Exponential Random Graph
Models Under Measurement
Error
Zoe Rehnberg
with Dr. Nan Lin
Washington University in St. Louis
ARTU 2014
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.
Network Data
• Nodes – individuals in a
mock high school
• Edges – mutual friendships
• Adjacency matrix (W )
• n = number of nodes
• wi,j = 1, if edge present
0, if edge absent
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:
P(W = w)µ exp{q T - g(w)}
• 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
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
ki = å wi, j
j
2.
Degree centrality
CD µ å (k* - ki )
i
3. Triangles
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)
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
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
• Simulated measurement
error by perturbing 100
networks at each
probability combination
• Calculated root mean
square error of the
perturbed ERGM
coefficients
1 m ˆ
RMSE =
(qi - qi )
å
m i
Method of Spectral Denoising
•
Wobs = Wtrue +Wnoise
• Naïve estimator:
p = probability of missing edge
q = probability of spurious edge
WKn =
• Empirical estimator: create Ŵobs through spectral
decomposition of Wobs
• 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.
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.
é
ê
Wobs = ê
ê
ê
ë
0
1
0
0
1
0
0
1
0
0
0
0
0
1
0
0
ù
ú
ú
ú
ú
û
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 Wtrue , we
want to simulate from the conditional distribution of
(i )
and
fit
ERGMs
to
each
W
Wtrue Wobs
true .
– The final estimation of q will then be based on this
simulated distribution.
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