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Carnegie 
Mellon 
University 
Making 
Sense 
of 
Large 
Graphs: 
Summarization 
and 
Similarity 
Danai Koutra 
Computer Science Department 
Carnegie Mellon University 
danai@cs.cmu.edu 
http://www.cs.cmu.edu/~dkoutra 
Mlconf 
‘14, 
Atlanta, 
GA
Making 
sense 
of 
large 
graphs 
Human 
Connectome 
Project 
>1.25B 
users! 
scalable algorithms and models 
for understanding massive graphs. 
Danai Koutra (CMU) 2
Understanding 
Large 
Graphs 
Part 1 
S u m m a r i z a t i o n 
Danai Koutra (CMU) 3
Ever 
tried 
visualizing 
a 
large 
79,870 email 
accounts 
288,364 emails 
graph? 
Danai Koutra (CMU) 4
Ever 
tried 
visualizing 
a 
large 
79,870 email 
accounts 
288,364 emails 
graph? 
Danai Koutra (CMU) 5
After 
this 
talk, 
you’ll 
know 
how 
to 
Cind… 
VoG Top-3 Stars 
klay@enron.com 
kenneth.lay@enron.com 
Danai Koutra (CMU) 6
Enron 
Summary 
VoG Top Near Bipartite Core 
Commenters CC’ed 
Danai Koutra (CMU) 7 
Ski 
excursion 
organizers 
participants 
“Affair”
Problem 
DeCinition 
Given: a graph 
Find: 
a succinct summary 
with possibly 
overlapping subgraphs 
≈ 
important graph 
structures. 
[Koutra, Kang, Vreeken, Faloutsos. SDM’14] 
Danai Koutra (CMU) 8 
Lady Gaga 
Fan Club
Main 
Ideas 
Idea 1: Use well-known structures (vocabulary): 
Idea 2: Best graph summary 
Shortest lossless description 
è optimal compression (MDL) 
Danai Koutra (CMU) 9
BACKGROUND 
Minimum 
Description 
Length 
~Occam’s razor 
min 
L(M) 
+ 
L(D|M) 
# bits 
for M 
a1 x + a0 
# bits for the 
data using M 
errors 
a10 x10 + a9 x9 + … + a0 
{ } 
simple & good 
explanations 
Danai Koutra (CMU) 10
Formally: 
Minimum 
Graph 
Description 
Given: - a graph G 
- vocabulary Ω 
Danai Koutra (CMU) 11 
Find: model M 
s.t. min L(G,M) = min{ L(M) + L(E) } 
Adjacency A Model M Error E
VoG: 
Overview 
≈? 
argmin 
≈ 
Danai Koutra (CMU) 12
VoG: 
Overview 
Danai Koutra (CMU) 13 
Pick best 
(with some criterion) 
Summary
Q: 
Which 
structures 
to 
pick? 
A: Those that 
min description length 
S of G 
2|S| combinations 
Danai Koutra (CMU) 14
Runtime 
1.25B 
users! 
VOG is near-linear on # edges of the input graph. 
Danai Koutra (CMU) 15
Understanding 
a 
wiki 
graph 
I don’t see 
anything! L 
Nodes: wiki editors 
Edges: co-edited 
Danai Koutra (CMU) 16
Wiki 
Controversial 
Article 
Danai Koutra (CMU) 17 
Stars: 
admins, 
bots, 
heavy users 
Bipartite cores: edit wars 
Kiev vs. Kyiv vandals vs. admins
VoG 
vs. 
other 
methods 
[Navlakha+’08] [Dunne+’13] [Chakrabarti+’03] 
Stars, cliques near-cliques 
Danai Koutra (CMU) 18 
VoG 
Bounded-­‐Error 
Summariza@on 
Mo@f 
Simplifica@on 
Clustering 
Methods 
Cross-­‐ 
Associa@ons 
Variety 
of 
Structures 
✔ 
✗ 
✗ 
✗ 
✗ 
Important 
Structures 
✔ 
✗ 
✗ 
✗ 
✗ 
Low 
Complexity 
✔ 
✗ 
✗ 
✔(?) 
✔ 
Visualiza@on 
✔ 
✔ 
✔ 
✗ 
✗ 
Graph 
Summary 
✔ 
✔ 
✔ 
✗ 
✗
VoG: 
summary 
• Focus on important 
• possibly-overlapping structures 
• with known graph-theoretic properties 
Danai Koutra (CMU) 19 
www.cs.cmu.edu/~dkoutra/SRC/vog.tar
Understanding 
Large 
Graphs 
Part 2 
S i m i l a r i t i e s 
Danai Koutra (CMU) 20
friendship 
graph 
≈ 
wall 
posts 
graph? 
VS. 
1 
Behavioral 
PaOerns 
Are 
the 
graphs 
/ 
behaviors 
similar? 
Danai Koutra (CMU) 21
Why 
graph 
similarity? 
Day 
1 
Day 
2 
Day 
3 
Day 
4 
Danai Koutra (CMU) 22 
2 Classification 
Temporal 
anomaly 
detec@on 
3 
4 
Intrusion 
detec@on 
! ! 12 13 14 22 23 
sim1 
sim2 
sim3
Problem 
DeCinition: 
Graph 
Similarity 
• Given: 
(i) 2 graphs with the 
same nodes and 
different edge sets 
(ii) node correspondence 
• Find: similarity score 
s [0,1] 
€ 
∈ 
GA 
GB 
Danai Koutra (CMU) 23
Obvious 
solution? 
Edge Overlap (EO) 
# of common edges 
(normalized or not) 
Danai Koutra 24 
GA 
GB
… 
but 
“barbell”… 
EO(B10,mB10) == EO(B10,mmB10) 
GA GA 
GB GB’ 
Danai Koutra 25
What 
makes 
a 
similarity 
function 
good? 
26 
• Properties: 
² Intuitive 
ProperFes 
like: 
“Edge-­‐importance” 
Danai Koutra
ProperFes 
like: 
“Weight-­‐awareness” 
✗ 
What 
makes 
a 
similarity 
function 
good? 
27 
• Properties: 
² Intuitive 
² Scalable 
Danai Koutra 
✗
MAIN 
IDEA: 
DELTACON 
28 
① Find the pairwise node influence, SA  SB. 
② Find the similarity between SA  SB. 
SA 
= 
SB = 
Danai Koutra (CMU) 
DETAILS
INTUITION 
How? 
Using 
Belief 
Propagation 
Attenuating Neighboring Influence for small ε: 
1-hop 2-hops … 
29 
S =[I+ε 2D−εA]−1 ≈ 
≈ [I −εA]−1 = I+εA+ε 2A2 +... 
Note: ε  ε2  ..., 0ε1 
Danai Koutra (CMU)
OUR 
SOLUTION: 
DELTACON 
DETAILS 
30 
① Find the pairwise node influence, SA  SB. 
② Find the similarity between SA  SB. 
Danai Koutra (CMU) 
sim( ) = 
1 
1+ Σ 
( 2 
s− s)i, j A,ij B,ij SA,SB 
SA 
= 
SB = 
“Root” 
Euclidean 
Distance
… 
but 
O(n2) 
… 
31 
f a s t e r ? 
O(m1+m2) 
in the paper J 
Danai Koutra (CMU)
32 
• Nodes: 
Temporal 
Anomaly 
Detection 
email 
accounts 
of 
employees 
• Edges: 
email 
exchange 
sim1 
sim2 
sim3 
sim4 
Day 
1 
Day 
2 
Day 
3 
Day 
4 
Day 
5 
Danai Koutra (CMU)
Temporal 
Anomaly 
Detection 
similarity 
Feb 
4: 
Lay 
resigns 
consecu@ve 
days 
Danai Koutra (CMU) 
33
Brain-­‐Connectivity 
Graph 
Clustering 
34 
• 114 brain graphs 
² Nodes: 70 cortical regions 
² Edges: connections 
• Attributes: gender, IQ, age… 
Danai Koutra (CMU)
Brain-­‐Connectivity 
Graph 
Clustering 
Danai Koutra (CMU) 35 
t-­‐test 
p-­‐value 
= 
0.0057
Graph 
Understanding 
via 
… 
• … Summarization … 
² VoG: to spot the important graph structures 
• … Comparison … 
² DeltaCon: to find the similarity between 
aligned networks 
² BiG-Align to align bi/uni-partite 
² Uni-Align graphs efficiently 
Danai Koutra (CMU) 36
Thank 
you! 
Understanding 
summarization similarities 
www.cs.cmu.edu/~dkoutra/pub.htm 
danai@cs.cmu.edu 
Danai Koutra (CMU) 37

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Danai Koutra – CMU/Technicolor Researcher, Carnegie Mellon University at MLconf ATL

  • 1. Carnegie Mellon University Making Sense of Large Graphs: Summarization and Similarity Danai Koutra Computer Science Department Carnegie Mellon University danai@cs.cmu.edu http://www.cs.cmu.edu/~dkoutra Mlconf ‘14, Atlanta, GA
  • 2. Making sense of large graphs Human Connectome Project >1.25B users! scalable algorithms and models for understanding massive graphs. Danai Koutra (CMU) 2
  • 3. Understanding Large Graphs Part 1 S u m m a r i z a t i o n Danai Koutra (CMU) 3
  • 4. Ever tried visualizing a large 79,870 email accounts 288,364 emails graph? Danai Koutra (CMU) 4
  • 5. Ever tried visualizing a large 79,870 email accounts 288,364 emails graph? Danai Koutra (CMU) 5
  • 6. After this talk, you’ll know how to Cind… VoG Top-3 Stars klay@enron.com kenneth.lay@enron.com Danai Koutra (CMU) 6
  • 7. Enron Summary VoG Top Near Bipartite Core Commenters CC’ed Danai Koutra (CMU) 7 Ski excursion organizers participants “Affair”
  • 8. Problem DeCinition Given: a graph Find: a succinct summary with possibly overlapping subgraphs ≈ important graph structures. [Koutra, Kang, Vreeken, Faloutsos. SDM’14] Danai Koutra (CMU) 8 Lady Gaga Fan Club
  • 9. Main Ideas Idea 1: Use well-known structures (vocabulary): Idea 2: Best graph summary Shortest lossless description è optimal compression (MDL) Danai Koutra (CMU) 9
  • 10. BACKGROUND Minimum Description Length ~Occam’s razor min L(M) + L(D|M) # bits for M a1 x + a0 # bits for the data using M errors a10 x10 + a9 x9 + … + a0 { } simple & good explanations Danai Koutra (CMU) 10
  • 11. Formally: Minimum Graph Description Given: - a graph G - vocabulary Ω Danai Koutra (CMU) 11 Find: model M s.t. min L(G,M) = min{ L(M) + L(E) } Adjacency A Model M Error E
  • 12. VoG: Overview ≈? argmin ≈ Danai Koutra (CMU) 12
  • 13. VoG: Overview Danai Koutra (CMU) 13 Pick best (with some criterion) Summary
  • 14. Q: Which structures to pick? A: Those that min description length S of G 2|S| combinations Danai Koutra (CMU) 14
  • 15. Runtime 1.25B users! VOG is near-linear on # edges of the input graph. Danai Koutra (CMU) 15
  • 16. Understanding a wiki graph I don’t see anything! L Nodes: wiki editors Edges: co-edited Danai Koutra (CMU) 16
  • 17. Wiki Controversial Article Danai Koutra (CMU) 17 Stars: admins, bots, heavy users Bipartite cores: edit wars Kiev vs. Kyiv vandals vs. admins
  • 18. VoG vs. other methods [Navlakha+’08] [Dunne+’13] [Chakrabarti+’03] Stars, cliques near-cliques Danai Koutra (CMU) 18 VoG Bounded-­‐Error Summariza@on Mo@f Simplifica@on Clustering Methods Cross-­‐ Associa@ons Variety of Structures ✔ ✗ ✗ ✗ ✗ Important Structures ✔ ✗ ✗ ✗ ✗ Low Complexity ✔ ✗ ✗ ✔(?) ✔ Visualiza@on ✔ ✔ ✔ ✗ ✗ Graph Summary ✔ ✔ ✔ ✗ ✗
  • 19. VoG: summary • Focus on important • possibly-overlapping structures • with known graph-theoretic properties Danai Koutra (CMU) 19 www.cs.cmu.edu/~dkoutra/SRC/vog.tar
  • 20. Understanding Large Graphs Part 2 S i m i l a r i t i e s Danai Koutra (CMU) 20
  • 21. friendship graph ≈ wall posts graph? VS. 1 Behavioral PaOerns Are the graphs / behaviors similar? Danai Koutra (CMU) 21
  • 22. Why graph similarity? Day 1 Day 2 Day 3 Day 4 Danai Koutra (CMU) 22 2 Classification Temporal anomaly detec@on 3 4 Intrusion detec@on ! ! 12 13 14 22 23 sim1 sim2 sim3
  • 23. Problem DeCinition: Graph Similarity • Given: (i) 2 graphs with the same nodes and different edge sets (ii) node correspondence • Find: similarity score s [0,1] € ∈ GA GB Danai Koutra (CMU) 23
  • 24. Obvious solution? Edge Overlap (EO) # of common edges (normalized or not) Danai Koutra 24 GA GB
  • 25. … but “barbell”… EO(B10,mB10) == EO(B10,mmB10) GA GA GB GB’ Danai Koutra 25
  • 26. What makes a similarity function good? 26 • Properties: ² Intuitive ProperFes like: “Edge-­‐importance” Danai Koutra
  • 27. ProperFes like: “Weight-­‐awareness” ✗ What makes a similarity function good? 27 • Properties: ² Intuitive ² Scalable Danai Koutra ✗
  • 28. MAIN IDEA: DELTACON 28 ① Find the pairwise node influence, SA SB. ② Find the similarity between SA SB. SA = SB = Danai Koutra (CMU) DETAILS
  • 29. INTUITION How? Using Belief Propagation Attenuating Neighboring Influence for small ε: 1-hop 2-hops … 29 S =[I+ε 2D−εA]−1 ≈ ≈ [I −εA]−1 = I+εA+ε 2A2 +... Note: ε ε2 ..., 0ε1 Danai Koutra (CMU)
  • 30. OUR SOLUTION: DELTACON DETAILS 30 ① Find the pairwise node influence, SA SB. ② Find the similarity between SA SB. Danai Koutra (CMU) sim( ) = 1 1+ Σ ( 2 s− s)i, j A,ij B,ij SA,SB SA = SB = “Root” Euclidean Distance
  • 31. … but O(n2) … 31 f a s t e r ? O(m1+m2) in the paper J Danai Koutra (CMU)
  • 32. 32 • Nodes: Temporal Anomaly Detection email accounts of employees • Edges: email exchange sim1 sim2 sim3 sim4 Day 1 Day 2 Day 3 Day 4 Day 5 Danai Koutra (CMU)
  • 33. Temporal Anomaly Detection similarity Feb 4: Lay resigns consecu@ve days Danai Koutra (CMU) 33
  • 34. Brain-­‐Connectivity Graph Clustering 34 • 114 brain graphs ² Nodes: 70 cortical regions ² Edges: connections • Attributes: gender, IQ, age… Danai Koutra (CMU)
  • 35. Brain-­‐Connectivity Graph Clustering Danai Koutra (CMU) 35 t-­‐test p-­‐value = 0.0057
  • 36. Graph Understanding via … • … Summarization … ² VoG: to spot the important graph structures • … Comparison … ² DeltaCon: to find the similarity between aligned networks ² BiG-Align to align bi/uni-partite ² Uni-Align graphs efficiently Danai Koutra (CMU) 36
  • 37. Thank you! Understanding summarization similarities www.cs.cmu.edu/~dkoutra/pub.htm danai@cs.cmu.edu Danai Koutra (CMU) 37