Exponential Random Graph Models (ERGM)

1
Exponential Random Graph Models(ERGM)

• Michael Beckman
• PAD777
• April 9, 2010

2
Introduction

• The purpose of ERGM, in a nutshell, is to
  describe parsimoniously the local selection
  forces that shape the global structure of a
  network.
• ERGM may then be used to understand a particular
  phenomenon or to simulate new random realizations
  of networks that retain the essential properties
  of the original. (Hunter et al 2008)
• General characteristics of ERGM
• Single observation rather than successive waves
• Change statistics compare observed network to
  random realizations
• Still computes Markov or Markov-like statistics
• Can model both structural and attribute
  parameters
• Assumptions and constraints are important to
  estimations
• Improved SEs even where pseudolikelihood
  produces acceptable estimates
• Goodness of fit statistics are reliable
• Significant move towards true stochastic modeling
  of networks

3
Agenda

• Wasserman and Robins (2005) An Introduction to
  Random Graphs, Dependence Graphs, and p
• Snijders ( 2002) Markov chain monte carlo
  estimation of ERGM
• Robins et al (2007) Recent developments in
  exponential random graph (p) models for social
  networks
• Hunter et al (2008) A Package to Fit, Simulate
  and Diagnose Exponential-Family Models for
  Networks
• Morris et al (2008) Specification of
  Exponential-Family Random Graph Models Terms and
  Computational Aspects
• Andrew (2009) Regional integration through
  contracting networks

4
Wasserman Robins - Intro

• Wasserman and Robins (2005) An Introduction to
  Random Graphs, Dependence Graphs, and p
• Historic development of p distribution for
  Markov random graphs
• Frank and Strauss 1986
• Strauss and Ikeda 1990 (estimation of
  distribution parameters)
• Wasserman and Pattison 1996 (extend parameter
  assumptions)
• Wasserman and Robins 2005 Family of models from
  dependence graphs
• Versus approximate autologistic regression
  (pseudo-likelihood)
• Standard network notation
• r1- single relation, dichotomous data
• Random variables, assumed interdependent
• Can use multivariate or valued relations
• Dependence graphs allows testing for independent
  elements in matrix X

5
Wasserman Robins - Intro

• Model parameters estimated from three new
  arrays converse, composition, intersection of
  measured relations

6
Wasserman Robins - Intro

• Consider the observed network as a subset of all
  possible configurations
• Dependence graphs help distinguish among possible
  distributions, by identifying ties that are
  statistically independent
• Dependence graph graph of nodes whose edges
  signify pairs of random variables that are
  assumed to be conditionally dependent

7
Wasserman Robins - Intro

• Three classes of dependence graphs
• Bernoulli assumption of conditional
  independence for each pair of ties
• Empty graph, due to complete independence
• Conditional uniform distribution
• Dyadic dependence assumes all dyads are
  statistically independent
• Dependence graph has edge set for each dyad
• Basis for p1 model of Holland and Leinhardt
  (1977,1981)
• General dependence graph arbitrary edge set
  with general probability distribution basis for
  p

8
Wasserman Robins - Intro

• Markov graphs and p
• Any two relational ties associated if they
  involve same actor
• Observed network considered a realization x of
  random array X
• Dependence graph D consists of any complete
  subgraphs, or cliques
• Hammersley-Clifford theorem characterizes Pr(Xx)
  in the form of an exponential family of
  distributions
• Set of non-zero parameters depends on maximal
  cliques

9
Wasserman Robins - Intro

• Estimating parameters can overwhelm the model, so
  constraints are needed
• Impose dependence assumptions on parameters
• Homogeneity ie, isomorphic dyads (MAN)
• Higher-order configurations typically set to zero
  (stars, triads etc)
• Constrained social settings
• Exact differentiation of log likelihood is
  mathematically challenging
• Pseudolikelihood measures of fit problematic
• MCMC model degeneracy may be a problem
• MCMC is normally preferred, improved algorithms
  are available and/or being developed

10
Snijders MCMC Estimation

• Snijders ( 2002) Markov chain Monte Carlo
  Estimation of ERGM
• Random graph is a Markov graph if number of nodes
  is fixed, and non-incident edges are independent
  conditional upon rest of graph
• Exponential family of probability functions (p)
• Where y is the adjacency matrix of a digraph and
  the sufficient statistic u(y) is any vector of
  statistics of the digraph
• Pseudolikelihood not a function of complete
  sufficient statistic u(Y) so not a suitable
  estimator
• Dahmstrom and Dahmstrom (1993) proposed MCMC

11
Snijders MCMC Estimation

• Random graph is a Markov graph if number of nodes
  is fixed, and non-incident edges are independent
  conditional upon rest of graph
• Gibbs Sampling all elements Yij are updated
  randomly, one element per draw, with all other
  elements left unchanged
• Assumes convergence at t -gt ?
• Conditional distribution toggles between Yij 1
  and Yij 0
• Can result in severe convergence problems
• Model may not simulate effects properly, or
• May result in an explosion of ties after
  significant stasis
• Bi-modal distribution results, consisting of
  high-density and low-density states or regimes
• Regime is defined as a subset of the outcome
  space
• Other regimes are possible (besides bi-modal)

12
Snijders MCMC Estimation

• Reciprocity p model of edges and reciprocity
• Assumes dyadic independence
• Probabilities calculated for MAN
• Independence assumption precludes the explosion
  effect
• Twostar p model - of edges and out-twostars
• Rows in adjacency matrix are statistically
  independent
• If total number of Y are fixed, number of
  out-twostars is a linear function of out-degree
  variance
• Combined reciprocity and twostar p model
  density, reciprocity, out-twostar
• Transforms digraph into its complement
• Changes Yij to (1 Yij)
• Density must be set to 0.5
• Simulates graphs equal to, less than or greater
  than 0.5 density
• Can result in the explosion effect
• In effect, results are determined by initial
  state ( high or low density)

13
Snijders MCMC Estimation

• Gibbs sampling algorithm
• For every two outcomes, there is a positive
  probability to go from one outcome to the other
  in finite steps, but
• It is possible one regime is dominant, so that
  sojourn time from one state to the other is
  practically infinite, so
• Initial state determines outcome with 0.5
  probability coin toss
• Three problems arise
• Bi-modal distribution is undesirable for single
  network observation
• Convergence with two regimes can be so slow that
  generating a random draw is practically
  impossible
• Expected values of sufficient statistics are
  extremely sensitive to parameter values, causing
  instability of estimation
• Other iteration procedures have been proposed and
  tested

14
Snijders MCMC Estimation

• Detailed balance technique
• Set of all adjacency matrices Yg
• Results in unique stationary distribution
• Small updating steps one element of Yij per
  step, as with Gibbs sampling
• Cell being updated is random, rather than
  deterministic
• Referred to as mixing, versus cycling
• Metropolis-Hastings algorithm - Changes Yij to (1
  Yij), all other ties constant
• Updates more frequently than Gibbs, so more
  efficient
• Dyadic or triplet updating steps update several
  elements per step
• Dyad or triplets chosen randomly
• Groupwise updating
• Slower to converge

15
Snijders MCMC Estimation

• Large updating steps update Yij from 0 to 1 or
  vice versa in blocks
• Biggest step is converting graph to its
  complement (inversion)
• Satisfies the detailed balance equation
• May be appropriate for bimodal distributions
• Inversion may reduce variance in estimation
  (conditioning)
• Fixed density only digraphs with given number
  of ties are drawn
• Random undirected graphs applied to half matrix
  of unique elements
• ML estimation not easily applied to exponential
  random graphs, due to problematic calculation for
  complex models
• Pseudolikelihood estimates can be good, but
  standard errors are too low
• Monte Carlo Markov Chain estimates
• Monte carlo simulation of Markov graph estimates
  moments
• Moments are used to estimate parameter effects
  for a neighborhood

16
Snijders MCMC Estimation

• MCMC Newton-Raphson Algorithm and Robbins-Monro
  Algorithm similar
• Robbins-Monro Algorithm three phases
• Estimate diagonal matrix using derivative of
  initial parameter estimate
• Iteratively determines provisional estimation
  values, leads quickly to solution of moment
  equation
• Large steps can lead to instability
• Parameter value is kept constant, then large
  number of steps used to check validity of
  equation
• Use of MC with Robbins-Monro yields, in theory,
  convergence probability of 1
• Snijders recommends use of inversion steps for
  models with triplet counts

17
Robins et al Recent Developments

• Robins et al (2007) Recent developments in
  exponential random graph (p) models for social
  networks
• Technically, MCMC estimation does not converge
  due to degeneracy problem near degenerate
• Problem is more acute as network size grows
  larger
• Inclusion of suitable constraints on parameters
  allows for estimation
• Parameters then provide information on structural
  effects
• Recall from Snijders problem of bimodal
  distribution/model degeneration
• Gradual increase in triangle parameter does not
  lead to gradual increase in graph triangulation,
  so inclusion of star/triangle parameters does not
  overcome problem

18
Robins et al Recent Developments
19
Robins et al Recent Developments

• Inclusion of higher-order structures
• Alternating k-stars
• Alternating k-triangles
• Alternating independent two-paths
• Alternating k-stars, technically only structure
  still a Markov random graph
• Assumption allows stars up to (n-1)
• Recall in previous models, higher-order stars
  normally set to 0
• In alternating k-star, higher-order stars are
  allowed
• Impact of higher-order stars is gradually
  diminished
• Essentially, there is weighting of structure from
  simple to complex
• Allows for interesting inference regarding
  network structure
• Positive parameter indicates hubs in node
  structure
• Negative parameter indicates smaller variance in
  degree (decentralized)

20
Robins et al Recent Developments

• Interpreting alternating k-star models
• Positive parameter tendency toward large number
  of low degree nodes, and small number of
  high-degree nodes
• Node degree may become saturated
• Increase in popularity plateaus additional
  ties do not add value
• Indicative of a loose core-periphery structure
• Alternation between positive and negative values
  helps prevent distribution graph from being
  forced to empty or complete graphs ( a la
  Snijders et al 06)

21
Robins et al Recent Developments

• Alternating k-triangles introduces conditional
  dependence
• In short, two possible edges in a graph, Yrs and
  Yuv, for distinct nodes r, s, u, v, are assumed
  to be conditionally dependent if Ysu Yuv 1.
• In other words, if the two possible edges in the
  graph were actually observed, they would create a
  4-cycle.
• Defines social circuit dependence
• Chance of Ysu is conditionally dependent on
  presence of Yuv
• Snijders et al (2006) combine k-triangles with
  Markov dependence
• K-triangle is combination of individual triangles
  that share one edge (base)
• Shared adjacency with other nodes are triangle
  sides
• Conditionally dependent structure, IF either
  Markov configuration (shared node), or Social
  Circuit Configuration (4-cycle)

22
Robins et al Recent Developments
23
Robins et al Recent Developments

• Interpreting k-triangles
• Positive parameter provides evidence of
  transitivity effects
• Also can suggest core-periphery structure, but
  due to triangulation rather than popularity
  influence
• More of a structural effect than an attribute
  effect
• IE, outcome of the triangulation process
• Alternating k-twopaths
• Lower order structure
• Combine with k-triangles
• Distinguish tendency to form ties at base versus
  side of triangle
• Side edges absent base edges indicates
  precondition to transitivity
• Presence of base edge indicates transitive
  closure
• Combination of parameters can indicate pressure
  towards closure

24
Robins et al Recent Developments

• Other possible parameters

25
Robins et al Recent Developments

• Estimating parameters
• MCMC is preferred method, when available
• When model converges, simulation produces
  distribution of graphs in which observed graph is
  typical for all effects
• Reliable standard errors
• Snijders et al (2006) conditioned on edges
• No density parameter
• Diminishes degeneracy problem with moderate
  impact on other parameters
• Robins et al find that, at least for smaller
  networks, conditioning on edges may not be needed

26
Robins et al Recent Developments

• Modeling with SIENA
• Output of estimates, standard error, t-stat for
  estimate (how well model converges)
• t-ratio close to zero good convergence of model
• Large ratios may indicate model has not
  converged, or is degenerate
• For non-degenerate models, absolute value of less
  than 0.1 is converged
• Other tests in SIENA
• Hysteresis analysis
• Simulate from estimates and compare with observed
  graph
• Modeling with statnet
• Newton-Raphson algorithm
• Fewer simulation runs, then weights graphs for
  estimating
• Incorporates advances from Metropolis-Hastings

27
Robins et al Recent Developments
28
Robins et al Recent Developments
Comparing pseduolikelihood to MCMC UCINET
datasets, SIENA modeling
29
Hunter et al Package to Fit

• Hunter et al (2008) A Package to Fit, Simulate
  and Diagnose Exponential-Family Models for
  Networks
• Implementing ERGM in R/statnet
• Specify ERGM
• Approximate/exact MLE
• Goodness of fit tests
• The purpose of ERGM, in a nutshell, is to
  describe parsimoniously the local selection
  forces that shape the global structure of a
  network.
• ERGM may then be used to understand a particular
  phenomenon or to simulate new random realizations
  of networks that retain the essential properties
  of the original.

30
Hunter et al Package to Fit

• Implementing ERGM in R/statnet variables
• Endogenous result of structure
• Exogenous attribute based (can serve as
  predictors)
• Attributes can be treated as functions of nodal
  covariates
• Statistics depend on attribute and relationship
  information
• Change statistics recall we are comparing
  conditional distribution toggled between Yij 1
  and Yij 0 (or some other Markov configuration)
• Particular choice g() of statistics
• Particular network y
• Particular pair of nodes (i,j)
• Seed can be specified for reproducibility

31
Hunter et al Package to Fit

• Dyadic independence models
• Dyadic independence term
• Term in an ERGM for which change statistics can
  be calculated regardless of value of (i,j) or any
  knowledge of y
• Dyadic independence ERGM
• All terms in the model are dyadic independence
  terms
• This model is purely stochastic
• For undirected models, unconditional or marginal
  probability is allowed
• Important to distinguish between dyadic and
  linear independence
• Linear dependencies can arise with either form
  above
• Implications for model specification
• Statnet eliminates/allows for elimination of
  statistics as needed

32
Hunter et al Package to Fit

• Dyadic dependence models
• Dyads that do not share a node are conditionally
  independent
• Analogous to nearest neighbor
• Homogeneity condition may be added as a
  constraint
• All isomorphic networks have same probability
• Problems with model as previously discussed
• Correctives suggested
• combine terms (endogenous and exogenous)
• Specify triad-based curved exponential family
  terms
• Geometrically weighted degree (GWD)
• Geometrically weighted edgewise shared partner
  (GWESP)
• Geometrically weighted dyadwise shared partner
  (GWDSP)

33
Hunter et al Package to Fit

• Curved exponential family model

34
Hunter et al Package to Fit

• Estimation and goodness of fit

• Parameters
• Edges
• Homophily term for grade
• Main effect for sex
• P. 23

35
Morris et al Specification of ERGM

• Morris et al (2008) Specification of
  Exponential-Family Random Graph Models Terms and
  Computational Aspects
• Where Hunter et al focused more on theory and
  statistical formulas, Morris et al provide basic
  instruction on implement ERGM in R/statnet
• Commands for basic effects, nodal attributes,
  relational attributes, structural configurations,
  higher-order configurations, actor specific
  effects, constraints
• Tips to fine-tune algorithm and processing
• Appendix A Table of Model Terms provides quick
  reference for what terms are appropriate to a
  particular model
• IE, directed/undirected, bipartite, dyadic
  independence etc.

36
Morris et al Specification of ERGM

• Constraints
• Model must include space of all possible networks
• Some networks are bipartite communication
  between but never within groups of nodes
• ERGM automatically implements these constraints
  as needed

37
Andrews Regional Integration

• Andrew (2009) Regional integration through
  contracting networks
• Research question Under what conditions do local
  governments choose to contract for services, or
  enter into regional agreements for the provision
  of services?
• Two hypotheses are advanced
• Bonding hypothesis in the presence of
  uncertainty and complexity of interjurisdictional
  activities, a highly dense network structure will
  emerge over time
• Bridging hypothesis for interjurisdictional
  activities involving high asset specificity, a
  sparse, core-periphery network is anticipated
• Institutional collective action framework
  transaction cost analysis, enforcement and
  monitoring, free-rider problem

38
Andrews Regional Integration

• Bonding local officials attracted to
  interjurisdictional, voluntary cooperation
  agreements
• Flexible, non-binding, fosters norm of
  reciprocity
• Can be constrained by local politics and
  coordination costs
• Bridging in asset-specific dilemma, local
  officials likely to choose strategic partner
• May produce services in-house
• Induce competition to attenuate opportunism of
  central actor
• Expected to contract with partner who already has
  ties with other jurisdictions

39
Andrews Regional Integration

• Research Design
• Contractual ties among law enforcement community
  in Orlando-Kissimmee
• Five waves from 1986 to 2003
• 66 total actors
• List of goods services derived from
  International City/County Management Association
  surveys
• Studying one metropolitan area controls for
  geographic variation and allows for in-depth
  analysis of regional integration

40
Andrews Regional Integration
41
Andrews Regional Integration

• Parameters
• Transitive triads
• Geodesic distance-2
• Covariate effects
• Importance of level of government, where
  municipality is coded 1 and higher level
  government is treated as benchmark
• Importance of professionalism, indicated by
  accreditation
• Both coded as dummy variables, treated as control
  variables
• Homophily effect
• Rate parameters were all positive and significant
• T-ration less than 0.3, indicating no problems
  with convergence (?)

42
Andrews Regional Integration
P.392
43
Andrews Regional Integration
P.392