343501800_B28","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1137\/1.9781611974010.63","article-title":"Fast eigen-functions tracking on dynamic graphs","author":"Chen,","year":"2015","journal-title":"Proceedings of the 2015 SIAM International Conference on Data Mining, SDM 2015"},{"key":"2021031102343501800_B29","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1002\/sam.11310","article-title":"On the eigen-functions of dynamic graphs: fast tracking and attribution algorithms","volume":"10","author":"Chen,","year":"2017","journal-title":"Stat. Anal. Data Mining"},{"key":"2021031102343501800_B30","first-page":"2981","article-title":"Out-of-sample extension of graph adjacency spectral embedding","volume":"80","author":"Levin,","year":"2018","journal-title":"Proceedings of the 35th International Conference on Machine Learning, ICML 2018"},{"key":"2021031102343501800_B31","first-page":"37","article-title":"Distributed large-scale natural graph factorization","author":"Ahmed,","year":"2013","journal-title":"22nd International Conference World Wide Web Conference, WWW 2013"},{"key":"2021031102343501800_B32","doi-asserted-by":"crossref","first-page":"1548","DOI":"10.1109\/TPAMI.2010.231","article-title":"Graph regularized nonnegative matrix factorization for data representation","volume":"33","author":"Cai,","year":"2011","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"2021031102343501800_B33","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1137\/1.9781611972825.10","article-title":"Symmetric nonnegative matrix factorization for graph clustering","author":"Kuang,","year":"2012","journal-title":"SDM. Proceedings of the 12th SIAM International Conference on Data Mining, SDM 2012"},{"key":"2021031102343501800_B34","first-page":"203","article-title":"Community preserving network embedding","author":"Wang,","year":"2017","journal-title":"AAAI. Proceedings of the 31st AAAI Conference on Artificial Intelligence"},{"key":"2021031102343501800_B35","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1145\/2623330.2623732","article-title":"DeepWalk: online learning of social representations","author":"Perozzi,","year":"2014","journal-title":"The 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2014"},{"key":"2021031102343501800_B36","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1145\/2939672.2939754","article-title":"node2vec: scalable feature learning for networks","author":"Grover,","year":"2016","journal-title":"Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2016"},{"key":"2021031102343501800_B37","first-page":"3111","article-title":"Distributed representations of words and phrases and their compositionality","author":"Mikolov,","year":"2013","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2021031102343501800_B38","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1145\/3159652.3159706","article-title":"Network embedding as matrix factorization: unifying DeepWalk, LINE, PTE, and node2vec","author":"Qiu,","year":"2018","journal-title":"WSDM"},{"key":"2021031102343501800_B39","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1145\/2939672.2939753","article-title":"Structural deep network embedding","author":"Wang,","year":"2016","journal-title":"KDD"},{"key":"2021031102343501800_B40","first-page":"1145","article-title":"Deep neural networks for learning graph representations","author":"Cao,","year":"2016","journal-title":"Proceedings of the 13th AAAI Conference on Artificial Intelligence"},{"key":"2021031102343501800_B41","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.dam.2013.05.032","article-title":"Hyperspherical embedding of graphs and networks in communicability spaces","volume":"176","author":"Estrada,","year":"2014","journal-title":"Discrete Appl. Math."},{"key":"2021031102343501800_B42","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.patcog.2018.09.018","article-title":"Machine learning analysis of complex networks in hyperspherical space","volume":"86","author":"Pereda,","year":"2019","journal-title":"Patt. Recogn."},{"key":"2021031102343501800_B43","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1038\/nature11459","article-title":"Popularity versus similarity in growing networks","volume":"489","author":"Papadopoulos,","year":"2012","journal-title":"Nature"},{"key":"2021031102343501800_B44","first-page":"3776","article-title":"Learning continuous hierarchies in the Lorentz model of hyperbolic geometry","volume":"80","author":"Nickel,","year":"2018","journal-title":"Proceedings of the 35th International Conference on Machine Learning, ICML 2018"},{"key":"2021031102343501800_B45","author":"Torres,","year":"2019","journal-title":"GLEE: Geometric Laplacian Eigenmap Embedding"},{"key":"2021031102343501800_B46","doi-asserted-by":"crossref","first-page":"1212","DOI":"10.1016\/j.cell.2014.10.050","article-title":"A proteome-scale map of the human interactome network","volume":"159","author":"Rolland,","year":"2014","journal-title":"Cell"},{"key":"2021031102343501800_B47","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1145\/1772690.1772756","article-title":"Predicting positive and negative links in online social networks","author":"Leskovec,","year":"2010","journal-title":"Proceedings of the 19th International Conference on World Wide Web, WWW 2010"},{"key":"2021031102343501800_B48","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1145\/1217299.1217301","article-title":"Graph evolution: densification and shrinking diameters","volume":"1","author":"Leskovec,","year":"2007","journal-title":"ACM Trans. Knowl. Discov. Data"},{"key":"2021031102343501800_B49","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","author":"Erd\u00f6s,","year":"1960","journal-title":"Publ. Math. Inst. Hung. Acad. Sci."},{"key":"2021031102343501800_B50","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Barab\u00e1si,","year":"1999","journal-title":"Science"},{"key":"2021031102343501800_B51","doi-asserted-by":"crossref","first-page":"036106","DOI":"10.1103\/PhysRevE.82.036106","article-title":"Hyperbolic geometry of complex networks","volume":"82","author":"Krioukov,","year":"2010","journal-title":"Phys. Rev. E"}],"container-title":["Journal of Complex Networks"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/comnet\/article-pdf\/8\/2\/cnaa007\/32817515\/cnaa007.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/comnet\/article-pdf\/8\/2\/cnaa007\/32817515\/cnaa007.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,25]],"date-time":"2023-09-25T13:31:54Z","timestamp":1695648714000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/comnet\/article\/doi\/10.1093\/comnet\/cnaa007\/5775302"}},"subtitle":[],"editor":[{"given":"Ernesto","family":"Estrada","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2020,3,6]]},"references-count":51,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,4,1]]}},"URL":"https:\/\/doi.org\/10.1093\/comnet\/cnaa007","relation":{},"ISSN":["2051-1310","2051-1329"],"issn-type":[{"value":"2051-1310","type":"print"},{"value":"2051-1329","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2020,4]]},"published":{"date-parts":[[2020,3,6]]},"article-number":"cnaa007"}}