Recommender Systems
Social Network Analysis
ERGM Internals
Robin Burke
DePaul University
Chicago, IL
Thanks to Carter Butts for some slides and illustrations
1
Admin
Milestone due
Visualization critique
Next week
5-7 visualizations
2 from Gephi, 2 from ggplot
Others can be any source
ERGM Review
g = the graph we observed
θ = [p1, p2, p3 … pk]
A vector of parameters
What we are trying to fit
t(g) = [t1(g), t2(g), t3(g), … tk(g)]
t(g) = a vector of computed properties of g
What we are measuring about g
γ = the set of all possible graphs of interest
Dot product of two vectors = scalar value
Conditional Log-odds
Useful implication
each unit change in the measurement tk when (i,j) edge is present
increases the conditional log-odds of (i,j) by k
What’s under the hood?
NOT like regression
Stochastic sampling process
approximating maximum likelihood
many parameters of the fitting process
Fitting can fail
Log-likelihood
Create a function κ to represent the ugly sum
Use a ratio with arbitrary vector
An improvement?
But, if Y has distribution governed by θ0
By law of large numbers
So, generate a lot of networks Yi based on a random θ0
Use them to estimate the log-likelihood of a given θ
Try to find θ of maximum likelihood
Problem
The farther away θ0 is from θ
the more samples are needed
exponentially many
Need a θ0 that is “pretty good”
otherwise it doesn’t work at all
Pseudolikelihood estimation
Local approximation of likelihood
Pretend edges are independent
Use logistic regression to estimate θ0
Talked about this last time
If your model is bad
your initial starting point will be bad
you start too far from the destination
Next problem: simulations
How to generate the networks?
We want to be able to sample the set of networks from the distribution given by θ0
but we don’t know the distribution
We also want to know the statistical properties of this distribution
to calculate standard errors
Sampling
Technique to approximate the expected value of a distribution
when the distribution is ugly / hard to integrate
idea: turn an integral into a sum
Also can be used to get the variance, etc.
Expected Values
Want to know the
expected value of a distribution.
We can calculate p(x)
remember that the ERGM is a probabilistic model
but integration is difficult
General method
We have a representation of p(x) and f(x), but integration is intractable
E[f] is difficult as an integral, but easy as a sum.
Randomly select points from distribution p(x) and use these as representative of the distribution of f(x).
It turns out that if correctly sampled, only 10-20 points can be sufficient to estimate the mean and variance of a distribution.
Samples must be independently drawn
Expectation may be dominated by regions of high probability, or high function values
Monte Carlo Example
Sampling techniques to solve difficult integration problems.
What is the area of a circle with radius 1?
What if you don’t know trigonometry?
Monte Carlo Estimation
Take a random x and a random y between 1 and -1
Sample from x and sample from y.
Determine if
Repeat many times.
Count the number of times that the inequality is true.
Divide by the area of the square
Rejection Sampling
The distribution p(x) is easy to evaluate at point x
But difficult to integrate.
Identify a simpler distribution, kq(x), which bounds p(x), and sample, x0, from it.
This is called the proposal distribution.
Generate another sample u from an even distribution between 0 and kq(x0).
If u ≤ p(x0) accept the sample
E.g. use it in the calculation of an expectation of f
Otherwise reject the sample
E.g. omit from the calculation of an expectation of f
This is the square
This is the circle
Rejection Sampling Example
Importance Sampling
One problem with rejection sampling is that you lose information when throwing out samples.
If we are only looking for the expected value of f(x), we can incorporate unlikely samples of x in the calculation.
Again use a proposal distribution to approximate the expected value.
Weight each sample from q by the likelihood that it was also drawn from p.
Graphical Example of Importance Sampling
Markov Chain Monte Carlo
Markov Chain:
p(x1|x2,x3,x4,x5,…) = p(x1|x2)
For MCMC sampling start in a state z(0).
At each step, draw a sample z(m+1) based on the previous state z(m)
Accept this step with some probability based on a proposal distribution.
If the step is accepted: state = z(m+1)
Else: z(m+1) = z(m)
Or only accept if the sample is consistent with an observed value
Markov Chain Monte Carlo
Goal: p(z(m)) = p*(z) as m →∞
MCMCs that have this property are ergodic.
Implies that the sampled distribution converges to the true distribution
Need to define a transition function to move from one state to the next.
How do we draw a sample at state m+1 given state m?
Often, z(m+1) is drawn from a gaussian with z(m) mean and a constant variance.
Metropolis-Hastings Algorithm
Assume the current state is z(m).
Draw a sample z* from q(z|z(m))
Accept probability function
Often use a normal distribution for q
Tradeoff between convergence and acceptance rate based on variance.
Application to simulation
Start with a random network
Modify the network by adding and removing an edge
Use the MH criterion to accept or reject the network
Markov chain converges (in the limit) to the set of networks defined by θ0
We sample this chain
to get networks that have high probability given θ0
Estimation
Being able to sample networks that match our distribution
Means we can calculate the expected values and variance of the target metrics
t(g) terms
Convergence
We need enough steps in the chain
so that the distribution has “mixed”
not contaminated by choice of θ0
We need enough samples from the mixed chain
so that we get good statistical properties
“snapshot”
Summing up
Little gnomes make an initial guess at θ0 using the MPLE
parameters of the model
More gnomes simulate y1,…,yn based on the initial guess
graphs that are likely given the parameters
The simulated sample is used to find θ using MLE
Possibly, the previous two steps are iterated a few times for good measure
since initial estimate may be incorrect
Simulation can fail
Insufficient burn-in
starting point still affects results
Insufficient post-burn samples
sample hasn’t converged
May be degenerate
almost all graphs are same
usually complete/empty
Sample does not cover observed graph
problematic for inference
bad θ0 due to bad model
P(g |θ, t,γ ) = e
θTt (g)
eθ
Tt (g ‘)
g ‘∈γ
∑
P(g|q,t,g)=
e
q
T
t(g)
e
q
T
t(g’)
g’Îg
å
log
P(M =mi, j
+ |θ, t,µ)
P(M =mi, j
− |θ, t,µ)
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=θ T [t(mi, j
+ )− t(mi, j
− )]=θ TΔij
Δij = t(mij
+ )− t(mij
− )
log
P(M=m
i,j
+
|q,t,m)
P(M=m
i,j
–
|q,t,m)
é
ë
ê
ê
ù
û
ú
ú
=q
T
[t(m
i,j
+
)-t(m
i,j
–
)]=q
T
D
ij
D
ij
=t(m
ij
+
)-t(m
ij
–
)
ℓ(θ ) =θ Tt(g)− log eθ
Tt (g ‘)
g ‘∈γ
∑
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
ℓ(q)=q
T
t(g)-loge
q
T
t(g’)
g’Îg
å
é
ë
ê
ê
ù
û
ú
ú
ℓ(θ ) =θ Tt(g)− logκ (θ,γ )
ℓ(q)=q
T
t(g)-logk(q,g)
ℓ(θ )− ℓ(θ0 ) = (θ −θ0 )
T t(g)− log κ (θ,γ )
κ (θ0,γ )
⎡
⎣
⎢
⎤
⎦
⎥
ℓ(q)-ℓ(q
0
)=(q-q
0
)
T
t(g)-log
k(q,g)
k(q
0
,g)
é
ë
ê
ù
û
ú
log κ (θ,γ )
κ (θ0,γ )
⎡
⎣
⎢
⎤
⎦
⎥= E e(θ−θ0 )
T t (Y ) |θ0( )
log
k(q,g)
k(q
0
,g)
é
ë
ê
ù
û
ú
=Ee
(q-q
0
)
T
t(Y)
|q
0
()
E e(θ−θ0 )
T t (Y ) |θ0( ) ≈ 1m e
(θ−θ0 )
T t (Yi )
i=1
m
∑
Ee
(q-q
0
)
T
t(Y)
|q
0
()
»
1
m
e
(q-q
0
)
T
t(Y
i
)
i=1
m
å
E[f ] =
�
f(x)p(x)dx
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