[PPT] - Monte Carlo algorithms for Bayesian social network models Alberto PowerPoint Presentation

SLIDE 1

Monte Carlo algorithms for Bayesian social network models

Alberto Caimo alberto.caimo@usi.ch

Social Network Analysis Research Centre InterDisciplinary Institute of Data Science University of Lugano, Switzerland

AMS Sectional Meeting, Loyola University Chicago October 2015

Alberto Caimo Bayesian computation for ERGMs

SLIDE 2

Outline

Exponential random graph models (ERGMs) Bayesian exponential random graph models (BERGMs) Computational approaches for BERGMs Examples

Alberto Caimo Bayesian computation for ERGMs

SLIDE 3

Exponential random graph models (ERGMs)

The relational structure of an observed network y can be explained by the relative prevalence of a set of overlapping sub-graph configurations s(y) called network statistics

Alberto Caimo Bayesian computation for ERGMs

SLIDE 4

Exponential random graph models (ERGMs)

The relational structure of an observed network y can be explained by the relative prevalence of a set of overlapping sub-graph configurations s(y) called network statistics The likelihood of an ERGM represents the probability distribution of y given a parameter θ: p(y|θ) = exp{θ ts(y)−γ(θ)}

Alberto Caimo Bayesian computation for ERGMs

SLIDE 5

Exponential random graph models (ERGMs)

The relational structure of an observed network y can be explained by the relative prevalence of a set of overlapping sub-graph configurations s(y) called network statistics The likelihood of an ERGM represents the probability distribution of y given a parameter θ: p(y|θ) = exp{θ ts(y)−γ(θ)} From a computational point of view we have an intractable likelihood problem

Alberto Caimo Bayesian computation for ERGMs

SLIDE 6

Bayesian exponential random graph models (BERGMs)

(1) Observed network y (2) Model specification p(y|θ, m) ∝ exp{θts(y)} (3) Doubly-intractable posterior p(θ, m|y) ∝ p(y|θ, m) p(θ|m) p(m) (4) Parameter inference p(θ|y, m) (5) Model choice across-model p(m|θ, y) within-model p(y|m) (6) Goodness of fit Alberto Caimo Bayesian computation for ERGMs

SLIDE 7

Bayesian exponential random graph models (BERGMs)

. . . . . .

. .

.parameter .network statistics .normalising constant likelihood: . . exp{θ t . s(y) − . . γ(θ) }.. prior . posterior: p(θ|y) = . . p(y|θ) . . p(θ) . . p(y) model evidence . .

Alberto Caimo Bayesian computation for ERGMs

SLIDE 8

Computational approaches

Let’s define: qθ(y) = exp{θ ts(y)} unnormalised likelihood z(θ) = exp{γ(θ)} normalising constant Metropolis-Hastings 1 Gibbs update of (θ ′)

i Draw θ ′ ∼ h(·|θ)

2 Accept move from θ to θ ′ with probability 1∧ qθ ′(y) qθ(y) p(θ ′) p(θ) h(θ|θ ′) h(θ ′|θ) × z(θ) z(θ ′)

Alberto Caimo Bayesian computation for ERGMs

SLIDE 9

Computational approaches

Approximate exchange algorithm (AEA)

(Murray et al., 2006; Caimo and Friel, 2011)

1 Gibbs update of (θ ′,y′)

(i) Draw θ ′ ∼ h(·|θ) (ii) Draw y′ ∼ p(·|θ ′) via MCMC

Alberto Caimo Bayesian computation for ERGMs

SLIDE 10

Computational approaches

Approximate exchange algorithm (AEA)

(Murray et al., 2006; Caimo and Friel, 2011)

1 Gibbs update of (θ ′,y′)

(i) Draw θ ′ ∼ h(·|θ) (ii) Draw y′ ∼ p(·|θ ′) via MCMC

2 Exchange move from θ to θ ′ with probability 1∧ qθ ′(y) qθ(y) p(θ ′) p(θ) h(θ|θ ′) h(θ ′|θ) qθ(y′) qθ ′(y′) × z(θ)z(θ ′) z(θ)z(θ ′)

1

Alberto Caimo Bayesian computation for ERGMs

SLIDE 11

Computational approaches

Computational challenges Posterior distribution p(θ|y) is difficult to sample efficiently from as ERGM parameters is typically very thin and highly correlated

Alberto Caimo Bayesian computation for ERGMs

SLIDE 12

Computational approaches

Improving chain mixing and convergence

(Caimo and Friel, 2011)

1(i) Parallel adaptive direction sampling (ADS) for Gibbs update

f θ ′

1(ii) Tie/no tie (TNT) sampler (as in the ergm package for R)

Alberto Caimo Bayesian computation for ERGMs

SLIDE 13

Computational approaches

θh2 θh1 θh θh

1

ADS sampler: “snooker move”

Alberto Caimo Bayesian computation for ERGMs

SLIDE 14

Bayesian exponential random graph models (BERGMs)

−15 −10 −5 5 10 −10 −5 5 10 Alberto Caimo Bayesian computation for ERGMs

SLIDE 15

Computational approaches

Adaptive approximate exchange algorithm with delayed rejection

(Caimo and Mira, 2015)

1(i) Adaptive strategies

Alberto Caimo Bayesian computation for ERGMs

SLIDE 16

Computational approaches

Adaptive approximate exchange algorithm with delayed rejection

(Caimo and Mira, 2015)

1(i) Adaptive strategies 2 Approximate exchange algorithm with delayed rejection

Alberto Caimo Bayesian computation for ERGMs

SLIDE 17

Computational approaches

Adaptive strategies for Gibbs update of θ ′

(Roberts and Rosenthal, 2007; Haario et al., 2001)

vertical adaptation: all past particles along the same chain (AAEA-1+DR) horizontal adaptation: all particles at the current time for all chains (AAEA-2+DR) rectangular adaptation: particles from all chains and all past simulations (AAEA-3+DR)

Alberto Caimo Bayesian computation for ERGMs

SLIDE 18

Computational approaches

Adaptive exchange algorithm with delayed rejection First stage move: α1(θ,θ ′) = 1∧ qθ ′(y) qθ(y) p(θ ′) p(θ) h1(θ|θ ′) h1(θ ′|θ) qθ(y′) qθ ′(y′)

Alberto Caimo Bayesian computation for ERGMs

SLIDE 19

Computational approaches

Adaptive exchange algorithm with delayed rejection First stage move: α1(θ,θ ′) = 1∧ qθ ′(y) qθ(y) p(θ ′) p(θ) h1(θ|θ ′) h1(θ ′|θ) qθ(y′) qθ ′(y′) If θ ′ rejected, try a second stage move to θ ′′ with probability α2(θ,θ ′,θ ′′) = 1∧ qθ(y′′) p(θ ′′) h1(θ ′|θ ′′) [1−α1(θ ′′,θ ′)] h2(θ|θ ′′,θ ′) qθ ′′(y) qθ(y) p(θ) h1(θ ′|θ) [1−α1(θ,θ ′)] h2(θ ′′|θ,θ ′) qθ ′′(y′)

Alberto Caimo Bayesian computation for ERGMs

SLIDE 20

Computational approaches

A hierarchy of proposal distributions can be exploited Moves (tennis-service strategy): “first bold” h1(·) proposal versus “second timid” h2(·)

Alberto Caimo Bayesian computation for ERGMs

SLIDE 21

Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Zachary Karate Club Network: Friendship relations between 34 members of a karate club at a US university in the 1970

Alberto Caimo Bayesian computation for ERGMs

SLIDE 22

Examples

ERGM: p(y|θ) ∝ exp{θ1 s1(y)+θ2 s2(y,φu)+θ3 s3(y,φv)}

Alberto Caimo Bayesian computation for ERGMs

SLIDE 23

Examples

ERGM: p(y|θ) ∝ exp{θ1 s1(y)+θ2 s2(y,φu)+θ3 s3(y,φv)} where: s1(y) = ∑i<j yij = number of edges s2(y,φv) = eφv ∑n−2

i=1

1−
1−e−φv i

EPi(y) geometrically weighted edgewise shared partners (gwesp) s3(y,φu) = eφu ∑n−1

i=1

1−
1−e−φui

Di(y) geometrically weighted degrees (gwdegree) EPi(y) = distribution of the number of unordered pairs of connected nodes having exactly k common neighbours Di(y) = degree distribution

Alberto Caimo Bayesian computation for ERGMs

SLIDE 24

Examples

for model 14 θ(1) (edges) θ(2) (gwesp) θ(3) (gwdegree) ADS-AEA

Post. mean

−3.51 0.74 1.18

Post. sd

0.62 0.21 1.12 AAEA-2+DR (horizontal adaptation + DR)

Post. mean

−3.44 0.72 1.01

Post. sd

0.59 0.21 1.07

Estimated posterior means and standard deviations

Alberto Caimo Bayesian computation for ERGMs

SLIDE 25

Examples

θ1 (edges)

6
5
4
3
2
1

0.0 0.1 0.2 0.3 0.4 0.5 θ2 (gwesp.fixed.0.693147180559945)

0.5

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 θ3 (gwdegree)

4
2

2 4 6 8 0.00 0.10 0.20 0.30 10 20 30 40 50

1.0
0.5

0.0 0.5 1.0 Lag 10 20 30 40 50

1.0
0.5

0.0 0.5 1.0 Lag 10 20 30 40 50

1.0
0.5

0.0 0.5 1.0 Lag θ1 (edges)

6
5
4
3
2

0.0 0.2 0.4 0.6 θ2 (gwesp.fixed.0.693147180559945) 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 θ3 (gwdegree)

2

2 4 6 0.0 0.1 0.2 0.3 0.4 10 20 30 40 50

1.0
0.5

0.0 0.5 1.0 Lag 10 20 30 40 50

1.0
0.5

0.0 0.5 1.0 Lag 10 20 30 40 50

1.0
0.5

0.0 0.5 1.0 Lag

Posterior density estimates for AEA (left) and AAEA+DR (right)

Alberto Caimo Bayesian computation for ERGMs

SLIDE 26

Examples

Effective sample size (ESS): AAEA-2+DR +70% Performance (= ESS/CPU time): AAEA-2+DR +40%

Alberto Caimo Bayesian computation for ERGMs

SLIDE 27

Examples

0.0 0.1 0.2 0.3 degree 2 4 6 8 10 12 14 16 18 0.0 0.2 0.4 0.6 0.8 minimum geodesic distance proportion of dyads 1 2 3 4 5 6 7 8 9 NR 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 edge-wise shared partners proportion of edges 2 4 6 8 10 12 14

Bayesian goodness-of-fit diagnostics: the observed network y is compared with a set of networks simulated from independent realisations of p(θ|y) in terms of high-level network statistics

Alberto Caimo Bayesian computation for ERGMs

SLIDE 28

Examples

Faux Mesa High School friendship 203-node network graph.

Alberto Caimo Bayesian computation for ERGMs

SLIDE 29

Examples

ERGM specification:

s1(y) = P

i<j yij number of edges

s2(y, x) = P

i<j yij(1(gradei=8) + 1(gradej=8))

node factor for “grade” = 8 s3(y, x) = P

i<j yij(1(gradei=9) + 1(gradej=9))

node factor for “grade” = 9 s4(y, x) = P

i<j yij(1(gredei=10) + 1(gradej=10))

node factor for “grade” = 10 s5(y, x) = P

i<j yij(1(gradei=11) + 1(gradej=11))

node factor for “grade” = 11 s6(y, x) = P

i<j yij(1(gradei=12) + 1(gradej=12))

node factor for “grade” = 12 s7(y, x) = P

i<j yij(1(sexi=M) + 1(sexj=M))

node factor for “sex = male” s8(y) = v(y, φv) GWESP s9(y) = u(y, φu) GWD

Alberto Caimo Bayesian computation for ERGMs

SLIDE 30

Examples

ADS-AEA AAEA-1 AAEA-2 AAEA-3 ESS 667 1041 1008 1094 Performance (per sec) 1.8 2.3 2.1 2.2 ADS-AEA+DR AAEA-1+DR AAEA-2+DR AAEA-3+DR ESS 873 1376 1320 1440 Performance (per sec) 1.4 2.6 2.6 2.6

Variance reduction of AAEA-based algorithms varies between 55% and 98% relative to the ADS-AEA. This translates into a better performance varying from 25% to 40%.

Alberto Caimo Bayesian computation for ERGMs