Joint work on higher-order link prediction using timestamped simplices datasets

1
Joint work with
Rediet Abebe & Jon Kleinberg (Cornell)
Michael Schaub & Ali Jadbabaie (MIT)
Higher-order link prediction
Austin R. Benson · Cornell University
Mining & Learning with Graphs · August 5, 2019
Slides. bit.ly/arb-MLG-19

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

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

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?

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 recommendation in social networks.
• Team formation.

6
First…
What are the basic organizational
principles of systems with
higher-order interactions?

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

How big are these datasets?
8
bit.ly/sc-holp-data
Not large in terms of
#(bytes) to store.
Large in terms of set
structure complexity.
1 size-5 simplex induces
31 sub-simplices.
Dataset # nodes # timestamped simplices
coauth-DBLP 1.92M 3.70M
coauth-MAG-Geology 1.26M 1.59M
coauth-MAG-History 1.01M 1.81M
music-rap-genius 56.8K 225K
tags-stack-overﬂow 50.0K 14.5M
tags-ask-ubuntu 3.00K 271K
tags-math-sx 1.63K 822K
threads-stack-overﬂow 2.7M 11.3M
threads-math-sx 176K 720K
threads-ask-ubuntu 126K 193K
NDC-substances 5.31K 112K
NDC-classes 1.16K 49.7K
DAWN 2.56K 2.27M
congress-bills 1.72K 261K
congress-committees 863 679
email-Eu 998 235K
email-Enron 143 10.9K
contact-high-school 327 172K
contact-primary-school 242 107K
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Thinking of higher-order data as a weighted projected
graph with“filled-in”structures is a convenient viewpoint.
9
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.

11
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

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.
12
See also [Patania-Petri-Vaccarino 17] for fraction of open
triangles on several arxiv collaboration networks.

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
Simplicial closure.
What are the common ways in which new simplices appear?
How do new closed triangles appear?

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.

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.
• Weak ties. Wij = 1 (one simplex contains i and j)
• Strong ties. Wij > 2 (at least two simplices contain i and j)

We can also study temporal dynamics in aggregate.
17
Coauthorship data of scholars publishing in history.
Most groups formed have no previous interaction.
Open triangle of all weak
ties more likely to form a
strong tie before closing.

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
Tension between edge
density and tie strength.
Left and middle observations are consistent with theory and empirical studies of social networks.
[Granovetter 73; Kossinets-Watts 06; Backstrom+ 06; Leskovec+ 08]
Closure probability Closure probability Closure probability

Simplicial closure on 4 nodes similar to on 3 nodes,
just“up one dimension”.
19
Increased edge density
Increased simplicial tie strength
Tension b/w edge density
simplicial tie strength.
Closure probability Closure probability Closure probability

20
Back to higher-order link prediction.
How can we predict which new simplices appear?
How can we predict which triangles become closed?

We propose“higher-order link prediction”as a similar
framework for evaluation of higher-order models.
21
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!

22
Our structural analysis tells us what we should be
looking at for prediction.
1. Edge density matters!
⟶ focus our attention on predicting which open
triangles become closed triangles
(intelligently reduce search space.)
2. Tie strength matters!
⟶ various ways of incorporating this information
i
j k
Wij
Wjk
Wjk

23
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.

24
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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1. Number of common neighbors of all 3 nodes
2. Generalized Jaccard coefficient
3. Preferential attachment
25
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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26
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
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S = (I W) 1
I
S = (I A) 1
I<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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27
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.

29
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
(only predicting on open triangles)
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

Generalized means of edges weights are often good
predictors of new 3-node simplices appearing.
30
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-math-sx
threads-ask-ubuntu
contact-high-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
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We often only need the top-k weighted triangles,and
we have fast algorithms for finding them.
31
Ongoing research with Raunak Kumar (Cornell),
Paul Liu (Stanford), and Moses Charikar (Stanford).
dataset # nodes # edges Fast enumeration Fast top-k (k = 1000)
(running time in seconds)
MAG co-authorship 173M 544M 596 16
AMINER co-authorship 93M 324M 255 10
Ethereum transactions 38M 103M 91 33
Stack Overﬂow interactions 2.3M 21M 31 0.3
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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
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Count
And some shameless plugs...
32
Graph-based Semi-Supervised & Active
Learning for Edge Flows.
Given traffic flows on some edges, how can
we interpolate to unobserved other edges?
RT16 10:00am-12:00n Thursday
Network Density of States.
How can we approximately compute the
entire spectrum of large, real-world graphs?
RT6 1:30pm-3:30pm Tuesday
Junteng Jia
Kun Dong

There are lots of opportunities in studying this data.
33
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. Please develop fancy embeddings to out-perform our baselines. 😁
3. Why does generalized mean between harm. and geom. work well?
4. Computation is more challenging and complex for 4-node patterns.
5. Generative models capturing summary statistics?
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34
THANKS! Austin R. Benson
Slides. bit.ly/arb-graphex-19
http://cs.cornell.edu/~arb
@austinbenson
arb@cs.cornell.edu
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

Most open triangles do not come from asynchronous
temporal behavior.
35
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

A simple model can account for open triangle variation.
36
• 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
0.00
0.25
0.50
0.75
1.00
Exactly 3 nodes per simplex (simulated)
n = 200
n = 100
n = 50
n = 25

We still have variety with only 3-node simplices.
37
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-math-sx
threads-ask-ubuntu
contact-high-school
10 5
10 4
10 3
10 2
10 1
0.00
0.25
0.50
0.75
1.00
Exactly 3 nodes per simplex

Joint work on higher-order link prediction using timestamped simplices datasets

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