Comparison of Elementary Dynamic Network Models Using Empirical Data

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Inspecting dynamics of networks opens up a new dimension in the understanding or mechanisms behind real-world systems. Involving the time factor may help identifying previously hidden (or otherwise …

Inspecting dynamics of networks opens up a new dimension in the understanding or mechanisms behind real-world systems. Involving the time factor may help identifying previously hidden (or otherwise hard to recognize) phenomenon and/or patterns compared to static analysis, like individuals periodically changing between groups within a community.

Concentrating on edge dynamics, we defined a set of dynamic network models with various rules (including creating new and relinking edges randomly, by using assortative mixing or preferential attachment strategies) to analyize the evolution of different network properties. Starting from an initial network created by classical network models (like the Erdos-Renyi model) we examined the evolution of basic structural network properties (including density, clustering, average path length, number of components, degree distribution and betweennes centralities). The structure of the snapshot network (i.e., the network that is actually observed in a given instant of time) and the cumulative network (i.e., the network that is constructed by collecting and aggregating several samples of snapshot networks over a period of time) is inherently different, but we also found that certain properties have a strong dependence on the sampling windows length: we made experiments through computer simulations with various aggregation time windows and found that it has a great impact on the results.

In our presentation, we would like to briefly introduce the key findings of our previous results regarding to the elementary dynamic network models, and compare the theoretical results obtained from evaluating different empirical data sets. The selected data sets used for the comparison include political event data compiled from English-language news reports and a dataset created to analyze internet-mediated sexual encounters in Brazil.

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  • 1. Richárd O. Legéndi, László GulyásEötvös Loránd University, RegionalKnowledge CentreAITIA International, Inc,rlegendi@aitia.ai, lgulyas@aitia.aiComparison of ElementaryDynamic Network Models UsingEmpirical DataThis work was partially supported by the European Union and the European SocialFund through project FuturICT.hu (grant no.: TÁMOP-4.2.2.C-11/1/KONV-2012-0013).TDN 2013, NetSci SatelliteCopenhagen, June 3-4, 2013
  • 2. Outline Motivation EDNs Previous results Comparison to empirical datasets Conclusion2013.06.03.TDN 2013 @ NetSci
  • 3. A Practical Problem with DynamicNetworksThe importance ofthe samplingwindow...∆t
  • 4. Life is about change...Elementary Models ofDynamic Networks
  • 5. Elementary Dynamic Networks Growing Networks Shrinking Networks Networks of Constant Size
  • 6. Elementary Dynamic Networks Growing Networks Shrinking Networks Networks of Constant Size Node set is fixed Edge set (unweighted, undirected)is changing about the sameconstant size
  • 7. Definitions Snapshot network (@t) The network at any single t moment in time.(Using the finest possible granularity available in the model) Cumulative network (@[t, t+T]) The union of snapshot networks(collected over the specified interval of time) Typically over the [0,T] interval in our studies Summation network (@[t, t+T]) The sum of snapshot networks(collected over the specified interval of time) Typically yields multi-nets
  • 8. DefinitionsSnapsott=0∆tt=1 t=2 t=2CumulativeSummation
  • 9. Elementary Models of DynamicNetworks (EDN’s) Starting from an initial G0 network ER1: Add each non-existing edge with pA. Deleteeach existing edge with pD. ER2: Add kA uniformly selected random new edges.Delete kD existing edges. SPA/CPA (Snapshot/Cumulative preferential):Add kA edges from a random node with preferentialattachment based on the snapshot or cumulativenetwork. Delete kD existing edges. AssortativeCPA/SPA Same as CPA/SPA, butedges are added with assortative mixing. DoubleCPA/SPA Same as CPA/SPA, but bothendpoints of an edge is chosen by weightedselection
  • 10. Elementary Dynamic Networks We defined simple dynamic models Similar in vein to models like: Erdős-Rényi, Watts-Strogatz or Barabási-Albert Starting from empty G0 networks (for the currentanalysis) Converging to the complete network Explore various sampling windows Through computer simulations We compare snapshot and cumulative networks
  • 11. Evolution of Structural Properties
  • 12. Evolution of Degree DistributionER1 DoubleCPA
  • 13. Sensitivity of Degree Distribution Normal, lognormal, even power lawdistribution For the same model Using different timeframes
  • 14. Comparison Against EmpiricalData2013.06.03.TDN 2013 @ NetSci
  • 15. The Gulf Dataset The Gulf „dataset covers the states of the Gulf regionand the Arabian peninsula for the period 15 April 1979to 31 March 1999. The Kansas Event Data Systemused automated coding of English-language newsreports to generate political event data focusing onthe Middle East, Balkans, and West Africa. Thesedata are used in statistical early warning models topredict political change. The ten-year project is basedin the Department of Political Science at theUniversity of Kansas; it has been funded primarily bythe U.S. National Science Foundation. There are twoversions of the data: a set coded from the leadsentences only (57,000 events), and a set coded fromfull stories (304,000 events)” 2013.06.03.TDN 2013 @ NetSciThe Kansas Event Data System: Gulfdata sethttp://web.ku.edu/~keds/data.dir/gulf.html
  • 16. The Gulf Dataset2013.06.03.TDN 2013 @ NetSci Connection of states within the Gulf region Over 20 years Annotated with imestamps Preprocessed, both monthly-daily granularity Relatively small network 174 nodes, 57 131 edges Compared to similarly parameterized EDN runs
  • 17. The Gulf Dataset2013.06.03.TDN 2013 @ NetSci
  • 18. 2013.06.03.TDN 2013 @ NetSci
  • 19. Sexual Network of Internet-MediatedProstitution2013.06.03.TDN 2013 @ NetSci „The community studied is a Brazilian, publiconline forum with free registration that is financedby advertisements. In this community, malemembers grade and categorize their sexualencounters with female escorts, both usinganonymous nicknames. The forum is oriented toheterosexual males.” Large, but sparse network 6,624 anonymous escorts and 10,106 sex buyers 50,632 edges
  • 20. Preliminary Results2013.06.03.TDN 2013 @ NetSci• Density is increasing linearly• Direct influence on other statistics(# of edges, betweennes, avg. degree, etc.)• Network is in the initial „evolution” phase• Average path length starts decreasing far before becomingconnected
  • 21. Degree Distribution2013.06.03.TDN 2013 @ NetSci
  • 22. Summary2013.06.03.TDN 2013 @ NetSci We defined a set of models Compared results to empirical data Models show identical trends to the ones exposed in thedatasets Gulf Dataset: A highly connected core evolves in the network(extremely high max. BC, clustering, but low avg. BC) Granuality does not yield significant difference Internet mediated sexual network: Density is in the initial stage of evolution Degree distribution is almost stable in the first 2000 turns, butindication of change is in the last 200 steps Preferential attachment plays a great role in real-worldsystems EDNs with PA are the closest to the data
  • 23. Future Works2013.06.03.TDN 2013 @ NetSci Include studies of richer (more realistic?) EDN’s Dedicating parts of the network as constant More extensive studies (e.g., parameterdependence)
  • 24. Questions?2013.06.03.TDN 2013 @ NetSciTHANK YOU!Richárd O. Legéndihttp://people.inf.elte.hu/legendi/June 3, 2013This work was partially supported by the European Union and the European SocialFund through project FuturICT.hu (grant no.: TÁMOP-4.2.2.C-11/1/KONV-2012-0013).