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.

1. Richárd O. Legéndi, László Gulyás
Eötvös Loránd University, Regional
Knowledge Centre
AITIA International, Inc,
rlegendi@aitia.ai, lgulyas@aitia.ai
Comparison of Elementary
Dynamic Network Models Using
Empirical Data
This work was partially supported by the European Union and the European Social
Fund through project FuturICT.hu (grant no.: TÁMOP-4.2.2.C-11/1/KONV-2012-0013).
TDN 2013, NetSci Satellite
Copenhagen, June 3-4, 2013

2. Outline
 Motivation
 EDNs
 Previous results
 Comparison to empirical datasets
 Conclusion
2013.06.03.TDN 2013 @ NetSci

3. A Practical Problem with Dynamic
Networks
The importance of
the sampling
window...
∆
t

4. Life is about change...
Elementary Models of
Dynamic 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 same
constant 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. Definitions
Snapsot
t=0
∆
t
t=1 t=2 t=2
Cumulative
Summation

9. Elementary Models of Dynamic
Networks (EDN’s)
 Starting from an initial G0 network
 ER1: Add each non-existing edge with pA. Delete
each 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 preferential
attachment based on the snapshot or cumulative
network. Delete kD existing edges.
 AssortativeCPA/SPA Same as CPA/SPA, but
edges are added with assortative mixing.
 DoubleCPA/SPA Same as CPA/SPA, but both
endpoints of an edge is chosen by weighted
selection

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 current
analysis)
 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 Distribution
ER1 DoubleCPA

13. Sensitivity of Degree Distribution
 Normal, lognormal, ev
en power law
distribution
 For the same model
 Using different time
frames

14. Comparison Against Empirical
Data
2013.06.03.TDN 2013 @ NetSci

15. The Gulf Dataset
 The Gulf „dataset covers the states of the Gulf region
and the Arabian peninsula for the period 15 April 1979
to 31 March 1999. The Kansas Event Data System
used automated coding of English-language news
reports to generate political event data focusing on
the Middle East, Balkans, and West Africa. These
data are used in statistical early warning models to
predict political change. The ten-year project is based
in the Department of Political Science at the
University of Kansas; it has been funded primarily by
the U.S. National Science Foundation. There are two
versions of the data: a set coded from the lead
sentences only (57,000 events), and a set coded from
full stories (304,000 events)” 2013.06.03.TDN 2013 @ NetSci
The Kansas Event Data System: Gulf
data set
http://web.ku.edu/~keds/data.dir/gulf.ht
ml

16. The Gulf Dataset
2013.06.03.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 Dataset
2013.06.03.TDN 2013 @ NetSci

18. 2013.06.03.TDN 2013 @ NetSci

19. Sexual Network of Internet-Mediated
Prostitution
2013.06.03.TDN 2013 @ NetSci
 „The community studied is a Brazilian, public
online forum with free registration that is financed
by advertisements. In this community, male
members grade and categorize their sexual
encounters with female escorts, both using
anonymous nicknames. The forum is oriented to
heterosexual males.”
 Large, but sparse network
 6,624 anonymous escorts and 10,106 sex buyers
 50,632 edges

20. Preliminary Results
2013.06.03.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 becoming
connected

21. Degree Distribution
2013.06.03.TDN 2013 @ NetSci

22. Summary
2013.06.03.TDN 2013 @ NetSci
 We defined a set of models
 Compared results to empirical data
 Models show identical trends to the ones exposed in the
datasets
 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, but
indication of change is in the last 200 steps
 Preferential attachment plays a great role in real-world
systems
 EDNs with PA are the closest to the data

23. Future Works
2013.06.03.TDN 2013 @ NetSci
 Include studies of richer (more realistic?) EDN’s
 Dedicating parts of the network as constant
 More extensive studies (e.g., parameter
dependence)

24. Questions?
2013.06.03.TDN 2013 @ NetSci
THANK YOU!
Richárd O. Legéndi
http://people.inf.elte.hu/legendi/
June 3, 2013
This work was partially supported by the European Union and the European Social
Fund through project FuturICT.hu (grant no.: TÁMOP-4.2.2.C-11/1/KONV-2012-0013).