Centrality informed embedding of networks for temporal feature extraction

Abstract

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.

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Correspondence to Anwitaman Datta.

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Appendix: Enron scandal—chronology of events

Appendix: Enron scandal—chronology of events

Timeline Event (non-exhaustive) summary
1999  
November Launch of EnronOnline, a global commodity trading web site
2000  
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
2001  
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
2002  
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)

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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

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Keywords

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