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Choosing to grow a graph: discrete choice models of network formation
1. 1
Joint work with
Jan Overgoor &
Johan Ugander
(Stanford)
Choosing to grow a graph
Austin R. Benson · Cornell University
NetSci SINM
May 27, 2019
Slides. bit.ly/arb-SINM-19
4. Knowing the ordering of edges provides much better
information for estimating mechanistic effects.
4
Power law distributions of
the World Wide Web.
Adamic and Huberman, 2000.
5. We can leverage temporal data through
the lens of discrete choice theory.
5
Discrete choice theory
preferential
attachment
fitnesshomophily
triadic closure
uniform attachment
6. Why discrete choice theory?
6
• More statistical perspective on network growth
(standard errors on estimates, likelihood ratio tests, …)
• Super flexible models with existing optimization routines.
• Easy to think of new models.
• Easy to incorporate covariates into growth models.
Model
#1 #2 #3 #4
log Citations 0.717* 0.794* 1.052* 1.044*
(0.008) (0.010) (0.012) (0.012)
Has degree 1.684* 1.677* 1.862* 1.830*
(0.053) (0.062) (0.063) (0.064)
Has same author 6.523* 5.928* 5.913*
(0.110) (0.114) (0.114)
log Age -1.096* -1.069*
(0.018) (0.021)
Max papers by author 0.029*
(0.011)
Observations 10,000 10,000 10,000 10,000
Log-likelihood -20,799 -16,600 -14,384 -14,390
Test accuracy 0.358 0.484 0.533 0.534
Note: *p<0.01
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The trick.
We use temporal information on edge
creation,so we don’t have to guess about
the formation model from summary
statistics (e.g.,degree distributions).
7. Background. Discrete choice and random utility models
form a workhorse framework in econometrics.
7
Uij = Vij + "ij, j 2 C<latexit 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• “Chooser” i makes a choice j from available alternatives C.
Random utility model.
base utility (constant)
random error
choice set
• i selects argmaxk Uik
random
utility
8. Vij = ✓T
xij
"ij ⇠ Gumbel(0, 1), i.i.d.<latexit 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Background. The conditional logit is one of the most
common random utility models.
8
Random utility model.
• “Chooser” i makes a choice from available alternatives C.
• Random utilities .
• i selects argmaxk Uik
Conditional logit.
feature vector
Uij = Vij + "ij, j 2 C<latexit 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Pr(i chooses j) = Pr(Uij > Uik, k 2 C{j}) =
e✓T
xij
P
k2C e✓T xik
<latexit 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Computational goals.
• Estimate 𝜃 from data (log likelihood is concave)
9. We think about each edge i → j as a choice made by i
with the conditional logit.
9
C
C
time 1 time 2