ERGM

1
ERGM SIENA

• Thus far networks have been described as static
  with indicators for structure derived at either
  the individual or network level.
• Behaviors have been described as being associated
  with a network position. E.g., popular students
  smoke
• Or behaviors have been described as something
  that happens on networks. E.g., diffusion more
  rapid in centralized networks.

2
Co-evolution

• Yet, we know that both behaviors and networks
  evolve and are related.
• And, we want an estimation of the likelihood for
  certain network, behavioral, and/or
  network-behavioral.
• The Exponential Random Graph Model (ERGM)
  framework provides the framework to address these
  issues.
• ERGM make extensive use of computer simulation
  and generates random networks are created to
  make statistical conclusions about the
  observed/empirical data.

3
ERGM

• ERGM Exponential Random Graph Models
• Random Graph creating randomly generated
  networks
• Exponential because they are based on an
  exponential distribution, that is the log of the
  ratio of probabilities

4
Key Features

• Statistical test is for the probability of a tie
  between 2 nodes.
• What is the likelihood a tie exists given a set
  of conditions.
• To calculate the probability, a large set of
  random networks need to be generated to make the
  comparison

5
ERGM

• Test hypotheses about network structure. E.g.,
  Does this network exhibit more reciprocity than
  would be expected by chance?
• Test hypotheses about behavior. E.g., is friend
  smoking associated with individual smoking?

6
P1, P, PNET, ERGM, SIENA

• Two competing teams developing software for
  hypotheses testing
• PNET/ERGM consist of Gary Robbins (Australia)
  and others
• SIENA (StocNet) consist of Tom Snijders (Oxford
  and U. Gronigen, Netherlands) and others

7
Network Characteristics Density
Reciprocity Transitivity 2-stars Other
Network Properties
A
B
Antecedents Sex Age Ethnicity
Socio-Economic Status Other Characteristics
Individual Behaviors Smoking Sexual Risk
Screening Other Behaviors
8
Crouch, Wasserman Contractor ()

• Explains how p works
• Quick review of logistic regression
• Provides hypothetical example
• Empirical example

9
Crouch et al., Example
10
Network
1 2 3 4 5 6 - - - - - - 1 0
1 1 0 0 0 2 1 0 1 0 0 0 3 0 1 0 1 0 1
4 0 0 0 0 1 1 5 0 0 0 1 0 0 6 0 0 1
1 0 0
N6 L12 Potential Links N(n-1)6530
11
Network is a function of

• Overall Density
• Mutuality
• Transitivity
• Cycles
• In other words, given certain densities,
  reciprocities, transitivities, etc., we can
  recreate the empirical network

12
Links are also function of properties

• Since the overall network is a function of
  density, mutuality, etc.
• We can examine any individual tie in the network
  as a function of these properties
• Tie is a function of choice, choice w/n
  attribute, mutuality, mutuality w/n attrib, etc.

13
And here is the tricky part

• To estimate the model we examine how each
  parameter changes when the links are changed
• Step through every dyadic relationship
• Calculate how the parameters (density, mutuality,
  etc.) change
• Then regress the links on these change parameters

14
Data are
15
Statistical Model is

• Tie Choice Choice_Within Mutuality
• Mutuality_Within Transitivity
• Ties are binary so we use logistic regression

16
Logit Analysis in STATA(note difference than
Crouch et al.)
logit tie l l_w m m_w t_t note l dropped due
to collinearity Iteration 0 log likelihood
-20.19035 Iteration 1 log likelihood
-11.068955 Iteration 2 log likelihood
-10.49291 Iteration 3 log likelihood
-10.446967 Iteration 4 log likelihood
-10.44628 Iteration 5 log likelihood
-10.44628 Logistic regression
Number of obs 30
LR
chi2(4) 19.49
Prob gt chi2
0.0006 Log likelihood -10.44628
Pseudo R2 0.4826 -------------
--------------------------------------------------
--------------- tie Coef. Std.
Err. z Pgtz 95 Conf.
Interval ---------------------------------------
--------------------------------------
l_w 2.740574 2.183488 1.26 0.209
-1.538985 7.020132 m 3.736891
1.82861 2.04 0.041 .1528821
7.3209 m_w -1.58782 2.921087
-0.54 0.587 -7.313046 4.137406
t_t -.408487 .6896983 -0.59 0.554
-1.760271 .9432968 _cons -2.20826
1.19699 -1.84 0.065 -4.554317
.1377973 -----------------------------------------
-------------------------------------
17
There are standard parameter settings
Note. See Snijders et al. (2006) and Robins,
Pattison, Wang (in press) for additional
information on model parameters.
18
From Static to Dynamic

• So in a single network, we can regress the
  probability of a tie between 2 actors as a
  function of several network properties.
• What about longitudinally?
• Can we model the probability of a tie at time 2
  based on these same types of network properties?

19
Yes, MCMC

• To model network dynamics, we employ Markov Chain
  Monte Carlo (MCMC)
• The Markov model states that a particular network
  configuration is a function of that network at
  the prior time period.
• We can generate a series of micro-steps which are
  small changes in the network and behavior to
  mimic how the data evolved from time 1 to time 2.

20
In SIENA

• One specifies the objective function The network
  tendencies (reciprocity, transitivity, etc.).
• One specifies the rate function The frequency of
  network and/or behavioral changes.

21
The Simulation

• SIENA generates hundreds of possible network and
  behavioral configurations at each step
• This dataset of randomly generated networks is
  compared to the empirical one and a t-test
  calculated.
• The average of these t-tests over the entire
  simulation is calculated to determine if there is
  a tendency in the data to conclude a structural
  or behavioral effect.

22
Exposure v. ERGM

• The exposure model we regress behavior on the
  number or percent of ties that engage in the
  behavior
• The dyadic model we regress behavior on whether
  the dyad engages in the behavior
• In ERGM we use the behavior as an attribute and
  determine whether links are more likely among
  nodes with the same attribute
• It is a homophily test.

23
Subtle but Important Difference

• The ERGM model allows the researcher to include
  higher order structural properties such as
  mutuality, transitivity, etc.
• The exposure model allows easier weighting of
  individual and alter attributes (e.g., is
  association between behaviors stronger for same
  sex dyads).
• Currently probably need to do both types of
  analysis

24
Recent Developments

• Special Issue of Social Networks May 2007 Vol. 29
• General introduction
• More parameters (2-stars, 2-,3-,4- triangles)
• Multiple networks

25




Empirical NxN Matrix of Ties
100 Randomly Generated NxN Matrices
Matched
Density Reciprocity Transitivity 2-Stars
Density Reciprocity Transitivity 2-Stars
Matched
Tested
Tested
26
The Actor-Oriented Co-evolution Model

• Provided the opportunity to control for network
  dependencies not previously controlled.
• Provided a means to compare selection and
  influence.