Synthetic Data Generation using exponential random Graph modeling
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This document discusses using exponential random graph modeling to generate synthetic network data. It introduces exponential random graph models and how they use log-linear models and conditional log-odds to model network formation. An example network of tobacco control agencies is analyzed, including adding node attributes, edge attributes, and structural effects to the model. Model diagnostics and goodness of fit are evaluated. Finally, the trained model can be used to simulate new synthetic networks.
![¡ Log-linear models of the form:
Pr( 𝑋= 𝑥) =exp{ 𝜃↑′ 𝑧( 𝑥)}/𝐾( 𝜃)
¡ The problem is the normalizing constant
¡ Solution: log-linear → logit.
¡ Consider conditional log-odds for a network x and a pair (i,j) of nodes:
§ 𝑋↓𝑖𝑗 ↓↑𝑐 : status of all pairs in x other than (i,j)
§ 𝑋↓𝑖𝑗 ↓↑+ : same network as x but 𝑥↓𝑖𝑗 =1
§ 𝑋↓𝑖𝑗 ↓↑− : same network as x but 𝑥↓𝑖𝑗 = 0
𝑃( 𝑋↓𝑖𝑗 =1| 𝑋↓𝑖𝑗 ↓↑𝑐 )/𝑃( 𝑋↓𝑖𝑗 =0| 𝑋↓𝑖𝑗 ↓↑𝑐 ) =exp{ 𝜃↑+ 𝑠( 𝑋↓𝑖𝑗 ↓↑+ )}/exp{ 𝜃↑+ 𝑠(
𝑋↓𝑖𝑗 ↓↑− )} =exp( 𝜃↑+ [𝑠( 𝑋↓𝑖𝑗 ↓↑+ )− 𝑠( 𝑋↓𝑖𝑗 ↓↑− )])
log( 𝑃 𝑋↓𝑖𝑗 =1 𝑋↓𝑖𝑗 ↓↑𝑐 /𝑃 𝑋↓𝑖𝑗 =0 𝑋↓𝑖𝑗 ↓↑𝑐 ) = 𝜃↑+ [𝑠( 𝑋↓𝑖𝑗 ↓↑+ )− 𝑠( 𝑋↓𝑖𝑗 ↓↑− )]
Exponential random graph modelling (p*)](https://image.slidesharecdn.com/burcupresentation-160316123322/85/Synthetic-Data-Generation-using-exponential-random-Graph-modeling-4-320.jpg)
Synthetic Data Generation using exponential random Graph modeling
- 1. Burcu Kolbay Pedro Delicado Arnau Prat Pérez SYNTHETIC DATA GENERATION USING EXPONENTIAL RANDOM GRAPH MODELING
- 2. ¡ The need of the synthetic data ¡ Exponential Random Graph Modeling (In theory) ¡ Going through the example ¡ Network simulation Contents
- 3. ¡ Internet & Social Media ¡ Data Privacy Issues ¡ The need for testing process The need of the synthetic data
- 4. ¡ Log-linear models of the form: Pr( 𝑋= 𝑥) =exp{ 𝜃↑′ 𝑧( 𝑥)}/𝐾( 𝜃) ¡ The problem is the normalizing constant ¡ Solution: log-linear → logit. ¡ Consider conditional log-odds for a network x and a pair (i,j) of nodes: § 𝑋↓𝑖𝑗 ↓↑𝑐 : status of all pairs in x other than (i,j) § 𝑋↓𝑖𝑗 ↓↑+ : same network as x but 𝑥↓𝑖𝑗 =1 § 𝑋↓𝑖𝑗 ↓↑− : same network as x but 𝑥↓𝑖𝑗 = 0 𝑃( 𝑋↓𝑖𝑗 =1| 𝑋↓𝑖𝑗 ↓↑𝑐 )/𝑃( 𝑋↓𝑖𝑗 =0| 𝑋↓𝑖𝑗 ↓↑𝑐 ) =exp{ 𝜃↑+ 𝑠( 𝑋↓𝑖𝑗 ↓↑+ )}/exp{ 𝜃↑+ 𝑠( 𝑋↓𝑖𝑗 ↓↑− )} =exp( 𝜃↑+ [𝑠( 𝑋↓𝑖𝑗 ↓↑+ )− 𝑠( 𝑋↓𝑖𝑗 ↓↑− )]) log( 𝑃 𝑋↓𝑖𝑗 =1 𝑋↓𝑖𝑗 ↓↑𝑐 /𝑃 𝑋↓𝑖𝑗 =0 𝑋↓𝑖𝑗 ↓↑𝑐 ) = 𝜃↑+ [𝑠( 𝑋↓𝑖𝑗 ↓↑+ )− 𝑠( 𝑋↓𝑖𝑗 ↓↑− )] Exponential random graph modelling (p*)
- 5. ¡ «Tcnetworks» data : Inter-organizational relationship among 25 agencies within the the Indiana State Tobacco Control Program (2010). ¡ 3 types of inter-organizational ties: § Frequency of contact § Level of collaboration § Whether each pair of agencies communicated with one another. ¡ The network data include: § a number of node characteristics (e.g., tob yrs, which records how long an agency has been working in tobacco control), § edge characteristics, § a sociomatrix (TCdist) which contains the geographic distance between each pair of agencies. ¡ Vertex attributes: ¡ Our vertex attributes are: § Agency_cat § Agency_lvl § Lead_agency § Tob_yrs Going through the example
- 6. ¡ 3 types of organizations (local, state, and national), is made up of 1 connected component that is fairly densely connected, and there is some variability of centrality across the network members. Going through the example
- 7. ¡ Start with base model: ¡ Then we include node attributes: Going through the example
- 8. ¡ Including dyadic predictors: Going through the example
- 9. Going through the example ¡ Including relational terms:
- 10. ¡ Including local structure predictors: Going through the example
- 11. ¡ We can check the goodness of fit our model. (e.g. With minimum geodesic distance, edgewise shared partner, triad census , degree etc.) ¡ We can check model diagnostics. ¡ An instance of the output for model diagnostics: Going through an example
- 12. ¡ Based on the model we can simulate new networks: Network simulation
- 13. ¡ We will use a social network data which includes several number of attributes. (Linkedin) ¡ From different type of attributes we can enrich the information we extract from the network. ¡ Based on these knowledge, we will be one step closer to generate synthetic data based on the dependency among the actors. Furthermore
- 14. ¡ A User's Guide to Network Analysis in R 1st ed. 2015 Edition 165-187. ¡ Newman, Mark. Networks: an introduction. OUP Oxford, 2010. ¡ Goodreau, Steven M. "Advances in exponential random graph (p*) models applied to a large social network." Social Networks 29.2 (2007): 231-248. References