András VörösEvidence for communication with assessors
Department of Social Statistics, University of Manchester
SOST71032 Social Network Analysis
Introduction to Exponential Random Graph Models (ERGMs)
Estimation
We do not know the theoretical model
We want to find out the parameters (= weights for local structural configurations)
We have many [n(n-1)] observations of ties and the local network structure
around them
These are not independent observations
The ties are assumed to be the outcomes of the same model (the theoretical
model, our ERGM)
We can use them to estimate the model parameters
Estimation
2
Estimation – the information carried by observed ties
3
Estimation – the information carried by observed ties
e: 1
m: 0
t: 0
4
Estimation – the information carried by observed ties
e: 0
m: 0
t: 0
5
Estimation – the information carried by observed ties
e: 1
m: 1
t: 0
6
Estimation – the information carried by observed ties
e: 1
m: 0
t: 0
7
Estimation – the information carried by observed ties
e: 1
m: 1
t: 1 (or 2 for tie pointing up)
8
When do we know that we found a good model (vector of parameters)?
If the networks simulated from our model reproduce well the modeled structural
patterns of the network (“target statistics”)
In the example: # ties, # reciprocated ties, # transitive triads
This is not Goodness of Fit yet. This is how the estimation works (every
converged model should provide good fit in case of the configurations which are
explicitly modeled; GoF is about other, non-modeled patterns)
Optimization
9
The specific methods vary by software, but all do some kind of approximate MLE in
an MCMC simulation framework
Main steps (very high level):
1. Get some initial values for the parameters, somehow
2. Generate random graphs under the current parameter vector
3. Update the parameter values by comparing the distribution of graphs against
the observed graph
4. Repeat steps 2-3 until the parameter estimates stabilize: convergence
Overview of the estimation process
10
11
Please continue with the next topic