Computational Frameworks for Higher-order Network Data Analysis

1
Computational Frameworks for
Higher-order Network Data Analysis
Austin R. Benson · Cornell University
Texas A&M Institute of Data Science · October 23, 2020
Slides. bit.ly/arb-tamu-20

Graph or network data modeling important complex
systems are everywhere.
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Commerce
nodes are products
edges link co-purchased
products
Communications
nodes are people/accounts
edges show info. exchange
Physical proximity
nodes are people
edges link those that interact
in close proximity
Drug compounds
nodes are substances
edge between substances that
appear in the same drug

Network data analysis studies the model to gain insight
and make predictions about these systems.
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1. Evolution / changes
What new connections will form? (email auto-fill suggestions, rec. systems)
2. Clustering / partitioning / community detection
How to find groups of related nodes? (similar products, protein functions)
3. Spreading and traversing
How does stuff move over the network? (viruses or misinformation)
4. Ranking
Which things are important? (PageRank and its variants)

Real-world systems are composed of“higher-order”
interactions that we often reduce to pairwise ones.
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Commerce
nodes are products
Several products
purchased at once
Communications
nodes are people/accounts
emails often have several
recipients,not just one.
Physical proximity
nodes are people
people gather in groups
Drug compounds
nodes are substances
Drugs are composed of
several substances

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What new insights does this give us?

We can ask the same network analysis questions while
taking into account the higher-order structure.
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1. Evolution / changes
What new connections will form? (email auto-fill suggestions, rec. systems)
2. Clustering / partitioning / community detection
How to find groups of related nodes? (similar products, protein functions)
3. Spreading and traversing
How does stuff move over the network? (viruses or misinformation)
4. Ranking
Which things are important? (PageRank and its variants)

Higher-order Network Data Analysis
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w/ R. Abebe, M. Schaub,
J. Kleinberg, A. Jadbabaie
1. Temporal evolution of higher-order interactions.
Simplicial Closure and Higher-order Link Prediction,PNAS 2018.
2. Clustering in large networks of higher-order interactions.
Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020.
3. Diffusions over higher-order interactions in networks.
Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.

We collected many datasets of timestamped simplices,
where each simplex is a subset of nodes.
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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

Thinking of higher-order data as a weighted projected
graph with filled-in structures is a convenient viewpoint.
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3
4
5
6
7
8
9
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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.
Projected graph W.
Wij = # of simplices containing nodes i and j.

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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.
12See also Patania-Petri-Vaccarino (2017) for similar ideas in collaboration networks.

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

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How do new simplices form?
Can we predict which simplices will form?

Groups of nodes go through trajectories until finally
reaching a“simplicial closure.”
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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.
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)

Simplicial closure depends on structure in projected graph.
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• 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 is similar to on 3 nodes,
just“up one dimension.”
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Increased edge density
Increased simplicial tie strength
Tension b/w edge density
simplicial tie strength.
Closure probability Closure probability Closure probability

We proposed“higher-order link prediction”as a
framework to evaluate models for closure.
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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 time t.
• Predict which groups of > 2
nodes will appear after time t.
t We predict structure that graph
models would not even consider!

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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

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For every open triangle,we assign a score function on
first 80% of data based on structural properties.
Score s(i, j, k)…
1. is a function of Wij, Wjk, Wjk
arithmetic mean, harmonic 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
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.
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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scorep(i, j, k)
= (Wp
ij + Wp
jk + Wp
ik)1/p
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23
A few lessons learned from applying these ideas.
1. We can predict pretty well on all datasets using some simple 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 with the harmonic mean.
3. Simple 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.
24
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

If we only need the top-k weighted triangles,
we have fast algorithms for finding them.
25
scorep(i, j, k)
= (Wp
ij + Wp
jk + Wp
ik)1/p
Simple (incorrect) algorithm.
1. Throw out edges with weight < t.
2. Find triangles in remainder.
i
j k
Wij
Wjk
Wjk
Better (correct) algorithm.
1. Dynamically choose threshold.
2. Careful pruning.
w/ R. Kumar, P. Liu, M. Charikar

We often only need the top-k weighted triangles,and
we have fast algorithms for finding them.
26
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dataset # nodes # edges Fast enumeration Fast top-k (k = 1000)
(running time in seconds)
Spotify co-listens 3.6M 1.93B too long 30
MAG co-authorship 173M 544M 596 16
AMINER co-authorship 93M 324M 255 10
Ethereum transactions 38M 103M 91 33

Higher-order data is pervasive!
27
• Simplicial Closure and Higher-order Link Prediction.Austin R.Benson,Rediet Abebe,Michael T.Schaub,Ali
Jadbabaie,and Jon Kleinberg. Proc.Natl.Acad.Sci.U.S.A.,2018. github.com/arbenson/ScHoLP-Tutorial
• Retrieving Top Weighted Triangles in Graphs. Raunak Kumar,Paul Liu,Moses Charikar,and Austin R.Benson.
Proc.Of WSDM,2020. github.com/raunakkmr/Retrieving-top-weighted-triangles-in-graphs
1. There are commonalities in temporal evolution. Generative models?
2. There is lots of signal in subsets! Unique to higher-order…
3. Please develop neural embeddings to out-perform our baselines. 😁
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5
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28
w/ Nate Veldt, J. KleinbergHigher-order Network Data Analysis
1. Temporal evolution of higher-order interactions.
Simplicial Closure and Higher-order Link Prediction,PNAS 2018.
2. Clustering in large networks of higher-order interactions.
Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020.
3. Diffusions over higher-order interactions in networks.
Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.

Graph minimum s-t cuts are fundamental.
29
minimizeS⇢V cut(S)
subject to s 2 S, t /2 S.<latexit 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1 3
2 4
5
6
7
8
s
t
• Maximum flow / min s-t cut [Ford, Fulkerson, Dantzig 1950s]
• Densest subgraph [Goldberg 84; Shang+ 18]
• Graph-based semi-supervised learning algorithms [Blum-Chawla 01]
• Local graph clustering [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16]
poly-time algorithms!

Real-world systems are composed of“higher-order”
interactions that we can model with hypergraphs.
30
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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What is a hypergraph minimum s-t cut?
31
s
t
Should we treat the 2/2 split
differently from the 1/3 split?
Historically, no. [Lawler 73, Ihler+ 93]
More recently, yes.
[Li-Milenkovic 17, Veldt-Benson-Kleinberg 20]
edge in a graph size-3 hyperedges
“Only one way to split a triangle”
[Benson+ 16; Li-Milenkovic 17; Yin+ 17]
Must be split 1/1.

We model hypergraph cuts with splitting functions.
32
s
t
Given a cut defined by S,
we incur penalty of
at each hyperedge e.
Hypergraph minimum s-t cut problem.
Cardinality-based splitting functions.
S<latexit 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cutH(S) = f (2) + f (1)<latexit 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<latexit sha1_base64="6FFH4JtCJ1Fb69WjugRCayyF5vI=">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</latexit>
we(e S)
<latexit sha1_base64="QjrhfsKLxaK/82LxnRurhFCzCik=">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</latexit>
minimizeS⇢V
P
e2E we(e S) ⌘ cutH(S)
subject to s 2 S, t /2 S.
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Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).

Cardinality-based splitting functions appear
throughout the literature.
33
[Lawler 73; Ihler+ 93; Yin+ 17]
[Hu-Moerder 85; Heuer+ 18]
[Agarwal+ 06; Zhou+ 06; Benson+ 16]
[Yaros- Imielinski 13]
[Li-Milenkovic 18]
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All-or-nothing we(A) =
(
0 if A 2 {e, ;}
1 otherwise
Linear penalty we(A) = min{|A|, |eA|}
Quadratic penalty we(A) = |A| · |eA|
Discount cut we(A) = min{|A|↵ , |eA|↵ }
L-M submodular we(A) = 1
2 + 1
2 · min
n
1, |A|
b↵|e|c , |eA|
b↵|e|c
o

We solve hypergraph cut problems with graph reductions.
34
1/21/2
1/2
1
1
1
1
∞
∞ ∞
∞
∞∞
Gadgets (expansions) model a hyperedge with a small graph.
clique expansion star expansion Lawler gadget [1973]hyperedge
In a graph reduction, we first replace all hyperedges with graph gadgets...
s
t
s
t
s
t
s
t
… then solve the (min s-t cut) problem exactly on the graph,
and finally convert the solution to a hypergraph solution.
Quadratic penalty
f(i) = i ( |e| – i )
Linear penalty
f(i) = i
All-or-nothing
f(0) = 0,o/w f(i) = 1

b
We made a new gadget for C-B splitting functions.
35
This gadget models min(|A|, |eA|, b).
Theorem [Veldt-Benson-Kleinberg 20a]. Nonnegative linear combinations of the
C-B gadget can model any submodular cardinality-based splitting function.
See also Graph Cuts for Minimizing Robust Higher Order Potentials,Kohli et al.,2008.
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C-B we(A) = f (min(|A|, |eA|)).
(F is submodular on X if F(A B) + F(A [ B)  F(A) + F(B) for any A, B ✓ X.)<latexit 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36
Theorem [Veldt-Benson-Kleinberg 20a]. The hypergraph min s-t cut problem
with a cardinality-based splitting function is graph-reducible (via gadgets)
if and only if the splitting function is submodular.
Cardinality-based splitting functions.
s
t
S<latexit 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cutH(S) = f (2) + f (1)<latexit 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Submodularity is key to efficient algorithms.
What happens when the splitting function isn’t submodular?
Can we use some other algorithm?
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Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).

Computational Frameworks for Higher-order Network Data Analysis

Computational Frameworks for Higher-order Network Data Analysis

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Computational Frameworks for Higher-order Network Data Analysis