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