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Link prediction in networks with core-fringe structure

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Slides from my talk at WWW 2019.
May 16, 2019

Published in: Data & Analytics
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Link prediction in networks with core-fringe structure

  1. 1. 1 Joint work with Jon Kleinberg (Cornell) Link prediction in networks with core-fringe structure. Austin R. Benson · Cornell University The Web Conference May 16, 2019 Slides. bit.ly/arb-WWW-19
  2. 2. Data is often collected by recording the interactions of a core set of specified individuals. 2 We call this core-fringe structure [Benson-Kleinberg,NeurIPS 2018] core fringe Enron emails
  3. 3. 3 How does knowledge about the fringe help with understanding the core?
  4. 4. We studied how including fringe connections affects the performance of link prediction algorithms. 4 1. Fix a link prediction algorithm. 2. Include fringe nodes and connections in some order, possibly changing the algorithm predictions. 3. Measure link prediction accuracy on core nodes as a function of the number of fringe nodes included. CommonNeighbors(u, v) = |N(u) N(v)| Jaccard(u, v) = |N(u) N(v)|/|N(u) [ N(v)|<latexit 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Link prediction. Given a graph, predict new links that will form in the future or links missing from the graph (hold-out 20% of edges by time or randomly). core fringe
  5. 5. 5
  6. 6. 6 Conventional data science and machine learning wisdom would say that more data is better... … but we observe rich and diverse behavior in how including the fringe affects link prediction performance.
  7. 7. 7 Datasets. 1. Email networks → core is some organization 2. Phone call and text networks → core is set of survey participants [Eagle-Pentland 2006] 3. Online social network → core is set of users in geographic region core fringe
  8. 8. Sometimes,including the entire fringe gives the best prediction performance. 8
  9. 9. Sometimes,including the entire fringe gives the best prediction performance. 9
  10. 10. Other times,including no fringe information leads to the best prediction performance. 10
  11. 11. In many cases,we want to include some intermediate amount of fringe data. 11
  12. 12. In many cases,we want to include some intermediate amount of fringe data. 12
  13. 13. We also observe saturation in performance when including a large enough amount of fringe information. 13
  14. 14. We also observe saturation in performance when including a large enough amount of fringe information. 14
  15. 15. It is far from obvious how we should include the fringe information to optimize for link prediction. 15 1. Sometimes, including all of the fringe gives the best performance. 2. Other times, including no fringe information is best. 3. Still, in other cases, we want some intermediate amount of fringe. 4. Performance can saturate as we gather more data.
  16. 16. 16 Should we expect this much diversity in results?
  17. 17. 17 It turns out that core-fringe link prediction performance diversity occurs in some simple random graph models.
  18. 18. Framework for theoretical analysis. 18 1. Random graph model with natural core-fringe structure. 2. Sample nodes u, v, w, z from core such that Pr((u, v)) > Pr((w, z)). 3. Amount of fringe parameterized by d, and algorithm measures 4. Correct if Xd > Yd, so want to choose d to maximize Pr(Xd > Yd) max d SNR(Zd) = E(Zd) p V(Zd) , Zd = Xd Yd <latexit 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Xd = CommonNeighbors(u, v) Yd = CommonNeighbors(w, z).<latexit 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  19. 19. Core-fringe stochastic block model. 19 1 3 4 2 q s rr s p p • Fringe parameterization. d = number of nodes from blocks 3 and 4 (any order). • u, v, w in block 1 • z in block 2 • Pr((u, v)) > Pr((w, z)) max d SNR(Zd) = E(Zd) p V(Zd) , Zd = Xd Yd <latexit 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Xd = CommonNeighbors(u, v) Yd = CommonNeighbors(w, z).<latexit 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• Optimization.
  20. 20. Core-fringe stochastic block model. 20 1 3 4 2 q s rr s p p Lemma (all-fringe optimality). If r > 0 and s = 0, then SNR(Zd) increases monotonically in d. s = 0
  21. 21. Core-fringe stochastic block model. 21 1 3 4 2 q s rr s p p Lemma (no-fringe optimality). If s = r, then SNR(Zd) decreases monotonically in d. s = 0 s = r
  22. 22. 1-D lattice small-world model. 22 0 1 2-1-2 c-c c+d-(c+d) • u v, w, z sampled from {-c, …, c} with |u – v| < |w – z| • Pr((u, v)) > Pr((w, z)) Pr((i, j)) = 1 / |i – j| max d SNR(Zd) = E(Zd) p V(Zd) , Zd = Xd Yd <latexit 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Xd = CommonNeighbors(u, v) Yd = CommonNeighbors(w, z).<latexit 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• Optimization.
  23. 23. 1-D lattice small-world model. 23 0 1 2-1-2 c-c c+d-(c+d) Pr((i, j)) = 1 / |i – j| If SNR(Z0) < SNR(Z1), then d⇤ = arg maxd SNR(Zd) satisfies 0 < d⇤ < 1.<latexit 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limd!1 SNR(Zd) = S⇤ > 0.<latexit 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Theorem (saturation). Theorem (intermediate-fringe optimality).
  24. 24. We should be aware of how“fringe”data effects our analysis of networks. 24 • Core-fringe structure is often a consequence of how we collect data. • Including fringe info. can have diverse effects on link prediction • Effects can be seen with random graph models, but we still do not know the underlying causes. • Can look at other analyses with core-fringe data. core fringe
  25. 25. 25 THANKS! Austin R. Benson Slides. bit.ly/arb-WWW-19 http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu Link prediction in networks with core-fringe structure. github.com/arbenson/cflp (code, reproducibility, and data)

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