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Choosing to grow a graph

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Slides from my talk at the SINM satellite at NetSci.
May 27, 2019.
Burlington, Vermont.

Published in: Data & Analytics
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Choosing to grow a graph

  1. 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
  2. 2. How do social networks grow? 2
  3. 3. 3 networksciencebook.com [Caldarelli+ 02] k [Barabási-Albert 99] Static graph Observe Heavy-tailed degree distribution! Model & reproduce
  4. 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. 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. 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 <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. 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. 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. 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
  10. 10. 10 Features xij actually change over time; notation omitted for readability.
  11. 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. 12. Example. Generalized Preferential Attachment. 12 Pr(i chooses j) = d↵ j P k2V d↵ k<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 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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. 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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×