research-article

Using Cliques with Higher-order Spectral Embeddings Improves Graph Visualizations

Authors:

Caitlin Kennedy,

Austin R. Benson,

David GleichAuthors Info & Claims

WWW '20: Proceedings of The Web Conference 2020

Pages 2927 - 2933

https://doi.org/10.1145/3366423.3380059

Published: 20 April 2020 Publication History

Abstract

In the simplest setting, graph visualization is the problem of producing a set of two-dimensional coordinates for each node that meaningfully shows connections and latent structure in a graph. Among other uses, having a meaningful layout is often useful to help interpret the results from network science tasks such as community detection and link prediction. There are several existing graph visualization techniques in the literature that are based on spectral methods, graph embeddings, or optimizing graph distances. Despite the large number of methods, it is still often challenging or extremely time consuming to produce meaningful layouts of graphs with hundreds of thousands of vertices. Existing methods often either fail to produce a visualization in a meaningful time window, or produce a layout colorfully called a “hairball”, which does not illustrate any internal structure in the graph. Here, we show that adding higher-order information based on cliques to a classic eigenvector based graph visualization technique enables it to produce meaningful plots of large graphs. We further evaluate these visualizations along a number of graph visualization metrics and we find that it outperforms existing techniques on a metric that uses random walks to measure the local structure. Finally, we show many examples of how our algorithm successfully produces layouts of large networks. Code to reproduce our results is available.

References

[1]

Alex Adai, Shailesh Date, Shannon Wieland, and Edward Marcotte. 2004. LGL: Creating a Map of Protein Function With an Algorithm for Visualizing Very Large Biological Networks. Journal of molecular biology 340 (07 2004), 179–90. https://doi.org/10.1016/j.jmb.2004.04.047

[2]

Réka Albert, Hawoong Jeong, and Albert-László Barabási. 1999. Internet: Diameter of the world-wide web. nature 401, 6749 (1999), 130.

[3]

David Auber, Daniel Archambault, Romain Bourqui, Maylis Delest, Jonathan Dubois, Antoine Lambert, Patrick Mary, Morgan Mathiaut, Guy Melançon, Bruno Pinaud, Benjamin Renoust, and Jason Vallet. 2017. TULIP 5. In Encyclopedia of Social Network Analysis and Mining, Reda Alhajj and Jon Rokne (Eds.). Springer, 1–28. https://doi.org/10.1007/978-1-4614-7163-9_315-1

[4]

Mathieu Bastian, Sebastien Heymann, and Mathieu Jacomy. 2009. Gephi: An Open Source Software for Exploring and Manipulating Networks. (2009). http://www.aaai.org/ocs/index.php/ICWSM/09/paper/view/154

[5]

Vladimir Batagelj and Andrej Mrvar. [n.d.]. Pajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/data/

[6]

Austin Benson, David F. Gleich, and Jure Leskovec. 2016. Higher-order organization of complex networks. Submitted. Science 353, 6295 (2016), 163–166. https://doi.org/10.1126/science.aad9029

[7]

Paolo Boldi, Marco Rosa, Massimo Santini, and Sebastiano Vigna. 2011. Layered Label Propagation: A Multiresolution Coordinate-free Ordering for Compressing Social Networks. In Proceedings of the 20th International Conference on World Wide Web (Hyderabad, India) (WWW ’11). ACM, New York, NY, USA, 587–596. https://doi.org/10.1145/1963405.1963488

Digital Library

[8]

P. Boldi and S. Vigna. 2004. The Webgraph Framework I: Compression Techniques. In Proceedings of the 13th International Conference on World Wide Web (New York, NY, USA) (WWW ’04). ACM, New York, NY, USA, 595–602. https://doi.org/10.1145/988672.988752

Digital Library

[9]

Fan R. L. Chung. 1992. Spectral Graph Theory. American Mathematical Society.

Digital Library

[10]

Darren Edge, Jonathan Larson, Markus Mobius, and Christopher White. 2018. Trimming the Hairball: Edge Cutting Strategies for Making Dense Graphs Usable. In 2018 IEEE International Conference on Big Data (Big Data). IEEE, 3951–3958.

[11]

Alessandro Epasto and Bryan Perozzi. 2019. Is a Single Embedding Enough? Learning Node Representations that Capture Multiple Social Contexts. CoRR abs/1905.02138(2019). arxiv:1905.02138http://arxiv.org/abs/1905.02138

[12]

Miroslav Fiedler. 1989. Laplacian of graphs and algebraic connectivity. Banach Center Publications 25, 1 (1989), 57–70. http://eudml.org/doc/267812

[13]

Nils Gehlenborg, Seán I. O’Donoghue, Nitin S. Baliga, Alexander Goesmann, Matthew A. Hibbs, Hiroaki Kitano, Oliver Kohlbacher, Heiko Neuweger, Reinhard Schneider, Dan Tenenbaum, and Anne-Claude Gavin. 2010. Visualization of omics data for systems biology. Nature Methods 7, 3 (2010), S56–S68. https://doi.org/10.1038/nmeth.1436

[14]

Aditya Grover and Jure Leskovec. 2016. node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 855–864.

Digital Library

[15]

Shweta Jain and C. Seshadhri. 2017. A Fast and Provable Method for Estimating Clique Counts Using TuráN’s Theorem. In Proceedings of the 26th International Conference on World Wide Web (Perth, Australia) (WWW ’17). International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, Switzerland, 441–449. https://doi.org/10.1145/3038912.3052636

[16]

Stephen G Kobourov. 2012. Spring embedders and force directed graph drawing algorithms. arXiv preprint arXiv:1201.3011(2012).

[17]

Martin Krzywinski, Inanc Birol, Steven JM Jones, and Marco A Marra. 2011. Hive plots—rational approach to visualizing networks. Briefings in Bioinformatics 13, 5 (12 2011), 627–644. https://doi.org/10.1093/bib/bbr069 arXiv:http://oup.prod.sis.lan/bib/article-pdf/13/5/627/1147399/bbr069.pdf

[18]

Martin Krzywinski, Inanc Birol, Steven JM Jones, and Marco A Marra. 2011. Hive plots—rational approach to visualizing networks. Briefings in Bioinformatics 13, 5 (12 2011), 627–644. https://doi.org/10.1093/bib/bbr069 arXiv:http://oup.prod.sis.lan/bib/article-pdf/13/5/627/1147399/bbr069.pdf

[19]

Jérôme Kunegis, Stephan Schmidt, Andreas Lommatzsch, Jürgen Lerner, Ernesto William De Luca, and Sahin Albayrak. 2010. Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization. In Proceedings of the SIAM International Conference on Data Mining, SDM 2010, April 29 - May 1, 2010, Columbus, Ohio, USA. 559–570. https://doi.org/10.1137/1.9781611972801.49

[20]

R. B. Lehoucq and D. C. Sorensen. 1996. Deflation Techniques for an Implicitly Restarted Arnoldi Iteration. SIAM J. Matrix Anal. Appl. 17, 4 (1996), 789–821. https://doi.org/10.1137/S0895479895281484

Digital Library

[21]

Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. 2005. Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations. In Proceedings of the Eleventh ACM SIGKDD International Conference on Knowledge Discovery in Data Mining (Chicago, Illinois, USA) (KDD ’05). ACM, New York, NY, USA, 177–187. https://doi.org/10.1145/1081870.1081893

Digital Library

[22]

Laurens van der Maaten and Geoffrey Hinton. 2008. Visualizing data using t-SNE. Journal of machine learning research 9, Nov (2008), 2579–2605.

[23]

Shawn Martin, W. Michael Brown, and Brian N. Wylie. 2007. Dr.L: Distributed Recursive (Graph) Layout, Version 00. https://www.osti.gov//servlets/purl/1231060

[24]

Huda Nassar, Austin R. Benson, and David F. Gleich. 2019. Pairwise Link Prediction. CoRR abs/1907.04503(2019). arxiv:1907.04503http://arxiv.org/abs/1907.04503

[25]

Matthew Richardson, Rakesh Agrawal, and Pedro Domingos. 2003. Trust Management for the Semantic Web. In Proceedings of the Second International Conference on Semantic Web Conference (Sanibel Island, FL) (LNCS-ISWC’03). Springer-Verlag, Berlin, Heidelberg, 351–368. https://doi.org/10.1007/978-3-540-39718-2_23

Digital Library

[26]

Matthew Suderman and Michael Hallett. 2007. Tools for visually exploring biological networks. Bioinformatics 23, 20 (08 2007), 2651–2659. https://doi.org/10.1093/bioinformatics/btm401 arXiv:http://oup.prod.sis.lan/bioinformatics/article-pdf/23/20/2651/16858965/btm401.pdf

[27]

Amanda L. Traud, Peter J. Mucha, and Mason A. Porter. 2011. Social Structure of Facebook Networks. CoRR abs/1102.2166(2011). arxiv:1102.2166

[28]

Christo Wilson, Bryce Boe, Alessandra Sala, Krishna P.N. Puttaswamy, and Ben Y. Zhao. 2009. User interactions in social networks and their implications. In Proceedings of the 4th ACM European conference on Computer systems (Nuremberg, Germany) (EuroSys ’09). ACM, New York, NY, USA, 205–218. https://doi.org/10.1145/1519065.1519089

Digital Library

[29]

Naganand Yadati, Madhav Nimishakavi, Prateek Yadav, Anand Louis, and Partha Pratim Talukdar. 2018. HyperGCN: Hypergraph Convolutional Networks for Semi-Supervised Classification. CoRR abs/1809.02589(2018). arxiv:1809.02589http://arxiv.org/abs/1809.02589

[30]

Jaewon Yang and Jure Leskovec. 2012. Defining and Evaluating Network Communities Based on Ground-truth. In Proceedings of the ACM SIGKDD Workshop on Mining Data Semantics (Beijing, China) (MDS ’12). ACM, New York, NY, USA, Article 3, 8 pages. https://doi.org/10.1145/2350190.2350193

Digital Library

[31]

Ziwei Zhang, Peng Cui, Xiao Wang, Jian Pei, Xuanrong Yao, and Wenwu Zhu. 2018. Arbitrary-Order Proximity Preserved Network Embedding. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM.

Digital Library

Cited By

Becker MNassar HEspinosa CStelzer IFeyaerts DBerson EBidoki NChang ASaarunya GCulos ADe Francesco DFallahzadeh RLiu QKim YMarić IMataraso SPayrovnaziri SPhongpreecha TRavindra NStanley NShome STan YThuraiappah MXenochristou MXue LShaw GStevenson DAngst MGaudilliere BAghaeepour N(2023)Large-scale correlation network construction for unraveling the coordination of complex biological systemsNature Computational Science10.1038/s43588-023-00429-y3:4(346-359)Online publication date: 13-Apr-2023
https://doi.org/10.1038/s43588-023-00429-y
Fu DHe J(2022)Natural and Artificial Dynamics in Graphs: Concept, Progress, and FutureFrontiers in Big Data10.3389/fdata.2022.10626375Online publication date: 2-Dec-2022
https://doi.org/10.3389/fdata.2022.1062637
Kapralov MMakarov MSilwal SSohler CTardos J(2022)Motif Cut Sparsifiers2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00044(389-398)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00044
Show More Cited By

Index Terms

Using Cliques with Higher-order Spectral Embeddings Improves Graph Visualizations

Index terms have been assigned to the content through auto-classification.

Recommendations

On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-...
(d+1,2) -Track Layout of Bipartite Graph Subdivisions

A ( k ,2- track layout of a graph G consists of a 2- track assignment of G and an edge k -coloring of G with no monochromatic X-crossing . This paper studies the problem of ( k ,2)-track layout of bipartite graph subdivisions. Recently V.Dujmović and ...
Queue Layout of Bipartite Graph Subdivisions

For an integer d > 0, a d-queue layout of a graph consists of a total order of the vertices, and a partition of the edges into d sets of non-nested edges with respect to the vertex ordering. Recently V. Dujmović and D. R. Wood showed that for every ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

WWW '20: Proceedings of The Web Conference 2020

April 2020

3143 pages

ISBN:9781450370233

DOI:10.1145/3366423

Editors:
Yennun Huang
Acadmica sinica, Taiwan
,
Irwin King
The Chinese University of Hong Kong, Hong Kong
,
Tie-Yan Liu
Microsoft Research Asia, China
,
Maarten van Steen
University of Twente, Netherlands

Copyright © 2020 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGWEB: ACM Special Interest Group on Hypertext, Hypermedia, and Web

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 20 April 2020

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed limited

Conference

WWW '20

Sponsor:

SIGWEB

WWW '20: The Web Conference 2020

April 20 - 24, 2020

Taipei, Taiwan

Acceptance Rates

Overall Acceptance Rate 1,899 of 8,196 submissions, 23%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
362
Total Downloads

Downloads (Last 12 months)11
Downloads (Last 6 weeks)3

Reflects downloads up to 02 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Becker MNassar HEspinosa CStelzer IFeyaerts DBerson EBidoki NChang ASaarunya GCulos ADe Francesco DFallahzadeh RLiu QKim YMarić IMataraso SPayrovnaziri SPhongpreecha TRavindra NStanley NShome STan YThuraiappah MXenochristou MXue LShaw GStevenson DAngst MGaudilliere BAghaeepour N(2023)Large-scale correlation network construction for unraveling the coordination of complex biological systemsNature Computational Science10.1038/s43588-023-00429-y3:4(346-359)Online publication date: 13-Apr-2023
https://doi.org/10.1038/s43588-023-00429-y
Fu DHe J(2022)Natural and Artificial Dynamics in Graphs: Concept, Progress, and FutureFrontiers in Big Data10.3389/fdata.2022.10626375Online publication date: 2-Dec-2022
https://doi.org/10.3389/fdata.2022.1062637
Kapralov MMakarov MSilwal SSohler CTardos J(2022)Motif Cut Sparsifiers2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00044(389-398)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00044
Tudisco FBenson AProkopchik K(2021)Nonlinear Higher-Order Label SpreadingProceedings of the Web Conference 202110.1145/3442381.3450035(2402-2413)Online publication date: 19-Apr-2021
https://dl.acm.org/doi/10.1145/3442381.3450035

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Figures

Tables

Media

View Table of Conten