Slides from my talk at the Applied Math Seminar at UB. October 30, 2018.

1. 1
Joint work with
Rediet Abebe & Jon Kleinberg (Cornell)
Michael Schaub & Ali Jadbabaie (MIT)
Gabor Lippner (Northeastern)
Paul Horn (Denver)
Simplicial closure &
simplicial diffusions
Austin R. Benson · Cornell
Buffalo Applied Math Seminar · Oct. 30, 2018
Slides. bit.ly/arb-UB18
1
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3
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5
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7
8
9

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]
• 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
What are the basic organizational
principles of systems with
higher-order interactions?

6. We collected many datasets of timestamped simplices,
where each simplex is a subset of nodes.
6
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

7. Thinking of higher-order data as a weighted projected
graph with“filled-in”structures is a convenient viewpoint.
7
1
2
3
4
5
6
7
8
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.

8. 8
5
23
16
20

9. 9
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

10. 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.
10

11. Dataset domain separation also occurs at the local level.
11
• 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.

12. Most open triangles do not come from asynchronous
temporal behavior.
12
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

13. A simple model can account for open triangle variation.
13
• 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 (sketch).
• 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

14. 14
Simplicial closure.
How do new simplices appear?
How do new closed triangles appear?

15. Groups of nodes go through trajectories until finally
reaching a“simplicial closure event.”
15
1
2
3
4
5
6
7
8
9
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}
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

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.
HIV protease
inhibitors
UGT1A1
inhibitors
Breast cancer
resistance protein 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.
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. 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.
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. 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
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

19. Simplicial closure on 4 nodes similar to on 3 nodes,
just“up one dimension”.
19
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
10 7
10 6
10 5
10 4
10 3
10 7
10 6
10 5
10 4
10 3
10 6
10 5
10 4
10 3
10 2
10 6
10 5
10 4
10 3
10 2
1
1
10 4
10 3
10 2
10 4
10 3
10 2
2+
2+
2+
2+
11
1
1
1
1

20. 20
How can we compare and evaluate
models for higher-order structure?

21. Link prediction is a classical machine learning problem
in network science that is used to evaluate models.
21
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 models/algorithms.
[Liben-Nowell-Kleinberg 03, 07; Lü-Zhau 11]

22. We propose“higher-order link prediction”as a similar
framework for evaluation of higher-order models.
22
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, 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.

23. 23
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

24. 24
For every open triangle,we assign a score function (model)
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.

25. 25
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
Bi,s = Ind[i in sth simplex]
W = BBT
diag(BBT
)
A = spones(W)
Open problem. Fast way to find top-k scores? Without forming W?

26. 26
2. s(i,j,k) is a function of is a function of A[:,[i,j,k]]
i
j k
Wij
Wjk
Wjk
1. Number of common fourth neighbors
2. Generalized Jaccard coefficient
3. Preferential attachment
s(i, j, k) =
eT
(A:,i A:,j A:,k)
eT (spones(A:,i + A:,j + A:,k))
s(i, j, k) = eT
(A:,i A:,j A:,k)
s(i, j, k) = (eT
A:,i)(eT
A:,j)(eT
A:,k)
Bi,s = Ind[i in sth simplex]
W = BBT
diag(BBT
)
A = spones(W)
Open problem. Fast way to find top-k scores (or just sample)?
Generalization of ideas in [Seshadhri-Pinar-Kolda 13; Ballard+ 15] ?

27. 27
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)
Bi,s = Ind[i in sth simplex]
W = BBT
diag(BBT
)
A = spones(W)S = [(I A) 1
I ] A
S = [(I W) 1
I ] A
S = (I A(diag(A · e)) 1
) 1
A
S = (I W(diag(W · e)) 1
) 1
A
Hadamard product with A is just reducing storage costs.

28. 28
How do we actually solve the systems?
Want to reduce
memory 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
s(i, j, k) = Sij + Sji + Sjk + Skj + Sik + SkiBi,s = Ind[i in sth simplex]
W = BBT
diag(BBT
)
A = spones(W)
First compute = 1/(2 1(W))
to guarantee solution.
S = [(I W) 1
I ] A
(I W)si = ei
(si ei) A:,i
Open software problem.We would like block iterative solvers in Julia!

29. 29
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.

30. 30

31. 31
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

32. Generalized means of edges weights are often good
predictors of new 3-node simplices appearing.
32
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, Katz) 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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33. 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. There is rich static and temporal structure in the datasets we collected.
3. We only briefly looked at 4-node patterns.
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.
arXiv:1802.0619, 2018.

34. 34
The diffusion-based (PageRank-based) score functions
are still based on reducing the problem to a graph.
i
j k
Wij
Wjk
Wjk
PageRank (unweighted or weighted)
Bi,s = Ind[i in sth simplex]
W = BBT
diag(BBT
)
A = spones(W)
S = (I A(diag(A · e)) 1
) 1
A
S = (I W(diag(W · e)) 1
) 1
A
s(i,j,k) is is built from“whole-network”similarity scores on edges
s(i,j,k) = Sij + Sji + Sjk + Skj + Sik + Ski

35. 35
Simplicial diffusions.
Can we construct a “higher-order” diffusion that respects the topology?
Does this model work well for predicting higher-order structure?

36. Background. Graph diffusions like (personalized) PageRank
can be interpreted through normalized Laplacians.
36
What we would like for a “higher-order” personalized PageRank.
1. a higher-order Laplacian operator (like L0).
2. a normalization scheme for the Laplacian (like D-1).
(I ↵AD 1
)x = (1 ↵)ei
() ( I + L0)x = ei, = (1 ↵)/↵
L0 = I AD 1
= L0D 1
(random walk normalized Laplacian)
L0 = D A (graph Laplacian)<latexit 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37. Background. By interpreting our data as a simplicial
complex,we can use the Hodge Laplacian.
37
• (Abstract) simplicial complex X: if A 2 X and B ✓ A, then B 2 X.
• Graph G = (V, E) as a SC: X = E [ V.
• Triangle complex: X = T [ E [ V, where T = triangles in G.
• Can induce a complex from higher-order interactions.<latexit 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38. 38
L0 = 0T
0 + B1BT
1 (graph Laplacian operates on nodes)
L1 = BT
1B1 + B2BT
2 (edge Laplacian operates on oriented edges)<latexit 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B1 : E ! V by B1(ei,j) = ej ei
B2 : T ! E by B2(ei,j,k) = ei,j ei,k + ej,k<latexit 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Background. By interpreting our data as a simplicial
complex,we can use the Hodge Laplacian.
1
-2
-2
B2 =
[1, 2, 3] [2, 3, 4]
[1, 2] 1 0
[1, 3] 1 0
[2, 3] 1 1
[2, 4] 0 1
[2, 6] 0 0
[3, 4] 0 1
[4, 5] 0 0
[4, 7] 0 0
[5, 6] 0 0
[5, 7] 0 0<latexit 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39. Getting a normalization scheme.
39
Our claim. A sensible normalization scheme for D1 is
D1({i, j}) = di + dj + dij,
where di = # edges containing i and dij = # triangles containing {i, j}.
Why? Theorem. Equivalent to “lift, Markov apply, and project”
I L0 = P<latexit 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sha1_base64="+rFV7of28TRVbrjLrQknMypzY2w=">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</latexit>
L0 = 0T
0 + B1BT
1
L1 = BT
1B1 + B2BT
2<latexit sha1_base64="5q20V1Bb8g9OzPntGICG3xptq2Q=">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</latexit><latexit sha1_base64="5q20V1Bb8g9OzPntGICG3xptq2Q=">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</latexit><latexit sha1_base64="5q20V1Bb8g9OzPntGICG3xptq2Q=">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</latexit><latexit sha1_base64="5q20V1Bb8g9OzPntGICG3xptq2Q=">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</latexit>
L0 = L0D 1
0
L1 = L1D 1
1<latexit 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V “lifts” an oriented flow on m edges to a symmetric flow on 2m edges.
P1 is a Markov operator on un-oriented edges.
VT “projects” back down to an oriented flow.
I L1 = VT
P1V
⇤(I L1) ✓ ⇤(P1)<latexit 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40. The lifted operated is a random walk on oriented edges.
40
• di = # edges containing i
• dij = # triangles containing {i, j}
• Fij = 2(di + dj + dij)
P1 transition from edge (i, j)
1. With prob. di / Fij, go to some (k, i)
2. With prob. dj / Fij, go to some (j, k)
3. With prob. (di + dj + dij) / Fij, go to some
(i, k) or (k, j) s.t. {i, j, k} is a triangle.
4. Otherwise, stay at (i, j)
B
initial position
A
Edge View
initial position
State-space
C
(self-loops not shown)
Transition types
initial position
self-loop
initial position
aligned
transition
(lower adjacent)
initial positionanti-aligned
transition
(upper adjacent)
Random walk
lifted state-space
We project back to un-oriented edges with stat. dist. !.
diffusion_value({i, j}) = !(i, j) - !(j, i)

41. Simplicial diffusions do not localize like graph diffusions.
41
( I + L0)x = ei, L0 = L0D 1
0 (graph PPR)
( I + L1)x = ei,j, L1 = L1D 1
1 (simplicial PPR)<latexit 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Seed node
Figure from PageRank Beyond the Web, David F. Gleich, SIAM Rev., 2015.

42. Simplicial diffusions do not localize like graph diffusions.
42
( I + L0)x = ei, L0 = L0D 1
0 (graph PPR)
( I + L1)x = ei,j, L1 = L1D 1
1 (simplicial PPR)<latexit 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Seed edge