Measurement Sensitivity

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Measurement Sensitivity

• It seems a reasonable approach to assessing the
  effect of measurement error on the ties in a
  network is to ask how would the network measures
  change if the observed ties differed from those
  observed. This question can be answered simply
  with Monte Carlo simulations on the observed
  network. Thus, the procedure I propose is to
• Generate a probability matrix from the set of
  observed ties,
• Generate many realizations of the network based
  on these underlying probabilities, and
• Compare the distribution of generated statistics
  to those observed in the data.
• How do we set pij?
• Range based on observed features (Sensitivity
  analysis)
• Outcome of a model based on observed patterns
  (ERGM)

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Measurement Sensitivity
As an example, consider the problem of defining
friendship ties in highschools. Should we
count nominations that are not reciprocated?
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Measurement Sensitivity
Reciprocated
All ties
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Measurement Sensitivity
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Measurement Sensitivity
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Measurement Sensitivity
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Measurement Sensitivity
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Measurement Sensitivity
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Measurement Sensitivity
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Statistical Analysis of Social Networks
Comparing multiple networks QAP

• The substantive question is how one set of
  relations (or dyadic attributes) relates to
  another.
• For example
• Do marriage ties correlate with business ties in
  the Medici family network?
• Are friendship relations correlated with joint
  membership in a club?

(review)
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Modeling Social Networks parametrically ERGM
approaches

• The earliest approaches are based on simple
  random graph theory, but theres been a flurry of
  activity in the last 10 years or so.
• Key historical references
• - Holland and Leinhardt (1981) JASA
• - Frank and Strauss (1986) JASA
• - Wasserman and Faust (1994) Chap 15 16
• Wasserman and Pattison (1996)
• Good practical overview http//www.jstatsoft.org/
  v24
• Great tutorial http//statnet.csde.washington.edu
  /workshops/SUNBELT/EUSN/ergm/ergm_tutorial.html
  (last years sunbelt)
• Or
• https//statnet.csde.washington.edu/trac/wiki/Sunb
  elt2014 (lots of how to slides)

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Modeling Social Networks parametrically ERGM
approaches

• The p1 model of Holland and Leinhardt is the
  classic foundation the basic idea is that you
  can generate a statistical model of the network
  by predicting the counts of types of ties (asym,
  null, sym). They formulate a log-linear model
  for these counts but the model is equivalent to
  a logit model on the dyads

Note the subscripts! This implies a distinct
parameter for every node i and j in the model,
plus one for reciprocity.
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Modeling Social Networks parametrically ERGM
approaches
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Modeling Social Networks parametrically ERGM
approaches
Results from SAS version on PROSPER datasets
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Modeling Social Networks parametrically ERGM
approaches
Once you know the basic model format, you can
imagine other specifications
Key is to ensure that the specification doesnt
imply a linear dependency of terms. Model fit
is hard to judge newer work shows that the ses
are approximate -)
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Modeling Social Networks parametrically ERGM
approaches
Where q is a vector of parameters (like
regression coefficients) z is a vector of network
statistics, conditioning the graph k is a
normalizing constant, to ensure the probabilities
sum to 1.
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Modeling Social Networks parametrically ERGM
approaches
The simplest graph is a Bernoulli random
graph,where each Xij is independent
Where qij logitP(Xij 1) k(q) P1 exp(ij
)
Note this is one of the few cases where k(q) can
be written.
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Modeling Social Networks parametrically ERGM
approaches
Typically, we add a homogeneity condition, so
that all isomorphic graphs are equally likely.
The homogeneous bernulli graph model
Where k(q) 1 exp(q)g
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Modeling Social Networks parametrically ERGM
approaches
If we want to condition on anything much more
complicated than density, the normalizing
constant ends up being a problem. We need a way
to express the probability of the graph that
doesnt depend on that constant. First some
terms
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Modeling Social Networks parametrically ERGM
approaches
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Modeling Social Networks parametrically ERGM
approaches
Note that we can now model the conditional
probability of the graph, as a function of a set
of difference statistics, without reference to
the normalizing constant. The model, then,
simply reduces to a logit model on the dyads.
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Modeling Social Networks parametrically ERGM
approaches
Consider the simplest possible model the
Bernoulli random graph model, which says the only
feature of interest is the number of edges in the
graph. What is the change statistic for that
feature?
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Modeling Social Networks parametrically ERGM
approaches
Consider the simplest possible model the
Bernoulli random graph model, which says the only
feature of interest is the number of edges in the
graph. What is the change statistic for that
feature? The Edges parameter is simply an
intercept-only model.
NODE ADJMAT 1 0 1 1 1 0 0 0 0 0
2 1 0 1 0 0 0 1 0 0 3 1 1 0 0 1 0
1 0 0 4 1 0 0 0 1 0 0 0 0 5 0 0
1 1 0 1 0 1 0 6 0 0 0 0 1 0 0 1 1 7
0 1 1 0 0 0 0 0 0 8 0 0 0 0 1 1 0 0
1 9 0 0 0 0 0 1 0 1 0
Density 0.311
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Modeling Social Networks parametrically ERGM
approaches
Consider the simplest possible model the
Bernoulli random graph model, which says the only
feature of interest is the number of edges in the
graph. What is the change statistic for that
feature? The Edges parameter is simply an
intercept-only model.
proc logistic descending datadydat model nom
run quit ---see results copy coef --- data
chk xexp(-0.5705)/(1exp(-0.5705)) run
proc print datachk run
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Modeling Social Networks parametrically ERGM
approaches
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Modeling Social Networks parametrically ERGM
approaches
The logit model estimation procedure was
popularized by Wasserman colleagues, and a good
guide to this approach is
Including A Practical Guide To Fitting p
Social Network Models Via Logistic
Regression The site includes the PREPSTAR
program for creating the variables of interest.
The following example draws from this work.
this bit nicely walks you through the logic of
constructing change variables, model fit and so
forth. But the estimates are not very good for
any parameters other than dyad independent
parameters!
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Modeling Social Networks parametrically ERGM
approaches

• Parameters that are often fit include
• Expansiveness and attractiveness parameters.
  dummies for each sender/receiver in the network
• Degree distribution
• Mutuality
• Group membership (and all other parameters by
  group)
• Transitivity / Intransitivity
• K-in-stars, k-out-stars
• Cyclicity
• Node-level covariates (Matching, difference)
• Edge-level covariates (dyad-level features such
  as exposure)
• Temporal data such as relations in prior waves.

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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
and there are LOTS of terms
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
In practice, logit estimated models are difficult
to estimate, and we have no good sense of how
approximate the PMLE is. The STATNET
generalization is to use MCMC methods to better
estimate the parameters. This is essentially a
simulation procedure working under the hood to
explore the space of graphs described by the
model parameters searching for the best fit to
the observed data.
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
You can specify a model as a simple statement on
terms
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
A simple example One of the schools in PROSPER
library(statnet) library(foreign) g lt-
read.paj("C/jwmdata/prosper/Network_data_files/PA
JEK/MATCHED/SC1C1W1Sch101.net") g v "indegree"
lt- degree(g,cmode"indegree") g v "outdegree"
lt- degree(g,cmode"outdegree") atrlt-read.table("C
/jwmdata/prosper/Network_data_files/Rfiles/ergmfi
les/n111101.txt") g v "sex" lt- atr,2 g v
"white" lt- atr,3 g v "slun" lt- atr,4 g
v "irtuse" lt- atr,5 g v "irtdev" lt-
atr,6 g v "tgrad" lt- atr,7 g v
"discip" lt- atr,8 g v "church" lt- atr,9
g v "sens" lt- atr,10 plot(g,vertex.col"
sex") plot(g,vertex.col"slun") plot(g,vertex.co
l"white")
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Dynamics 1
Simple time-lag model Prosper Peers
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Modeling Social Networks parametrically Exponenti
al Random Graph Models
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Complete Network Analysis Stochastic Network
Analysis
An example
Panel model in PROSPER
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Complete Network Analysis Stochastic Network
Analysis
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Modeling Social Networks parametrically Exponenti
al Random Graph Models Degeneracy
"Assessing Degeneracy in Statistical Models of
Social Networks" Mark S. Handcock, CSSS Working
Paper 39
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Modeling Social Networks parametrically Exponenti
al Random Graph Models Quick example (demo)
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Modeling Social Networks parametrically Latent
Space Models
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Modeling Social Networks parametrically Latent
Space Models
Z a dimension in some unknown space that, once
accounted for makes ties independent. Z is
effectively chosen with respect to some latent
cluster-space, G. These groups define
different social sources for association.
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Modeling Social Networks parametrically Latent
Space Models
Z a dimension in some unknown space that, once
accounted for makes ties independent. Z is
effectively chosen with respect to some latent
cluster-space, G. These groups define
different social sources for association.
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Modeling Social Networks parametrically Latent
Space Models
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Modeling Social Networks parametrically Latent
Space Models
Prosper data, with three groups
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Modeling Social Networks parametrically Latent
Space Models
Prosper data, with three groups (posterior
density plots)
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Modeling Social Networks parametrically Latent
Space Models
note there is a non-R option.,..
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Generating Random Graph Samples
A conceptual merge between exponential random
graph models and QAP/sensitivity models is to
attempt to identify a sample of graphs from the
universe you are trying to model.
That is, generate X empirically, then compare
z(x) to see how likely a measure on x would be
given X. The difficulty, however, is generating
X.
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Generating Random Graph Samples
The first option would be to generate all
isomorphic graphs within a given
constraint. This is possible for small graphs,
but the number gets large fast. For a network
with 3 nodes, there are 16 possible directed
graphs. For a network with 4 nodes, there are
218, for 5 nodes 9608, for 6 nodes1,540,944, and
so on So, the best approach is to sample from
the universe, but, of course, if you had the
universe you wouldnt need to sample from it.
How do you sample from a population you havent
observed? (a) use a construction algorithm
that generates a random graph with known
constraints (b) use a ERGM model like above.
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Generating Random Graph Samples
Romantic Networks
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Generating Random Graph Samples
Romantic Networks
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Generating Random Graph Samples
Romantic Networks
A draw from the simulation, this is what appeared
in Glamour
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Generating Random Graph Samples
Edge-matching random permutation
Can easily generate networks with appropriate
degree distributions by generating edge stems
and sorting
di1
di2
di3
Degree 1 2 2 2 3 1
(need to ensure you have a valid edge list!)
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Generating Random Graph Samples
Edge-matching random permutation
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Generating Random Graph Samples
Emergent Connectivity in low-degree networks
Partner Distribution
Component Size/Shape
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Complete Network Analysis Network Connections
Connectivity
Development of STD cores in low-degree networks
rapid transition without stars.
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Complete Network Analysis Network Connections
Connectivity
Extend this view across the space of low-degree
distributions defined by shape and volume...
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Complete Network Analysis Network Connections
Connectivity
Extend this view across the space of low-degree
distributions defined by shape and volume...
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Complete Network Analysis Network Connections
Connectivity

• ERGMs make it (fairly) easy to simulate networks
  from models.
• Simple simulation from an estimated ERGM (this
  is how the GOF function works)
• Simple II simulate from a pre-defined ERGM
  formula (i.e. set the parameters by hand)
• A little harder Simulate from EGO networks.
  Here you can use ERGM to match the observed
  distribution for mixing by node characteristics
  reported in an ego-network survey.
• Can use degree, attribute mixing,
• A bit harder fit global structure features using
  ego-nets by modeling distribution of
  sub-structures (see Jeff Smiths work)

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Generating Random Graph Samples Model based
estimates
ERGM to simulate networks from Add Health
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Modeling Network Dynamics Rule-based simulation
models
Rule-Based simulation models The network-science
approach to dynamic networks has been to identify
toy behavioral models and play out the
implications of these models for network
dynamics. Focus is typically on how the network
evolves (or reaches a steady stat). dynamics OF
networks Balance, preferential attachment,
voter models dynamics ON networks diffusion
simulations These are usually agent-based
models, difficult to specify tradeoff in
simplicity realism.
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Modeling Network Dynamics Descriptive dynamic
techniques
Goal here is to make sense of how networks
change or how things flow through them using a
clear measurement / metrics approach. Challenge
is defining the network.
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Time and Social Networks
Examples of looking at change in networks Roy
and interlocking directorates (ASR 1983, 248-257)
Non-financial interlocks 1886 - 1890
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Time and Social Networks
Examples of looking at change in networks Roy
and interlocking directorates (ASR 1983, 248-257)
Non-financial interlocks 1891 - 1895
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Time and Social Networks
Examples of looking at change in networks Roy
and interlocking directorates (ASR 1983, 248-257)
Non-financial interlocks 1896 - 1900
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Time and Social Networks
Examples of looking at change in networks Roy
and interlocking directorates (ASR 1983, 248-257)
Non-financial interlocks 1901 - 1905
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Bearman and Everett The Structure of Social
Protest
7
5
6
(61-63)
(66-68)
(71-73)
(76-78)
See paper for group compositions
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Data on drug users in Colorado Springs, over 5
years
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Data on drug users in Colorado Springs, over 5
years
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Data on drug users in Colorado Springs, over 5
years
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Data on drug users in Colorado Springs, over 5
years
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Data on drug users in Colorado Springs, over 5
years
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Representing dynamic networks?
Animation captures much of the dynamism we care
about
STD Diffusion
http//csde.washington.edu/statnet/movies/Concurre
ncyAndReachability.mov
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Representing dynamic networks?
Animation captures much of the dynamism we care
about
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Representing dynamic networks?
Animation captures much of the dynamism we care
about
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Modeling Network Dynamics Random Graph models
Panel ERGM Simply want to account for effect of
past structures, you can add temporal covariates
to the standard ERGM. Really only good for two
waves. STERGM Separable Temporal ERGM. This is
a two-equation model, with one equation for the
formation of ties, a 2nd for the dissolution of
ties. Goal is like ERGM, to explain the dynamics
of the network. http//statnet.csde.washington.edu
/workshops/SUNBELT/current/tergm/tergm_tutorial.pd
f RELEVENT Relational Events Model. This is
really a model of action on a network ? think of
conversation events or similar. Dynamic networks
of very short duration events. http//statnet.csde
.washington.edu/workshops/SUNBELT/current/relevent
/statnet_sunbelt2014_relevent.pdf SIENA
Stochastic Actor Oriented Model (SAOM). Used to
disentangle selection from influence, by jointly
modeling both as functions of each other.
Multi-equation model, simplest is one for
behavior one for network formation. Intro
https//www.stats.ox.ac.uk/snijders/siena/Snijder
sSteglichVdBunt2009.pdf Manual
https//www.stats.ox.ac.uk/snijders/siena/RSiena_
Manual.pdf
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Modeling Network Dynamics Random Graph models
STERGM
http//statnet.csde.washington.edu/workshops/SUNBE
LT/current/tergm/tergm_tutorial.html
slides adapted from the workshop materials
http//statnet.csde.washington.edu/EpiModel/nme/in
dex.html
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Modeling Network Dynamics Random Graph models
STERGM
Under certain assumptions, you can model a single
network w. average duration information (assumes
an equilibrium process)
http//statnet.csde.washington.edu/workshops/SUNBE
LT/current/tergm/tergm_tutorial.html
slides adapted from the workshop materials
http//statnet.csde.washington.edu/EpiModel/nme/in
dex.html
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Modeling Network Dynamics Random Graph models
STERGM
samp.fit lt- stergm(samp, formation
edgesmutualcyclicaltiestransitiveties,
dissolution edgesmutualcyclicaltiestransiti
veties, estimate "CMLE", times13 )
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SIENA
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SIENA Key Assumptions of the model
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SIENA
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SIENA
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SIENA
Key element is how actors make changes. This is
based on an evaluation of utility functions,
similar to discrete choice models. The model is
then implemented as an actor-simulation, where
actors are striving to maximize their utility.
note Tom is adamant that this is an as if
model no clear ontological commitment to a
choice model!
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Modeling Network Dynamics Random Graph models
Siena
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Modeling Network Dynamics Random Graph models
Siena
Osgood, D. W., Ragan, D. T., Wallace, L., Gest,
S. D., Feinberg, M. E., Moody, J. 2013. Peers
and the emergence of alcohol use Influence and
selection processes in adolescent friendship
networks. Journal of Research on Adolescence
23500512.
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Modeling Network Dynamics Random Graph models
RelEvent
For repeated interactions amongst nodes