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Simplicial closure & higher-order link prediction

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Slides from my talk at ICIAM.
July 16, 2019.

Published in: Data & Analytics
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Simplicial closure & higher-order link prediction

  1. 1. 1 Joint work with Rediet Abebe & Jon Kleinberg (Cornell) Michael Schaub & Ali Jadbabaie (MIT) Simplicial closure & higher-order link prediction Austin R. Benson · Cornell University ICIAM · July 16, 2019 Slides. bit.ly/arb-ICIAM-19 bit.ly/combos-CC
  2. 2. Canonical networks are everywhere slide, but with a hidden purpose! 2 Collaboration nodes are people/groups edges link entities working together Communications nodes are people/accounts edges show info.exchange Physical proximity nodes are people/animals edges link those that interact in close proximity Drug compounds nodes are substances edge between substances that appear in the same drug
  3. 3. Real-world systems are composed of“higher-order” interactions that we often reduce to pairwise ones. 3 Collaboration nodes are people/groups teams are made up of small groups Communications nodes are people/accounts emails often have several recipients,not just one Physical proximity nodes are people/animals people often gather in small groups Drug compounds nodes are substances drugs are made up of several substances
  4. 4. There are many ways to mathematically represent the higher-order structure present in relational data. 4 • Hypergraphs [Berge 89] • Set systems [Frankl 95] • Tensors [Kolda-Bader 09] • Affiliation networks [Feld 81,Newman-Watts-Strogatz 02] • Multipartite networks [Lambiotte-Ausloos 05,Lind-Herrmann 07] • Abstract simplicial complexes [Lim 15,Osting-Palande-Wang 17] • Multilayer networks [Kivelä+ 14,Boccaletti+ 14,many others…] • Meta-paths [Sun-Han 12] • Motif-based representations [Benson-Gleich-Leskovec 15,17] • … Data representation is not the problem. Some problems: • What are the basic organizational principles of systems? • How can we evaluate and compare models?
  5. 5. We propose“higher-order link prediction”as a similar framework for evaluation of higher-order models. 5 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} Data. Observe simplices up to some time t. Using this data, we want to predict what groups of > 2 nodes will appear in a simplex in the future. t We predict structure that classical link prediction would not even consider! Possible applications • Novel combinations of drugs for treatments. • Group chat recommendation in social networks. • Team formation.
  6. 6. 6 What are the basic organizational principles of systems with higher-order interactions?
  7. 7. We collected many datasets of timestamped simplices, where each simplex is a subset of nodes. 7 1. Coauthorship in different domains. 2. Emails with multiple recipients. 3. Tags on Q&A forums. 4. Threads on Q&A forums. 5. Contact/proximity measurements. 6. Musical artist collaboration. 7. Substance makeup and classification codes applied to drugs the FDA examines. 8. U.S. Congress committee memberships and bill sponsorship. 9. Combinations of drugs seen in patients in ER visits. https://math.stackexchange.com/q/80181 bit.ly/sc-holp-data
  8. 8. Thinking of higher-order data as a weighted projected graph with“filled-in”structures is a convenient viewpoint. 8 1 2 3 4 5 6 7 8 9 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} Data. Pictures to have in mind. Projected graph W. Wij = # of simplices containing nodes i and j.
  9. 9. 9 5 23 16 20
  10. 10. 10 i j k i j k Warm-up. What’s more common in data? or “Open triangle” each pair has been in a simplex together but all 3 nodes have never been in the same simplex “Closed triangle” there is some simplex that contains all 3 nodes
  11. 11. music-rap-genius NDC-substances NDC-classes DAWN coauth-DBLP coauth-MAG-geology coauth-MAG-history congress-bills congress-committees tags-stack-overflow tags-math-sx tags-ask-ubuntu email-Eu email-Enron threads-stack-overflow threads-math-sx threads-ask-ubuntu contact-high-school contact-primary-school 10 5 10 4 10 3 10 2 10 1 Edge density in projected graph 0.00 0.25 0.50 0.75 1.00 Fractionoftrianglesopen There is lots of variation in the fraction of triangles that are open,but datasets from the same domain are similar. 11 See also [Patania-Petri-Vaccarino 17] for fraction of open triangles on several arxiv collaboration networks.
  12. 12. A simple model can account for open triangle variation. 12 • n nodes; only 3-node simplices; {i, j, k} included with prob. p = 1 / nb i.i.d. • ⟶ always get ϴ(pn3) = ϴ(n3 - b) closed triangles in expectation. b = 0.8, 0.82, 0.84, ..., 1.8 Larger b is darker marker. Proposition (rough). • b < 1 ⟶ ϴ(n3) open triangles in expectation for large n. • b > 1 ⟶ ϴ(n3(2-b)) open triangles in expectation for large n. • The number of open triangles grows faster for b < 3/2. 10 1 100 Edge density in projected graph 0.00 0.25 0.50 0.75 1.00 Fractionoftrianglesopen Exactly 3 nodes per simplex (simulated) n = 200 n = 100 n = 50 n = 25
  13. 13. Dataset domain separation also occurs at the local level. 13 • Randomly sample 100 egonets per dataset and measure log of average degree and fraction of open triangles. • Logistic regression model to predict domain (coauthorship, tags, threads, email, contact). • 75% model accuracy vs. 21% with random guessing.
  14. 14. 14 Simplicial closure. How do new simplices appear? How do new closed triangles appear?
  15. 15. Groups of nodes go through trajectories until finally reaching a“simplicial closure event.” 15 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} For this talk, we will focus on simplicial closure on 3 nodes.
  16. 16. Groups of nodes go through trajectories until finally reaching a“simplicial closure event.” 16 Substances in marketed drugs recorded in the National Drug Code directory. We bin weighted edges into “weak” and “strong ties” in the projected graph W. Wij = # of simplices containing nodes i and j. • Weak ties. Wij = 1 (one simplex contains i and j) • Strong ties. Wij > 2 (at least two simplices contain i and j)
  17. 17. We can also study temporal dynamics in aggregate. 17 Coauthorship data of scholars publishing in history. Wij = # of simplices containing nodes i and j. Most groups formed have no previous interaction. Open triangle of all weak ties more likely to form a strong tie before closing.
  18. 18. Simplicial closure depends on structure in projected graph. 18 • First 80% of the data (in time) ⟶ record configurations of triplets not in closed triangle. • Remainder of data ⟶ find fraction that are now closed triangles. Increased edge density increases closure probability. Increased tie strength increases closure probability. Tension between edge density and tie strength. Left and middle observations are consistent with theory and empirical studies of social networks. [Granovetter 73; Leskovec+ 08; Backstrom+ 06; Kossinets-Watts 06] Closure probability Closure probability Closure probability
  19. 19. 19 Higher-order link prediction. How can we predict which new simplices appear? How can we predict which triangles become closed?
  20. 20. We propose“higher-order link prediction”as a similar framework for evaluation of higher-order models. 20 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} Data. Observe simplices up to some time t. Using this data, we want to predict what groups of > 2 nodes will appear in a simplex in the future. t We predict structure that classical link prediction would not even consider!
  21. 21. 21 Our structural analysis tells us what we should be looking at for prediction. 1. Edge density is a positive indicator. ⟶ focus our attention on predicting which open triangles become closed triangles. 2. Tie strength is a positive indicator. ⟶ various ways of incorporating this information i j k Wij Wjk Wjk
  22. 22. 22 For every open triangle,we assign a score function on first 80% of data based on structural properties. Four broad classes of score functions for an open triangle. Score s(i, j, k)… 1. is a function of Wij, Wjk, Wjk arithmetic mean, geometric mean, etc. 2. looks at common neighbors of the three nodes. generalized Jaccard, Adamic-Adar, etc. 3. uses “whole-network” similarity scores on projected graph sum of PageRank or Katz scores amongst edges 4. is learned from data train a logistic regression model with features i j k Wij Wjk Wjk After computing scores, predict that open triangles with highest scores will be closed triangles in final 20% of data.
  23. 23. 23 1. s(i,j,k) is a function of Wij,Wjk,and Wjk i j k Wij Wjk Wjk 1. Arithmetic mean 2. Geometric mean 3. Harmonic mean 4. Generalized mean s(i, j, k) = 3/(W 1 ij + W 1 ik + W 1 jk ) s(i, j, k) = (WijWikWjk)1/3 s(i, j, k) = (Wij + Wik + Wjk)/3 s(i, j, k) = mp(Wij, Wjk, Wik) = (Wp ij + Wp jk + Wp ik)1/p Wij = #(simplices containing i and j)<latexit 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  24. 24. 1. Number of common neighbors of all 3 nodes 2. Generalized Jaccard coefficient 3. Preferential attachment 24 2. s(i,j,k) is a function of is a function of neighbors s(i, j, k) = |N(i) N(j) N(k)| |N(i) [ N(j) [ N(k)|<latexit 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s(i, j, k) = |N(i) N(j) N(k)|<latexit 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s(i, j, k) = |N(i)| · |N(j)| · |N(k)|<latexit 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i j k l m x y r z N(i) = {j, k, l, m, x, y, z} N(j) = {i, k, l, m, r} N(k) = {i, j, l, m}<latexit 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  25. 25. 25 3. s(i,j,k) is is built from“whole-network”similarity scores on edges: s(i,j,k) = Sij + Sji + Sjk + Skj + Sik + Ski i j k Wij Wjk Wjk 1. PageRank (unweighted or weighted) 2. Katz (unweighted or weighted) S = (I ↵WD 1 W ) 1 S = (I ↵AD 1 A ) 1 <latexit 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Wij = #(simplices containing i and j) A = min(W, 1) DW = diag(W1) DA = diag(A1)<latexit 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  26. 26. 26 4. s(i,j,k) is learned from data. 1. Split data into training and validation sets. 2. Compute features of (i, j, k) from previous ideas using training data. 3. Throw features + validation labels into machine learning blender → learn model. 4. Re-compute features on combined training + validation → apply model on the data.
  27. 27. 27
  28. 28. 28 A few lessons learned from applying all of these ideas. 1. We can predict pretty well on all datasets using some method. → 4x to 107x better than random w/r/t mean average precision depending on the dataset/method 2. Thread co-participation and co-tagging on stack exchange are consistently easy to predict. 3. Simply averaging Wij, Wjk, and Wik consistently performs well. i j k Wij Wjk Wjk
  29. 29. Generalized means of edges weights are often good predictors of new 3-node simplices appearing. 29 music-rap-genius NDC-substances NDC-classes DAWN coauth-DBLP coauth-MAG-geology coauth-MAG-history congress-bills congress-committees tags-stack-overflow tags-math-sx tags-ask-ubuntu email-Eu email-Enron threads-stack-overflow threads-math-sx threads-ask-ubuntu contact-high-school contact-primary-school harmonic geometric arithmetic p 4 3 2 1 0 1 2 3 4 0 20 40 60 80 Relativeperformance 4 3 2 1 0 1 2 3 4 p 2.5 5.0 7.5 10.0 12.5 Relativeperformance 4 3 2 1 0 1 2 3 4 p 1.0 1.5 2.0 2.5 3.0 3.5 Relativeperformance Good performance from this local information is a deviation from classical link prediction,where methods that use long paths (e.g.,PageRank) perform well [Liben-Nowell & Kleinberg 07]. For structures on k nodes,the subsets of size k-1 contain rich information only when k > 2. i j k Wij Wjk Wjk i j k ? scorep(i, j, k) = (Wp ij + Wp jk + Wp ik)1/p <latexit 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  30. 30. There are lots of opportunities in studying this data. 30 1. Higher-order data is pervasive! We have ways to represent data, and higher-order link prediction is a general framework for comparing comparing models and methods. 2. There is rich static and temporal structure in the datasets we collected. 3. Computation becomes much more challenging for 4-node patterns! Code. bit.ly/sc-holp-code Data. bit.ly/sc-holp-data Simplicial closure and higher-order link prediction. Benson, Abebe, Schaub, Jadbabaie, & Kleinberg. Proceedings of the National Academy of Sciences, 2018.
  31. 31. 31 THANKS! Austin R. Benson Slides. bit.ly/arb-graphex-19 http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu Simplicial closure & higher-order link prediction Simplicial closure and higher-order link prediction. Benson, Abebe, Schaub, Jadbabaie, & Kleinberg. Proceedings of the National Academy of Sciences, 2018. github.com/arbenson/ScHoLP-Tutorial bit.ly/sc-holp-data
  32. 32. Simplicial closure on 4 nodes similar to on 3 nodes, just“up one dimension”. 32 Increased edge density increases closure probability. Increased simplicial tie strength increases closure probability. Tension b/w edge density simplicial tie strength. Closure probability Closure probability Closure probability coauth-DBLP coauth-MAG-geology coauth-MAG-history congress-bills congress-committees email-Eu email-Enron threads-stack-overflow threads-math-sx threads-ask-ubuntu music-rap-genius DAWNtags-stack-overflow tags-math-sx tags-ask-ubuntu NDC-substances NDC-classes contact-high-school contact-primary-school
  33. 33. Most open triangles do not come from asynchronous temporal behavior. 33 i j k In 61.1% to 97.4% of open triangles, all three pairs of edges have an overlapping period of activity. ⟶ there is an overlapping period of activity between all 3 edges (Helly’s theorem). # overlaps Dataset # open triangles 0 1 2 3 coauth-DBLP 1,295,214 0.012 0.143 0.123 0.722 coauth-MAG-history 96,420 0.002 0.055 0.059 0.884 coauth-MAG-geology 2,494,960 0.010 0.128 0.109 0.753 tags-stack-overflow 300,646,440 0.002 0.067 0.071 0.860 tags-math-sx 2,666,353 0.001 0.040 0.049 0.910 tags-ask-ubuntu 3,288,058 0.002 0.088 0.085 0.825 threads-stack-overflow 99,027,304 0.001 0.034 0.037 0.929 threads-math-sx 11,294,665 0.001 0.038 0.039 0.922 threads-ask-ubuntu 136,374 0.000 0.020 0.023 0.957 NDC-substances 1,136,357 0.020 0.196 0.151 0.633 NDC-classes 9,064 0.022 0.191 0.136 0.652 DAWN 5,682,552 0.027 0.216 0.155 0.602 congress-committees 190,054 0.001 0.046 0.058 0.895 congress-bills 44,857,465 0.003 0.063 0.113 0.821 email-Enron 3,317 0.008 0.130 0.151 0.711 email-Eu 234,600 0.010 0.131 0.132 0.727 contact-high-school 31,850 0.000 0.015 0.019 0.966 contact-primary-school 98,621 0.000 0.012 0.014 0.974 music-rap-genius 70,057 0.028 0.221 0.141 0.611
  34. 34. We still have variety with only 3-node simplices. 34 music-rap-genius NDC-substances NDC-classes DAWN coauth-DBLP coauth-MAG-geology coauth-MAG-history congress-bills congress-committees tags-stack-overflow tags-math-sx tags-ask-ubuntu email-Eu email-Enron threads-stack-overflow threads-math-sx threads-ask-ubuntu contact-high-school contact-primary-school 10 5 10 4 10 3 10 2 10 1 Edge density in projected graph 0.00 0.25 0.50 0.75 1.00 Fractionoftrianglesopen Exactly 3 nodes per simplex

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