1. Exponential Random Graph Models Under Measurement Error Zoe Rehnberg with Dr. Nan Lin Washington University in St. Louis ARTU 2014

2. Examples of Social Networks Wikipedia pages – Individual pages are connected when there is a reference for one on the other. Article authorship – Two statisticians are connected when they co-author a paper. Friendship – Two high school students are connected when they indicate that they are friends with each other.

3. Network Data Nodes – individuals in a mock high school Edges – mutual friendships Adjacency matrix (W ) n = number of nodes w i,j = 1, if edge present 0, if edge absent

4. Exponential Random Graph Models The combination of nodes and edges in an adjacency matrix is random. Exponential random graph models explain how likely it is that a specific configuration of edges will occur: w – a given set of edges in an adjacency matrix θ – vector of model coefficients g(w) – a vector of statistics for the given adjacency matrix

5. Descriptive Statistics These statistics summarize how nodes are related to each other within the larger graph as a whole. These are used to form the ERG model for a given adjacency matrix. Examples 1.Degree 2.Degree centrality 3.Triangles

6. Estimating Model Coefficients The ERGM function in the statnet package of R uses a maximum likelihood approach to estimate , the vector of model coefficients. library("statnet") data(faux.mesa.high) dat <- faux.mesa.high # fit the original ERG model orig.model <- ergm(dat ~ edges + nodematch("Grade") + nodematch ("Race") + nodematch("Sex") + gwesp(0.4, fixed = TRUE), control = control.ergm(MCMC.samplesize = 1e+5, seed = 123)) # simulate from the original model sim.net <- simulate(orig.model, seed = 1534)

7. Possible Measurement Error Measurement error refers to how well an observed network reflects the true network. We focused on missing (false negative) and spurious (false positive) edges in the network. Possible sources of error: – Mistakes in collecting or coding data – Differences in perception

8. Goal of Our Study Goal: understanding ERGMs under measurement error Method: study by simulation 1.Model and simulate friendship network g(w) – edges, assortative mixing, shared partners 2.Imitate measurement error Adding probability: q = 0.001, 0.005, 0.01, 0.05 Removing probability: p = 0.01, 0.02, …, 0.20 3.Estimate ERGM coefficients and statistics

9. Simulated measurement error by perturbing 100 networks at each probability combination Calculated root mean square error of the perturbed ERGM coefficients

12. Method of Spectral Denoising Naïve estimator: Empirical estimator: create through spectral decomposition of There is a continuity requirement for statistics. P. Balachandran, E. M. Airoldi, and E. D. Kolaczyk. Inference of network summary statistics through nonparametric network denoising. Annals of Statistics, 2013. arXiv:1310.0423v3. p = probability of missing edge q = probability of spurious edge =

13. Challenges and Future Work The estimated adjacency matrices have non-integer values, which causes practical computational problems. – The R function ergm( ) only accepts adjacency matrices with 0/1 entries.

14. Challenges and Future Work The estimated adjacency matrices have non-integer values, which causes practical computational problems. – The R function ergm( ) only accepts adjacency matrices with 0/1 entries. Instead of obtaining a single estimate of, we want to simulate from the conditional distribution of and fit ERGMs to each. – The final estimation of will then be based on this simulated distribution.

15. References [1]A. Caimo and N. Friel. Bayesian inference for exponential random graph models. Social Networks, 2010. http://arxiv.org/abs/ 1007.5192. [2] Hanneman, Robert A. and Mark Riddle. 2005. Introduction to social network methods. Riverside, CA: University of California, Riverside. http://faculty.ucr.edu/~hanneman/nettext/. [3]P. Balachandran, E. M. Airoldi, and E. D. Kolaczyk. Inference of network summary statistics through nonparametric network denoising. Annals of Statistics, 2013. arXiv:1310.0423v3. [4]Wang, D.J., et al., Measurement error in network data: A reclassification. Soc. Netw. (2012), doi:10.1016/j.socnet.2012.01.003