Efficient Markov Chain Monte Carlo Estimation of Exponential Random Graph models

21 Pages·2017·1.24 MB·English

Efficient Markov Chain Monte Carlo Estimation of
Exponential Random Graph models
Maksym Byshkin1, Alex Stivala2, Antonietta Mira1,3,
Garry Robins2, Alessandro Lomi1,2
1Università della Svizzera italiana, Lugano, Switzerland
2University of Melbourne, Australia
3Università dell’Insubria, Como, Italy
Third European Conference on Social Networks, Mainz, Germany,
September 26 - 29, 2017
Introduction
We have developed efficient Markov chain-based algorithm to
perform Maximum Likelihood parameter estimation for probability
distributions with intractable normalizing constants
ERGMs are exponential family of probability distributions for
dependent network data
1 æ ö
å
p(x,q) = exp q z (x)
ç ÷
A A
k
è ø
A
æ ö
å å
k = exp q z (x)
ç ÷
A A
è ø
x A
z (x) are networks statistics (e.g. number of ties, triangles, stars ..)
A
Robins G, Snijders T, Wang P, Handcock M, & Pattison P (2007) Recent developments in
exponential random graph (p*) models for social networks. Social Networks 29(2):192-215.
Introduction
We want to find MLE of the model parameters q
( )
E z (x) = z (x )
p(q) A A obs
å are expected statistics
E (z (x)) = z (x)p(x,q)
p(q) A A
x
Metropolis-Hastings algorithm may be used to compute
E (z (x)) and to generate simulated data x(q)
p(q) A
And we want to solve inverse problem: find q(x )
obs
Lehmann, E.L., Casella, G. Theory of point estimation (2006)
Existing estimation approaches
1) Bayesian
2) Geyer-Thompson MCMCMLE
3) Method of Moments
(Stochastic approximation)
- Iteratively update q
A
q
- At many different values perform MCMC
A
simulations to draw simulated data x(q )
A
MLE is computationally expensive
We can find MCMCMLE without MCMC simulation
Metropolis-Hastings algorithm
Given the state x the state x’ is proposed with probability q(x ® x ')
ì q(x' ® x)p(x',q )ü
Acceptance probability a(x ®=x',q ) min 1, A
í ý
A
q(x ® x')p(x,q )
î þ
A
Transition probability P(x ®®=x',q ) ®q(x x')a(x x',q )
A A
The algorithm generates a Markov chain x that converges to 𝜋 𝑥, 𝜃
t
if the algorithm step t is larger then the burn-in time
Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, & Teller E Equation of state
calculations by fast computing machines. The journal of chemical physics 21, 1087-1092 (1953)
Hastings WK, Monte Carlo sampling methods using Markov chains and their applications.
Biometrika 57, 97-109 (1970)
New approaches for MLE
𝜋(𝑥, 𝜃)
I can show that for simulated networks drawn from
MLE may be found from Equilibrium Expectation (EE):
( )
E dz (x,q) = 0
p(q) A
�
- -
𝑑𝑧 (𝑥, 𝜃) = + P 𝑥 → 𝑥 , 𝜃 (𝑧 𝑥 −𝑧 𝑥 )
𝐴 1 1
0
/
The left part may be computed by Monte Carlo
integrations, without time consuming MCMC simulation
New approaches for MLE
If observed network x is large then one sample is enough
obs
ˆ
å ( )
P(x ® x ',q) z (x=') - z (x ) 0
obs A A obs
x'
But may be used also for small networks
x , x ,.., x
If we have data samples iid from 𝜋 ( 𝑥 , 𝜃 ) :
S S S
1 2 n
�
1
𝐸 (dz (𝑥, 𝜃)) = + dz (𝑥 , 𝜃)
4 5 𝐴 𝐴 :
𝑛
;
;
Very fast MLE !
MLE for empirical data
How to apply this methodology for empirical data x ?
obs
We can draw simulated data x so that z (x) = z (x )
A A obs
å ( )
P(x ® x',q ) z=(x') - z (x) 0
A A A
( )
E z (x) = z (x )
x'
q A A obs
z (x) = z (x )
A A obs
We have developed MCMC algorithm to solve this system of equation
EE algorithm for empirical data
MCMC to constrain the values of all the statistics
EE algorithm for empirical data
MoM (Snijders 2002) and MCMCMLE (Geyer CJ & Thompson
1992) require many converged outputs of Metropolis-Hastings algorithm
EE algorithm does not need such outputs. Instead it generates one
converged output
EE algorithm is similar to Metropolis-Hastings algorithm, but allows
Monte Carlo simulation to be performed while constraining the values of
statistics z (x) and in such a way that the EE condition is satisfied
A