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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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 |
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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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DOI: https://doi.org/10.1007/s13278-021-00720-8