András Vörös
Department of Social Statistics, University of Manchester
SOST71032 Social Network AnalysisInternally Assure the Quality of Assessment
Introduction to Exponential Random Graph Models (ERGMs)
Model specification
Model specification is not a simple issue in ERGMs
Simple ERGMs: modeled statistics are not interrelated
Nested ERGMs: modeled statistics are interrelated
Note that dependence assumptions are nested too:
Bernoulli < dyadic < Markov < social circuit
In nested models: multiple mechanisms explain same structural outcome
All interesting real-life cases require nested ERGMs
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Model specification in ERGMs: simple and nested models
e: 1
m: 1
t: 1
Competing mechanisms are represented by different subgraphs
Subgraphs: structural configurations (also called “motifs” in network science)
Typical subgraph statistics for Markov dependence:
Edge: number of edges in the network
Mutuality: number of reciprocated ties or dyads (in directed networks)
Triangles: number of closed triangles (in directed networks: transitive and cyclic triangles)
Stars: number of 2-stars (2-paths), 3-stars, 4-stars, … k-stars (in directed: in-, out-stars)
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Model specification in ERGMs: “classic” Markov models
edge
(1-star) triangle 2-star 3-star 4-star
Why are nested models complicated?
Issue with Markov specification
Sometimes networks generated by the model look like the observed one, but only on average
But most specific draws from the ERGM are unrealistic networks
Thus, we think the estimated model is a good representation of the observed network, but no!
Degeneracy and multi-modality
Convergence slow or cannot be achieved
Even if converged, the model generates unrealistic outcomes
(Near-)degeneracy: probability distribution (nearly) collapses into empty and complete graph
Multi-modality: probability mass centers around a few extreme network structures
The problem occurs in many interesting cases
larger, denser networks with strong clustering real-life social networks
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Model specification in ERGMs: degeneracy and multi-modality
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Model specification in ERGMs: degeneracy and multi-modality
Model terms Simulated networks The problem
From: Handcock et al. (2008)
“Target statistics”:
1. number of edges
2. mean clustering
coefficient
Simulate from
parameters that on
average reproduce
the observed targets.
Nested terms:
an edge may also contribute
to a large number of triangles.
As a result, the model is
always “drawn towards”
sparse and clustered or
dense and random outcomes.
This is not an issue of the
estimation algorithm, the
model specification is wrong.
What causes degeneracy and multi-modality?
1) “Nestedness” or “embeddedness” of configurations
one tie can complete many triangles
2) Assumption of simple additive effects
e.g. 2 shared friends has twice the effect on closing a triad than 1; 10 have 10 times the effect
We need to look at the context of ties in the network beyond the triad
Diminishing returns: some ties should have lower effects than others
e.g. 2 shared friends should not count twice as much as 1 for closing a triad
Further relax Markov dependence social circuit model
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Model specification in ERGMs: degeneracy and multi-modality
Social circuit dependence (Snijders et al. 2006):
Ties are interdependent if they are in configurations with 4 or more nodes
Proposed structural effects to replace the Markov versions:
So-called “alternating” stars/paths, triangles – diminishing effects
GWESP (geometrically weighted edgewise shared partners) for triangles
GWDegree (geometrically weighted degrees) for stars/paths
Decay parameter, 0-1: closer to 1, bigger discount on additional ties
When are these preferred: practically for all interesting social networks
Or whenever problems occur with convergence, degeneracy, multi-modality – specification!
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Model specification in ERGMs: “new” social circuit models
Many other effects to consider when specifying models
Individual covariates (e.g. gender)
Sender – activity
Receiver – popularity
Similarity – homophily
…
Dyadic covariates (e.g. other networks)
Tie coincidence
Mixed reciprocity
Mixed transitivity
…
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Model specification in ERGMs: covariates and multiple networks
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