This document summarizes research on simplicial closure and higher-order link prediction in network science. It finds that groups of nodes often interact through complex trajectories before reaching "simplicial closure" where all nodes are jointly present in a simplex. Predicting these closed simplices is framed as a higher-order link prediction problem. Various score functions are proposed based on edge weights, node neighborhoods, and similarity measures. Scores combining local edge weight information consistently perform well, outperforming classical link prediction approaches. The results provide insights into higher-order structure and a framework for evaluating models of complex relational data.

1. Simplicial closure and
higher-order link
predictionAustin R. Benson · Cornell
Linear Algebra and Network Science
SIAM Annual · July 11, 2018
1
bit.ly/sc-holp-code bit.ly/arb-SIAMAN18
bit.ly/sc-holp-data arXiv 1802.06916
Joint work with
Rediet Abebe & Jon Kleinberg (Cornell)
Michael Schaub & Ali Jadbabaie (MIT)

2. Networks are sets of nodes and edges (graphs)
that model real-world systems.
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. 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. There are many ways to mathematically represent
the higher-order structure present in relational
data.
4
• Hypergraphs [Berge 89]
• Set systems [Frankl 95]
• 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.
The problem is how do we evaluate and compare models and algorithms?

5. Link prediction is a classical machine learning
problem in network science that is used to
evaluate models.
5
We observe data which is a list of edges in a graph up to some point t.
We want to predict which new edges will form in the future.
Shows up in a variety of applications
• Predicting new social relationships and friend recommendation.
[Backstrom-Leskovec 11; Wang+ 15]
• Inferring new links between genes and diseases.
[Wang-Gulbahce-Yu 11; Moreau-Tranchevent 12]
• Suggesting novel connections in the scientific community.
[Liben-Nowell-Kleinberg 07; Tang-Wu-Sun-Su 12]
Link prediction is also used as a framework to compare
[Liben-Nowell-Kleinberg 03, 07; Lü-Zhau 11]

6. We propose “higher-order link prediction” as a
similar framework for evaluation of higher-order
models.
6
Data.
Observe simplices up to some
time t. Using this data, want to
predict what groups of > 2
nodes will appear in a simplex
in the future.
t
1
2
3
4
5
6
7
8
9
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.

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. 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
Data. Pictures to have in mind.

9. 9
5
23
16
20

10. 10
i
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i
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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. There is lots of variation in the fraction of triangles
that are open, but datasets from the same domain
are similar.
11See also [Patania-Petri-Vaccarino 17] for fraction of open triangles on arxiv collaboration networks.
Open problem.What is a generative model that
captures the diversity in these two features?

12. Dataset domain separation also occurs at the local
level.
12
• 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.

13. 13
Two remaining parts of this talk.
1. Simplicial closure.
How do new simplices (e.g., closed triangles) appear?
2. Higher-order link prediction.
How can we predict which new simplices appear?
Algorithms boil down to some fun linear algebra!

14. Groups of nodes go through trajectories until
finally reaching a “simplicial closure event.”
14
1
2
3
4
5
6
7
8
9
1
2 6
1
2 6
1
2 6
1
2 6
1
2 6
{ 1, 2, 3, 4}
t1
{ 1, 6}
t3
{ 2, 6}
t4
{ 1,2,6}
t8
For this talk, we will just look at simplicial closure on 3 nodes.

15. Groups of nodes go through trajectories until
finally reaching a “simplicial closure event.”
15
Substances in marketed drugs recorded in the National Drug Code
directory.HIV protease
inhibitors
UGT1A1
inhibitors
Breast cancer
resistanceprotein inhibitors
1
2+
2+
1
2+
1
1
2+
2+
1
2+
2+
2+
Reyataz
RedPharm
2003
Reyataz
Squibb & Sons
2003
Kaletra
Physicians
Total Care
2006
Promacta
GSK (25mg)
2008
Promacta
GSK (50mg)
2008
Kaletra
DOH Central
Pharmacy
2009
Evotaz
Squibb & Sons
2015
We bin weighted edges into “weak” and
“strong” ties in the projected graph W.
• Weak ties. Wij = 1 (one simplex contains i and j)
• Strong ties. Wij > 2 (at least two simplices contain i and j)

16. 1
2+1
1
2+
1
1
1
1
2+
2+
2+
1
1
2+
2+
1
2+
2+
2+
245,996
74,219
14,541
7,575
5,781
773
3,560
952
445
389
618
285
2,732,839
157,236
66,644
7,987
8,844
328
3,171
779
722
285
We can also study temporal dynamics in
aggregate.
16
Coauthorship data of scholars publishing in
history. Nodes in most new closed
triangles have had no previous
interaction.
Open triangle of all weak
ties more likely to form a
strong tie before closing.

17. Simplicial closure depends on structure in projected
graph.
17
• 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

18. 18
Two remaining parts of this talk.
1. Simplicial closure.
How do new simplices (e.g., closed triangles) appear?
2. Higher-order link prediction.
How can we predict which new simplices appear?
Algorithms boil down to some fun linear algebra!

19. We propose “higher-order link prediction” as a
similar framework for evaluation of higher-order
models.
19
Data.
Observe simplices up to some
time t. Using this data, want to
predict what groups of > 2
nodes will appear in a simplex
in the future.
t
1
2
3
4
5
6
7
8
9
We predict structure that classical link
prediction would not even consider!

20. 20
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
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Wij
Wjk
Wjk

21. 21
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
2. is a function of A[:, [i, j, k]]
3. is built from “whole-network”
similarity scores on edges
4. is learned from data
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.

22. 22
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
Open problem. Fast way to find top-k scores? Without forming W?

23. 23
2. s(i, j, k) is a function of is a function of A[:, [i, j,
k]]
i
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Wij
Wjk
Wjk
1. Number of common fourth neighbors
2. Generalized Jaccard coefficient
3. Preferential attachment
Open problem. Fast way to find top-k scores (or just
sample)?

24. 24
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)
Hadamard product with A is just reducing storage costs.

25. 25
How do we actually solve the systems?
Want to reduce
computational cost
(only need scores on
open triangles).
For each node i that participates in an open triangle
1. Solve
2. Store ith column
using iterative solver with low tolerance
Open software problem. We would like block iterative solvers in Julia!

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

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. Generalized means of edge weights are good
predictors of new closed triangles.
29
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
?

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. We only looked at 3-node patterns.
Computation becomes much more challenging for 4-node patterns.
4. Do you have a better model, algorithm, or data representation?
Great! Show us by out-performing these baselines.
Data. bit.ly/sc-holp-data
Code. bit.ly/sc-holp-code

31. Simplicial closure and higher-order link prediction.
31
Paper. arXiv 1802.06916
Data. bit.ly/sc-holp-data
Code. bit.ly/sc-holp-code
Slides. bit.ly/arb-SIAMAN18
Austin R. Benson
http://cs.cornell.edu/~arb
@austinbenson
arb@cs.cornell.edu
THANKS!
i
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