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
How do social networks grow?
2
3
networksciencebook.com [Caldarelli+ 02]
k
[Barabási-Albert 99]
Static graph
Observe
Heavy-tailed degree distribution!
Model &
reproduce
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.
We can leverage temporal data through
the lens of discrete choice theory.
5
Discrete choice theory
preferential
attachment
fitnesshomophily
triadic closure
uniform attachment
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).
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
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)
We think about each edge i → j as a choice made by i
with the conditional logit.
9
C
C
time 1 time 2
10
Features xij actually change over time;
notation omitted for readability.
Example. Uniform (random) attachment.
11
Uniform attachment.
Conditional logit.
Vij = ✓T
xij, Pr(i chooses j) =
e✓T
xij
P
k2C e✓T xik
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Uniform attachment as conditional logit.
Pr(i chooses j) =
1
|V|<latexit 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sha1_base64="YtaAq5P2I37bhkDLtUkju925lNM=">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</latexit><latexit sha1_base64="YtaAq5P2I37bhkDLtUkju925lNM=">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</latexit>
xij = 0, C = V
e✓T
xij
= 1<latexit 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sha1_base64="1xxMBcr8gCkU6HQkPv1XFU156jA=">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</latexit><latexit sha1_base64="1xxMBcr8gCkU6HQkPv1XFU156jA=">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</latexit>
Example. Generalized Preferential Attachment.
12
Pr(i chooses j) =
d↵
j
P
k2V d↵
k<latexit 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sha1_base64="91TRelHBKwn8ECvbrf4O5DvPSOI=">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</latexit><latexit 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Generalized PA. [Albert-Barabási 99,Krapivsky+ 00,many others…]
Conditional logit.
Vij = ✓T
xij, Pr(i chooses j) =
e✓T
xij
P
k2C e✓T xik
<latexit sha1_base64="KUn/oCGVzWiKUIHnRKNHCLzGo5k=">AAAHpXicfVVdb9s2FFW6Le60j6br417YBca6QnbsFFmabRkMtChWoMGy2UkLRK5LSVcWa1LSSKqxQ/CH7mn7Kbu0nCVysgmwRV3ecw7vB8mo5EzpXu/PjTsfffzJZuvup/5nn3/x5b2t+1+dqqKSMZzEBS/km4gq4CyHE800hzelBCoiDq+j2TM3//oDSMWKfKQXJYwFneYsZTHVaJpsXZxODHtvySEJdQaavh3Nl4Yg/BEtMNfmWNpHbDUmcVYUChSx5P13DpNKGht4a5pga02oKjExMxKynDyzZM1lhi7+ZGu71+0tH3Jz0F8Ntgc/7/418DzveHJ/80GYFHElINcxp0qd9XulHhsqNYs5WD+sFJQ0ntEpnOEwpwLU2CyTZEkbLQlJC4m/XJOl1b8OQR5JFw0Wo2lUcSrnTWtUFDOcUdb3m5o6fTo2LC8rDXlcS6YVJ7ogLvckYRJizRekqavZ7CLIWQwumwEVSlCdBSVz6wz07KIzlbTMAkFnEAPnV6Z6VQ7OWSSpXLgQinMVRMg8lUWVJyooqdYgc4V4Ldk8UBktQQUp00FMeey+E4cpeaEFlTP1X6xdgfXDyWXmOGgzqlINv0NijYTk4dPew4ij7nUPLPlUAuTWLF/O5zxjGtZ8Il6BNe7/moffJpnWpfphZwd7r6s0csM8zmg+hW5ciJ0/KlCuidVO//u9g92DHQWCYa9H2Nqic8501nFBdFjeiXBHgFz6Pdnfrl9+6BJKcce4/PjhlBcR5SF+hg42gFxVEgZJwbEBBrhf4iKBw1ACp/NLbIGLbzbR2ag/Nq5wrgEaVT4eDWnukishh3MMQNA8MWFKBeOLBFJace32TXo5bjaJSl1XWL99XUxhBSE57HUPglgwFMW24NjyKKDnKnUUzSCRO8z13FENarBRj89wr+2N7XpQzwE3mYThQkQFf4EhmZpFWfPr0StrcichmDXCGobLDYegb3NGQ7IOiVaQlYYDDKsIy6krV9LbBdYVhi+OXEouBUb9RvpMNLdG8SsR51yjzUv0dDmgvMyovVrqu5drWU+mHFicderc3zaDhVZ4vDTPB+ForldZDNlUoFJYd5WjM2EkTFjb7Y22EK/wDE9uQ6wmbFPicTiPqDzD5guzqJib8IP7b/thJisOJAM2zTServt7pSZtMsqA0FhXlBOE+eEMT4hed3cP5m1y+bTJc7x/aB4DiUCf4/51vgTFiFqm0a+l2j4hS4JOr9sH0b5ED7NCYnZYPiVFTrCpCIdUE8UScIhrcW337b8keAE8+V8SuYxkyWJX10h//dK4OTjd7fZxeb/hffKTVz93va+9b7xHXt/b9wbeL96xd+LF3t8bmxv3NrZa37aOWqPWae16Z2OFeeA1ntbkH6IxseU=</latexit><latexit 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sha1_base64="ZCfo+t3jDqg2ywS3qWVTZ+UK4vA=">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</latexit><latexit sha1_base64="VAq6YYMOOy2E5kipfnw1DJqFE/s=">AAAHpXicfVVdb9s2FFW6Le60j6br417YBca6QnbsFFmabQEMtChWoMGy2U0LRK5LSVcWa1LSSKqRQ/CH7m0/ZZe2s1hONgG2qMt7zuH9IBmVnCnd6/21deeTTz/bbt393P/iy6++vrdz/5szVVQyhtdxwQv5NqIKOMvhtWaaw9tSAhURhzfR7Jmbf/MRpGJFPtLzEsaCTnOWsphqNE12Ls8mhn2w5JiEOgNN343qhSEIf0YL1NqcSvuIrcYkzopCgSKWfPjBYVJJYwPvTBNsrQlVJSZmRkKWk2eWbLjM0MWf7Oz2ur3FQ24O+qvBrrd6Tif3tx+ESRFXAnIdc6rUeb9X6rGhUrOYg/XDSkFJ4xmdwjkOcypAjc0iSZa00ZKQtJD4yzVZWP11CPJIOm+wGE2jilNZN61RUcxwRlnfb2rq9OnYsLysNOTxUjKtONEFcbknCZMQaz4nTV3NZpdBzmJw2QyoUILqLCiZW2egZ5edqaRlFgg6gxg4vzYtV+XgnEWSyrkLobhQQYTMU1lUeaKCkmoNMleI15LVgcpoCSpImQ5iymP3nThMyQstqJyp/2LtCqwfTi4yx0GbUZVq+AMSayQkD5/2HkYcddc9sORTCZBbs3g5n4uMadjwiXgF1rj/NQ+/TTKtS/XT3h72Xldp5IY6zmg+hW5ciL0/K1CuidVe/8eDo/2jPQWCYa9H2Nqic8F01nFBdFjeiXBHgFz4PTncXb780CWU4o5x+fHDKS8iykP8DB1sALmqJAySgmMDDHC/xEUCx6EETusrbIGLbzbR+ag/Nq5wrgEaVT4dDWnukishhwsMQNA8MWFKBePzBFJace32TXo1bjaJSl1XWL+9LqawgpAc97pHQSwYimJbcGx5FNC1Sh1FM0jkDnNdO6rBEmzU43PcawdjuxnUc8BNJmE4F1HBX2BIZsmirPnt5JU1uZMQzBphDcPlhkPQtzmjIdmERCvISsMBhlWE5dSVK+ntApsKwxcnLiVXAqN+I30mqq1R/FrEOS/R5iV6uhxQXmbUXi/1/cuNrCdTDizOOsvc3zaDhVZ4vDTPB+Fo1qsshmwqUClcdpWjM2EkTLi02xttIV7hGZ7chlhN2KbE47COqDzH5guzqKhN+NH9t/0wkxUHkgGbZhpP18ODUpM2GWVAaKwrygnC/HCGJ0Svu38AdZtcPW3yHO8fmsdAItAXuH+dL0ExohZp9JdSbZ+QBUGn1+2DaF+hh1khMTssn5IiJ9hUhEOqiWIJOMRaXLt9+y8JXgBP/pdELiJZsNjVNdLfvDRuDs72u31c3u+93cEvqwvlrvet9533yOt7h97A+9U79V57sff31vbWva2d1vetk9aodbZ0vbO1wjzwGk9r8g+t+rAW</latexit>
Generalized PA as conditional logit.
xij = log(dj), C = V, ✓ = ↵<latexit 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e✓T
xij
= e↵ log dj
= d↵
j<latexit 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sha1_base64="lNAqm/a7QdPQwEosVEH6zkLsdzs=">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</latexit>
Pr(i chooses j) =
fj(dj)
P
k2V fk(dk)<latexit 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sha1_base64="YCI8GUQcQauaQTgALe99+WgFMqc=">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</latexit><latexit sha1_base64="YCI8GUQcQauaQTgALe99+WgFMqc=">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</latexit>
Example. Non-parametric Preferential Attachment.
13
Non-parametric PA.
Conditional logit.
Vij = ✓T
xij, Pr(i chooses j) =
e✓T
xij
P
k2C e✓T xik
<latexit 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Non-parametric PA as conditional logit.
xij = edj
2 Rn
, ✓T
edj
= ✓dj
, C = V<latexit 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fj(dj) = e
✓dj
<latexit 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14
Process Vi,j C
Uniform attachment 0 V
Preferential attachment ↵ log dj V
Non-parametric PA ✓dj
V
Triadic closure 0 {j : FoFi,j}
FoF attachment ↵ log ⌘i,j V
PA, FoFs only ↵ log dj {j : FoFi,j}
Individual node fitness ✓j V
PA with fitness ↵ log dj + ✓j V
Latent space · d(i, j) V
Stochastic block model !gi,gj
V
Homophily h · I
⇥
gi = gj
⇤
V
<latexit 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Estimating model parameters from data.
15
• Log likelihood is concave.
• Choice set can be large (all nodes), so we use negative sampling
(unbiased and consistent if sampled uniformly at random).
• Get standard errors from , where H is the hessian
p
H 1
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LL(✓; i chooses j from C) = ✓T
xij log(
P
k2C e✓T
xik
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The logit model provides a clean way to estimate
preferential attachment kernels.
16
(see also PAFit by Pham,Sheridan,and Shimodaira,2015)
2001
17
The next most common discrete
choice model is the mixed logit…
Mixed logits provide additional modeling flexibility.
18
• Two modes with latent parameters 𝜃1 and 𝜃2,mixture
probabilities 𝜋1 and 1 - 𝜋1, and choice sets C1 and C2.
• With probability 𝜋1,
• With probability 1 - 𝜋1,
• Can learn 𝜃1,𝜃2,and 𝜋1 with EM.
Pr(i chooses j) =
e✓T
1 xij
P
k2C1
e✓T
1
xik
<latexit 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Pr(i chooses j) =
e✓T
2 xij
P
k2C2
e✓T
2
xik
<latexit 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Brief interlude on the power of mixed logits.
19
Uij = Vij + "ij, j 2 C<latexit 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Random utility model.
• i selects argmaxk Uik
• Universality Theorem [McFadden-Train 00].
Any random utility model can be represented as a mixture of logits
(but possibly a large number of mixture components).
Example. Local search as a mixed logit.
20
• Node i is making a connection.
With probability r, i connects to a random node.
With probability 1 – r, i connects to a friend of a friend.
Local search [Holme-Kim 02; Jackson-Rogers 07]
Mixed logit.
xij = 0, ✓i arbitrary (same as uniform attachment)
⇡1 = r
C1 = V, C2 = {k | (i, j), (j, k) 2 E, (i, k) /2 E}<latexit 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Example. Copy model as a mixed logit.
21
• New node i arrives and selects random node j.
With probability p, connects to j.
With probability 1 – p, connects to a random neighbor of j.
Copy model. [Kumar+ 00]
Mixed logit.
xij =

0
log(dj)
, ✓1 =

1
0
, ✓2 =

0
1
✓T
1 xij = 0, ✓T
2 xij = log(dj)
⇡1 = p, C1 = C2 = V<latexit 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It’s easy to create new growth models in this framework.
22
• New (r, p) model (mixed logit with 4 modes).
• Node i is making a choice.
• Choice set determined by r:
w/p r, C = all nodes not connected to i
w/p 1 – r, C = friends of friends of i not connected to i
• Formation determined by p:
w/p p, uniform attachment
w/p 1 – p, linear preferential attachment
• p = 1 → local search
• r = 1 → copy model
23
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3
4
5
0 0.25 0.50 0.75 1
r
Estimateofg p
0.01
0.25
0.50
0.75
1.00
Power law exponent estimated with MLE. 20,000 nodes.
Lesson. We shouldn’t estimate a formation model
from an observed degree distribution.
24
What actually happens on some real data?
Flickr analysis.
25
Model
#1 #2 #3 #4
log Followers 1.149* 0.715* 0.536*
(0.007) (0.009) (0.010)
Has degree -0.580* -0.631* -1.745*
(0.202) (0.190) (0.234)
Reciprocal 8.419* 8.347* 8.197* 7.903*
(0.220) (0.220) (0.240) (0.244)
Is FoF 6.12* 3.955*
(0.045) (0.050)
2 Hops 6.290*
(0.190)
3 Hops 2.851*
(0.185)
4 Hops 0.583*
(0.189)
5 Hops -0.585*
(0.218)
6 Hops -1.122*
(0.266)
Observations 20,000 20,000 20,000 20,000
Log-likelihood -16,448 -14,685 -10,728 -9,789
Test accuracy 0.758 0.722 0.853 0.855
Note: *p<0.01
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Citation analysis.
26
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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There are some similar approaches from the more
sociological literature.
27
• Relational Event Models [Butts 08]
• Stochastic Actor Oriented Models [Snijders+ 10]
• Temporal Exponential Random Graph Models [Hanneke-Xing 06]
• Dynamic Network Actor Models [Stadtfeld-Block 17]
Learning parameters is costly. Mostly limited to small datasets.
Designed for “snapshot” data.
Some limitations of our work.
28
• Requires directionality of edges so there is a “chooser”.
Collect this information if possible! (e.g., Facebook)
• Requires “fine-grained” temporal information.
Collect data on fine time scales! (avoid lumping)
• Conditional logit requires i.i.d. errors.
Could look at other discrete choice models (e.g., probit).
Choosing to grow a graph.
29
• Temporal data and discrete choice theory provide a convenient
lens through which to view network growth models.
• Many common growth models can be phrased in terms of
discrete choice models.
• Learning these generalized linear models is flexible and scalable.
30
THANKS! Austin R. Benson
Slides. bit.ly/arb-SINM-19
http://cs.cornell.edu/~arb
@austinbenson
arb@cs.cornell.edu
Choosing to
grow a graph
bit.ly/c2g-code
(code, reproducibility, and data)

Choosing to grow a graph

  • 1.
    1 Joint work with JanOvergoor & Johan Ugander (Stanford) Choosing to grow a graph Austin R. Benson · Cornell University NetSci SINM May 27, 2019 Slides. bit.ly/arb-SINM-19
  • 2.
    How do socialnetworks grow? 2
  • 3.
    3 networksciencebook.com [Caldarelli+ 02] k [Barabási-Albert99] Static graph Observe Heavy-tailed degree distribution! Model & reproduce
  • 4.
    Knowing the orderingof 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 leveragetemporal data through the lens of discrete choice theory. 5 Discrete choice theory preferential attachment fitnesshomophily triadic closure uniform attachment
  • 6.
    Why discrete choicetheory? 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 <latexit 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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 choiceand 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 abouteach edge i → j as a choice made by i with the conditional logit. 9 C C time 1 time 2
  • 10.
    10 Features xij actuallychange over time; notation omitted for readability.
  • 11.
    Example. Uniform (random)attachment. 11 Uniform attachment. Conditional logit. Vij = ✓T xij, Pr(i chooses j) = e✓T xij P k2C e✓T xik <latexit 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Uniform attachment as conditional logit. Pr(i chooses j) = 1 |V|<latexit 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xij = 0, C = V e✓T xij = 1<latexit 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  • 12.
    Example. Generalized PreferentialAttachment. 12 Pr(i chooses j) = d↵ j P k2V d↵ k<latexit 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sha1_base64="tF5kcHidwiXpa3nGjdRzPxardGU=">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</latexit> Generalized PA. [Albert-Barabási 99,Krapivsky+ 00,many others…] Conditional logit. Vij = ✓T xij, Pr(i chooses j) = e✓T xij P k2C e✓T xik <latexit 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Generalized PA as conditional logit. xij = log(dj), C = V, ✓ = ↵<latexit 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e✓T xij = e↵ log dj = d↵ j<latexit 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  • 13.
    Pr(i chooses j)= fj(dj) P k2V fk(dk)<latexit 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Example. Non-parametric Preferential Attachment. 13 Non-parametric PA. Conditional logit. Vij = ✓T xij, Pr(i chooses j) = e✓T xij P k2C e✓T xik <latexit 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Non-parametric PA as conditional logit. xij = edj 2 Rn , ✓T edj = ✓dj , C = V<latexit 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fj(dj) = e ✓dj <latexit 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  • 14.
    14 Process Vi,j C Uniformattachment 0 V Preferential attachment ↵ log dj V Non-parametric PA ✓dj V Triadic closure 0 {j : FoFi,j} FoF attachment ↵ log ⌘i,j V PA, FoFs only ↵ log dj {j : FoFi,j} Individual node fitness ✓j V PA with fitness ↵ log dj + ✓j V Latent space · d(i, j) V Stochastic block model !gi,gj V Homophily h · I ⇥ gi = gj ⇤ V <latexit 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  • 15.
    Estimating model parametersfrom data. 15 • Log likelihood is concave. • Choice set can be large (all nodes), so we use negative sampling (unbiased and consistent if sampled uniformly at random). • Get standard errors from , where H is the hessian p H 1 <latexit 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LL(✓; i chooses j from C) = ✓T xij log( P k2C e✓T xik )<latexit 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  • 16.
    The logit modelprovides a clean way to estimate preferential attachment kernels. 16 (see also PAFit by Pham,Sheridan,and Shimodaira,2015) 2001
  • 17.
    17 The next mostcommon discrete choice model is the mixed logit…
  • 18.
    Mixed logits provideadditional modeling flexibility. 18 • Two modes with latent parameters 𝜃1 and 𝜃2,mixture probabilities 𝜋1 and 1 - 𝜋1, and choice sets C1 and C2. • With probability 𝜋1, • With probability 1 - 𝜋1, • Can learn 𝜃1,𝜃2,and 𝜋1 with EM. Pr(i chooses j) = e✓T 1 xij P k2C1 e✓T 1 xik <latexit 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Pr(i chooses j) = e✓T 2 xij P k2C2 e✓T 2 xik <latexit 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  • 19.
    Brief interlude onthe power of mixed logits. 19 Uij = Vij + "ij, j 2 C<latexit 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Random utility model. • i selects argmaxk Uik • Universality Theorem [McFadden-Train 00]. Any random utility model can be represented as a mixture of logits (but possibly a large number of mixture components).
  • 20.
    Example. Local searchas a mixed logit. 20 • Node i is making a connection. With probability r, i connects to a random node. With probability 1 – r, i connects to a friend of a friend. Local search [Holme-Kim 02; Jackson-Rogers 07] Mixed logit. xij = 0, ✓i arbitrary (same as uniform attachment) ⇡1 = r C1 = V, C2 = {k | (i, j), (j, k) 2 E, (i, k) /2 E}<latexit 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  • 21.
    Example. Copy modelas a mixed logit. 21 • New node i arrives and selects random node j. With probability p, connects to j. With probability 1 – p, connects to a random neighbor of j. Copy model. [Kumar+ 00] Mixed logit. xij =  0 log(dj) , ✓1 =  1 0 , ✓2 =  0 1 ✓T 1 xij = 0, ✓T 2 xij = log(dj) ⇡1 = p, C1 = C2 = V<latexit 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  • 22.
    It’s easy tocreate new growth models in this framework. 22 • New (r, p) model (mixed logit with 4 modes). • Node i is making a choice. • Choice set determined by r: w/p r, C = all nodes not connected to i w/p 1 – r, C = friends of friends of i not connected to i • Formation determined by p: w/p p, uniform attachment w/p 1 – p, linear preferential attachment • p = 1 → local search • r = 1 → copy model
  • 23.
    23 ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3 4 5 0 0.250.50 0.75 1 r Estimateofg p 0.01 0.25 0.50 0.75 1.00 Power law exponent estimated with MLE. 20,000 nodes. Lesson. We shouldn’t estimate a formation model from an observed degree distribution.
  • 24.
    24 What actually happenson some real data?
  • 25.
    Flickr analysis. 25 Model #1 #2#3 #4 log Followers 1.149* 0.715* 0.536* (0.007) (0.009) (0.010) Has degree -0.580* -0.631* -1.745* (0.202) (0.190) (0.234) Reciprocal 8.419* 8.347* 8.197* 7.903* (0.220) (0.220) (0.240) (0.244) Is FoF 6.12* 3.955* (0.045) (0.050) 2 Hops 6.290* (0.190) 3 Hops 2.851* (0.185) 4 Hops 0.583* (0.189) 5 Hops -0.585* (0.218) 6 Hops -1.122* (0.266) Observations 20,000 20,000 20,000 20,000 Log-likelihood -16,448 -14,685 -10,728 -9,789 Test accuracy 0.758 0.722 0.853 0.855 Note: *p<0.01 <latexit 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  • 26.
    Citation analysis. 26 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 <latexit 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  • 27.
    There are somesimilar approaches from the more sociological literature. 27 • Relational Event Models [Butts 08] • Stochastic Actor Oriented Models [Snijders+ 10] • Temporal Exponential Random Graph Models [Hanneke-Xing 06] • Dynamic Network Actor Models [Stadtfeld-Block 17] Learning parameters is costly. Mostly limited to small datasets. Designed for “snapshot” data.
  • 28.
    Some limitations ofour work. 28 • Requires directionality of edges so there is a “chooser”. Collect this information if possible! (e.g., Facebook) • Requires “fine-grained” temporal information. Collect data on fine time scales! (avoid lumping) • Conditional logit requires i.i.d. errors. Could look at other discrete choice models (e.g., probit).
  • 29.
    Choosing to growa graph. 29 • Temporal data and discrete choice theory provide a convenient lens through which to view network growth models. • Many common growth models can be phrased in terms of discrete choice models. • Learning these generalized linear models is flexible and scalable.
  • 30.
    30 THANKS! Austin R.Benson Slides. bit.ly/arb-SINM-19 http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu Choosing to grow a graph bit.ly/c2g-code (code, reproducibility, and data)