Centrality informed embedding of networks for temporal feature extraction


We propose a two-step methodology for exploring the temporal characteristics of a network. First, we construct a graph time series, where each snapshot is the result of a temporal whole-graph embedding. The embedding is carried out using the degree, Katz and betweenness centralities to characterize first and higher order proximities among vertices. Then a principal component analysis is performed over the collected temporal graph samples, which exhibits eigengraphs, graphs whose temporal weight variations model the sampled graph series. Analysis of the temporal timeline of each of the main eigengraphs reveals moments of importance in terms of structural graph changes. Parameters such as the dimension of the embeddings and the number of temporal samples are explored. Two case studies are presented: a Bitcoin subgraph, where findings are cross-checked by looking at the subgraph behavior itself, and the Enron email network, which allows us to compare our findings with prior studies. In both cases, the proposed methodology successfully identified temporal structural changes in the graph evolution.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. Akcora C, Kantarcioglu M, Gel Y (2018) Blockchain data analytics. In: IEEE International Conference on Data Mining (ICDM)

  2. Batagelj V, Praprotnik S (2016) An algebraic approach to temporal network analysis based on temporal quantities. Soc Netw Anal Min 6(28):1–22

    MATH  Google Scholar 

  3. Bourgain J (1985) On Lipschitz embedding of finite metric spaces in Hilbert space. Isr J Math 52(1–2):46–52

    MathSciNet  MATH  Article  Google Scholar 

  4. Cai H, Zheng VW, Chang KC-C (2018) A comprehensive survey of graph embedding: problems, techniques and applications. IEEE Trans Knowl Data Eng 30:1616–1637

    Article  Google Scholar 

  5. Estrada E (2011) The structure of complex networks: theory and applications. Oxford University Press Inc., Oxford

    Google Scholar 

  6. Fraiman D, Fraiman N, Fraiman R (2017) Nonparametric statistics of dynamic networks with distinguishable nodes. Test 26(3):546–573

    MathSciNet  MATH  Article  Google Scholar 

  7. Goyal P, Ferrara E (2018) Graph embedding techniques, applications, and performance: a survey. Knowl-Based Syst 151:78–94

    Article  Google Scholar 

  8. Goyal P, Kamra N, He X, Liu Y (2018) Dyngem: deep embedding method for dynamic graphs. arXiv preprint arXiv:1805.11273

  9. Harrigan M, Fretter C (2016) The unreasonable effectiveness of address clustering. Tech. Rep. arXiv:1605.06369, arXiv

  10. Holme P, Saramäki J (eds) (2013) Temporal networks. Springer, Berlin

    Google Scholar 

  11. Klimt B, Yang Y (2004) The enron corpus: a new dataset for email classification research. In: European Conference on Machine Learning

  12. Kondor D, Csabai I, Szüle J, Pósfai M, Vattay G (2014) Inferring the interplay between network structure and market effects in bitcoin. New J Phys 16:125003

    Article  Google Scholar 

  13. Narayanan A, Chandramohan M, Venkatesan R, Chen L, Liu Y, Jaiswal S (2017) graph2vec: learning distributed representations of graphs. Workshop on mining and learning with graphs

  14. Park Y, Priebe CE, Youssef A (2013) Anomaly detection in time series of graphs using fusion of graph invariants. IEEE J Sel Top Signal Process 7(1):67–75

    Article  Google Scholar 

  15. Peel L, Clauset A (2015) Detecting change points in the large-scale structure of evolving networks. In: Proceedings of the twenty-ninth AAAI conference on artifical intelligence

  16. Sarkar S, Guo R, Shakarian P (2019) Using network motifs to characterize temporal network evolution leading to diffusion inhibition. Soc Netw Anal Min 9(1):14

    Article  Google Scholar 

  17. Sirovich L, Kirby M (1987) Low-dimensional procedure for the characterization of human faces. J Opt Soc Am A 4(3):519–524

    Article  Google Scholar 

  18. Ye C, Wilson RC, Hancock ER (2018) Network analysis using entropy component analysis. J Complex Netw 6(3):404–429

    MathSciNet  Article  Google Scholar 

  19. Zhu L, Guo D, Yin J, Ver Steeg G, Galstyan A (2016) Scalable temporal latent space inference for link prediction in dynamic social networks. IEEE Trans Knowl Data Eng 28(10):2765–2777

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Anwitaman Datta.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Enron scandal—chronology of events

Appendix: Enron scandal—chronology of events

Timeline Event (non-exhaustive) summary
November Launch of EnronOnline, a global commodity trading web site
January Launch of Enron Broadband Service (EBS)
19 July Announcement of EBS joining forces with Blockbuster
23 August Enron stock reached all time high price of $90.75
3 October Enron attorney discussed Timothy Belden’s strategies
1 November FERC investigation exonerated Enron
December At end of 2000, Enron declared $53 million earnings for Broadband
13 December Announcement that Jeffrey Skilling would take over as CEO
17 January Rolling blackouts in California
1 February\(^\dagger \) State lawmakers legislate to spend up to $10 billion for power
12 February\(^\dagger \) Skilling was named CEO of Enron, replacing Lay
23 March Enron conference call with analysts to boost stock
17 April The ‘asshole” call: Jeffrey Skilling response to an analyst query
15 May\(^\dagger \) California energy regulators adopted the highest rate increase in the state’s history
17 May\(^\dagger \) California energy regulators uncovered evidence that some electrical power companies repeatedly shut down generating plants for unnecessary maintenance
26 May Schwarzenegger, Lay, Milken meeting
5 June\(^\dagger \) Karl Rove divested his stocks in energy, defense and pharmaceutical companies (including Enron)
11/12 July Quarterly conference call
24-25 July Skilling met analysts and investors in NY
14 August\(^\dagger \) Skilling resigned; Lay named CEO again
22 August Sherron Watkins met Lay to discuss accounting irregularities
16 October\(^\dagger \) Enron announces $638 million in third-quarter losses
19 October\(^\dagger \) Securities and Exchange Commission launches inquiry
23 October\(^\dagger \) Lay professes confidence in Fastow to analysts
24 October\(^\dagger \) Fastow ousted
9 November Dynegy Inc. announced an agreement to buy Enron
19 November Enron restated its 3rd-quarter earnings disclosing $690M debt
28 November Dynegy called off its $8.4B merger with Enron
  Enron stock plunged below $1
2 December Enron Corp. under CEO Kenneth Lay filed for bankruptcy
29 January Stephen Cooper took over as interim Enron CEO
5 February Lay cancelled senate committee appearance invoking the 5th
  Fastow, Kopper, Lay invoked the 5th
7 February Skilling testified
  Fastow and Kopper invoked the 5th
14 February Sharon Watkins testified
14 March Former Enron auditor Arthur Andersen LLP indicted
  1. \(^\dagger \)Events not considered in Peel and Clauset (2015)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Oggier, F., Datta, A. Centrality informed embedding of networks for temporal feature extraction. Soc. Netw. Anal. Min. 11, 12 (2021). https://doi.org/10.1007/s13278-021-00720-8

Download citation


  • Social network dynamics
  • Time series of graphs
  • Dynamic network
  • Centrality
  • Graph embedding