Slides from poster session of KFOS 2020. July 21, 2020.

1. 1
Higher-order link prediction and other hypergraph modeling
Austin R. Benson · Department of Computer Science · Cornell University
Kavli Frontiers of Science · July 21, 2020
Slides. bit.ly/arb-KFOS-20
@austinbenson

2. Networks are everywhere.
2
Collaboration
• nodes are people
• edges link people
working together
Communications
• nodes are accounts
• edges show info.exchange
Physical proximity
• nodes are people
• edges link those that interact
Drug compounds
• nodes are substances
• edge between substances
co-appear in drugs

3. But real-world systems are composed of“higher-order”
interactions that we often reduce to pairwise ones.
3
Collaboration
• nodes are people
• Teams are groups of
people acting as one
Communications
• nodes are accounts
• emails often have multiple
recipients,not just one
Physical proximity
• nodes are people
• students gather in groups
within classrooms
Drug compounds
• nodes are substances
• drugs are made up of
several substances

4. What information is in our data?
Hypergraphs give us better models.
4
i
j k
i
j k
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
H = (V, E), edge e 2 E is a subset of V (e ⇢ V)<latexit 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1 2
3
4
5
V = {1, 2, 3, 4, 5}
E = {{1, 2, 3}, {2, 4, 5}}<latexit 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5. We analyze hypergraph to gain insights into the
systems that they model.
5
1. Evolution
What new connections will form?
2. Clustering/partitioning/community detection
How to find groups of nodes that form functional modules?
3. Spreading and traversing
How do viruses & social influences spread?
4. Ranking
What things are important? (e.g., PageRank and its variants)

6. Evolution.Using network structure features,we can
predict which groups will form in the future.
6
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.
Strategy.
• Observe up to time t.
• Predict which groups
of more than 2 nodes
will appear together
in the future.
t
Applications
• Novel combinations of drugs for treatments.
• Predicting new groups in social networks ⟶ recommender systems.
Simplicial closure and higher-order link prediction.
A.R.Benson,R.Abebe,M.T.Schaub,A.Jadbabaie,and J.Kleinberg.PNAS,2018.

7. Clustering. What is a hypergraph cut?
7
s
t
Should we treat the 2/2 split
differently from the 1/3 split?
1 3
2 4
5
6
7
8
s
t
There is only one way to
split an edge (1/1).
Hypergraph Cuts with General Splitting Functions.
N.Veldt,A.R.Benson,and J.Kleinberg.arXiv:2001.02817,2020.

8. 8
Cluster |T| time (s) HyperLocal Baseline1 Baseline2
Amazon Fashion 31 3.5 0.83 0.77 0.6
All Beauty 85 30.8 0.69 0.60 0.28
Appliances 48 9.8 0.82 0.73 0.56
Gift Cards 148 6.5 0.86 0.75 0.71
Magazine Subscriptions 157 14.5 0.87 0.72 0.56
Luxury Beauty 1581 261 0.33 0.31 0.17
Software 802 341 0.74 0.52 0.24
Industrial & Scientific 5334 503 0.55 0.49 0.15
Prime Pantry 4970 406 0.96 0.73 0.36
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• 2.3M Amazon products (nodes),reviewed by 4.3M users (hyperedges).
• mean hyperedge size > 17,max hyperedge size ~9.3k.
• Product categories provide ground truth cluster labels.
F1 recovery scores given a handful of nodes from the ground truth cluster T.
Clustering. We can analyze big data fast.
Minimizing Localized Ratio Cut Objectives in Hypergraphs.
N.Veldt,A.R.Benson,and J.Kleinberg.Proc.of KDD,2020.

9. Traversing. Low-dim. embeddings of flows.
9
A B
A B C
Harmonic trajectory
embedding of
synthetic data.
Harmonic trajectory
embedding of ocean
drifters near
Madagascar.
Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian.
M.T.Schaub,A.R.Benson,P.Horn,G.Lippner,and A.Jadbabaie. SIAM Review,2020.

10. Ranking.Tensor computations for centrality.
10
CEC ZEC HEC
1 alcohol cephalothin alcohol
2 cocaine naloxone alprazolam
3 marijuana meclizine acet.-hydrocodone
4 acet.-hydrocodone cyclosporine clonazepam
5 alprazolam desipramine cocaine
6 clonazepam donnatal elixir marijuana
7 ibuprofen pyridostigmine quetiapine
8 quetiapine amoxapine lorazepam
9 acetaminophen aspirin ibuprofen
10 lorazepam bicalutamide zolpidem
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T(i, j, k) =
(
1 i, j, and k connected by a hyperedge
0 otherwise
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Three hypergraph eigenvector centralities.
A.R.Benson.SIAM J.on Mathematics of Data Science,2019.