Introduction to ERGMp model

1
Introduction to ERGM/p model

• Kayo Fujimoto, Ph.D.
• Based on presentation slides by Nosh Contractor
  and Mengxiao Zhu

2
Four parts of ERGM

• Observed network data
• Network statistics (or counts) of each
  configuration
• ERG Modeling
• Conditional probability and Change statistics
• Estimation and Simulation
• Estimate Parameters by Simulation
• Method MCMC ML estimation
• Goodness of fit test (convergence t-test)
• Compare observed and simulated graphs
• Recent development in ERGM
• New model specification

3
Exponential Random Graph Model(ERGM)

• ERGMs take the form of a probability distribution
  of graphs
• Y is a set of tie indicator variables Y
• y is a realization, the observed network
• g(y) is a vector of network statistics
• ? is a parameter vector corresponding to g(y)
• k(?) is a normalizing factor calculated by
  summing up
• exp?g(y) over all possible network
  configurations

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Observed network

• Graph statistics (or counts) of each configuration

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Network Statistics Examplesfor Undirected
Networks
Example
Edge 6 2-Star 1316011 3-Star
010405 4-Star 1 Triangle 2
b
c
a
d
e
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A Simple Example of ERGM
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A Simple ERG model

• Predict network using edge count
• ? can take different values
• ? 0, ? -0.69, ? 0.69
• L(y) can the following values
• L(y) 0, L(y) 1, L(y) 2, L(y) 3

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Example 1 ? 0, L0

• Model
• ERGM Formula

? 0
9
Example 1 ? 0, L1

• Model
• ERGM Formula

? 0
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Example 1 ? 0, L2

• Model
• ERGM Formula

? 0
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Example 1 ? 0, L3

• Model
• ERGM Formula

? 0
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Example 1 ? 0

• Model
• ERGM Formula

? 0
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Example 2 ? -0.69

• Model
• ERGM Formula

? -0.69
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Example 3 ? 0.69

• Model
• ERGM Formula

?0.69
? 0.69
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Why Change Statistics?

• Huge Sample Space

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ERG modeling

• Conditional Probability and Change Statistics

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Conditional Probability vs. Total Probability

• Total probability of the whole network
• It is impossible to calculate when the
  size of the network gets large
• Introduce the Conditional Probability of edges
• Reduce sample space

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Avoid the Calculation on Sample Space

• Conditional Probability of an Edge to exist
• Conditional Probability of an Edge to be absent
  is
• Logit p model model log odds ratio of Yij exists

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Change Statistics (logit p model)

• From the end of last slide, we have
• Define Change Statistics as
• Model log odds of a tie being present to absent

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Estimation and Simulation

• (Monte Carlo Markov Chain Maximum Likelihood
  Method)

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Review Maximum Likelihood Estimation (MLE)

• Likelihood functions
• Estimate parameter ? given the observed network.
• Maximum Likelihood Estimation
• Find ? values such that the observed statistics
  are equal to the expected statistics
• Approximate MLE by simulation

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Procedures for simulating ERG distribution

• Markov Chain Monte Carlo Maximum Likelihood
  Estimation (MCMCMLE)
• 1. Simulate a distribution of random graphs from
  a starting set of parameter values
• 2. Refine the parameter values by comparing the
  distribution of graphs against the observed graph
• 3. Repeat this process until the parameter
  estimate stabilize

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Convergence T-statistics

• Test adequacy of parameter values estimated
• T-statistics for each configuration
• T lt.1? good fit
• NOTE If the parameter estimates do not converge,
  the model is degenerate

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A Simple Example of MCMCMLE

• Model
• Observed Network y
• Goal Find ? value such that the observed number
  of edges are equal to the expected number of
  edges

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If ? can be chosen from the following 3 cases,
?-0.69 is preferred because it gives the highest
probability for the observed network

• Given the observed Network y

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Markov dependence (Frank and Strauss, 1986)

• Potential ties are dependent only if they share a
  common actor
• Two possible network ties are conditionally
  independent unless they share a common actor
• Once homogeneity assumption is imposed, we obtain
  the following configurations

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Markov random graph models(non-directed networks)
Two-star(?2)
Density or edge(?)
Triangle(?)
Three-star(?3)
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Problems of degeneracy for Markov random models

• Certain parameter values place almost all of the
  probability mass on either the empty or the full
  graph
• Simulation studies showed that Markov random
  graph models are degenerate for many empirical
  networks with high level of clustering
• A few very high degree nodes
• Some regions of high triangulation

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Two possibilities for the degeneracy problem
(Snijders, et al 2006)

• Makov dependence assumption may be too
  restrictive
• The representation of transitivity by the total
  number of triangles might be too simplistic
• ? New specification of higher order network
  dependency

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New development in ERGM

• Partial conditional dependence assumption and
• new model specification

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Partial conditional dependence(Social circuit
dependence)

• Two possible network ties being conditionally
  dependent if their observation would lead to a
  4-cycle

i
k
possible edges observed edges
j
l
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Partial conditional dependence(Example)
Daughter B
Daughter A
Father B
Father A
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Difference between the two types of dependence
assumptions
Markov dependence assumptions
Partial conditional dependence assumptions
i
k
k
i
j
l
j
l
potential tie ties which affect the formation
of the potential tie ties with no effect on the
potential tie
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New Specifications of ERGM

• Represent structural parameters similar to the
  Markov parameters
• Effects are incorporated within the one
  configuration parameter
• Three new statistics for non-directed network
• Alternating k-stars
• Alternating k-triangles
• Alternating independent two-paths

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Examples of new specifications

• Alternating k-star configuration (degree distn)
• Alternating k-triangle (tendency to form triads)
• Alternating k-two-path (tendency to form cycles)

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Interpretation of the parameter

• Positive alternating k-star parameter
• Networks with some higher degree nodes are highly
  probable. ? Core-periphery structure
• Positive alternating k-triangle parameter
• Triangulation in the network as well as
  tendencies for triangles themselves group
  together in larger higher order clump
• Positive alternating k-path parameter
• Tendency for 4-cycles in the network

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Summary for model construction

• Random variables
• Each network tie (Yij) among nodes of a network
• A random tie variable Yij1 if a tie form i to j
  exist, Yij0 otherwise
• yij the observed value of the variable Yij
• Dependence assumptions
• Define contingencies among network variables
• Determine the type of parameters in the model
• Ties also depends on node-level attributes
  (homophily)
• Homogeneity assumption
• Simplify parameters by imposing homogeneity
  constraints.
• Estimation procedures
• Find the best parameter values based on the
  observed network
• Use simulation (MCMLE)

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Software for ERGM

• SIENA (Snijders, and colleagues)
• PNet (Robbins, and colleagues)
• Statnet (Butts, and colleagues)

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Reference

• Harrigan, Nicholas. Exponential Rnadom Graph
  (ERG) models and their application to the study
  of corporate elites.
• Robins, Garry (manuscript). Exponential Random
  Graph (p) models for social Networks, published
  in Melnet website.
• Robins, G., Pattison, P. Kalish, y. Lusher, D.
  (2007). An introduction to exponential random
  graph (p) models for social networks. Social
  Networks, 29, 173-191.
• Snijders, T.A.B., Pattison, P., Robins, G,
  Hancock M. (2006). New specifications for
  exponential random graph models. Sociological
  Methodology, 36 99-153.

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Thank you for your attention

• Any questions?