mining large dynamic graphs and tensorskijungs/defense/slides.pdf · 2019-02-09 · thesis...
TRANSCRIPT
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Thesis Committee• Prof. Christos Faloutsos (Chair)
• Prof. Tom M. Mitchell
• Prof. Leman Akoglu
• Prof. Philip S. Yu
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What Do Real Graphs Look Like?• Part 1 (Chapters 3 - 8)
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How to Spot Anomalies?• Part 2 (Chapters 9 - 13)
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How to Model Behavior?
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• Part 3 (Chapters 14 - 15)
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Graphs are Everywhere!
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Graphs are Large and Dynamic• Large: many nodes, more edges
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2B+ active users
500M+ products
•Dynamic: additions/deletions of nodes and edges
40B+ web pages
5M+ articles
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.. and with Rich Side Information•Rich: timestamps, scores, text, etc.
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Poor Quality
Satisfied ☺
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Simple Graphs are Matrices
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0 0
1 1
1 0
0
1 1
Graph Adjacency Matrix
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Rich Graphs are Tensors
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1
00
0
0
•Tensors: multi-dimensional array
3-order tensor(3-dimensional array)
+ Stars(4-order tensor)
+ Text(5-order tensor)
Satisfied ☺
0
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Thesis Goal and Focus•Goal:
•Our Focus: To Develop Scalable Algorithms for
▪T1. Structure Analysis (Part 1)
▪T2. Anomaly Detection (Part 2)
▪T3. Behavior Modeling (Part 3)
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To Fully Understand and Utilize Large Dynamic Graphs and Tensors
on User Behavior
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Tasks and Their Relation
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“How do they look like?”
•Given large dynamic graphs or tensors,
“How to spot anomalies?”
“How to model behavior?”
T1.Structure
T2.Anomaly
T3.Behavior
contrast
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Our Tools for Scalability •We design (sub) linear algorithms
•Running on big data platforms
• Exploiting empirical patterns in data◦ locality, power-laws, etc.
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Approx. Sampling Streaming Out-of-core Parallel
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Organization of the Thesis
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Part1.Structure Analysis
Part2. AnomalyDetection
Part3. BehaviorModeling
GraphsTriangle Count
(§§ 3-6) Anomalous Subgraph
(§ 9)
PurchaseBehavior
(§ 14)Summarization(§ 7)
Tensors Summarization(§ 8)
Dense Subtensors(§§ 10-13)
Progression(§ 15)
Chapter
Chapters
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Focuses of This Presentation
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Part1.Structure Analysis
Part2. AnomalyDetection
Part3. BehaviorModeling
GraphsTriangle Count
(§§ 3-6) Anomalous Subgraph
(§ 9)
PurchaseBehavior
(§ 14)Summarization(§ 7)
Tensors Summarization(§ 8)
Dense Subtensors(§§ 10-13)
Progression(§ 15)
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Roadmap• T1. Structure Analysis (Part 1) <<
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
• Future Directions
•Conclusion
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T1. Structure Analysis (Part 1)
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“Given a large graph (or tensor), how can we analyze its structure?
𝑎
𝑏
𝑐
𝑑
𝑒
𝑓
𝑔
Structure measures- density- clustering coefficients- transitivity ratio- triangle connectivity× 4
× 9
× 7
Basic statisticsInput graph
T1-1. Triangle Counting
T1-2. Summarization𝑎, 𝑏 𝑐, 𝑑, 𝑒 𝑓, 𝑔
Summary graph
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T1-1. Triangle Counting (§§ 4-6)
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“Given a large dynamic graph, how can we track the count of triangles
accurately with sub-linear memory?”
× 4 × 6
…
, + , , − , , +
Sources Destination
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T1-1. Triangle Counting (§§ 4-6)
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…
, + , , − , , +
Sources Destination
Created at9:21 AM
Created at9:08 AM
Created at9:02 AM
How can we exploit temporal patterns? (§4)
How can we make good use of multiple machines? (§5)
How can we handle removed edges? (§6)
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Roadmap• T1. Structure Analysis (Part 1)
◦ T1.1 Triangle Counting
▪Handling Deletions (§6) <<▪…
◦ …
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
• Future Directions
•Conclusions
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K. Shin, J. Kim B. Hooi, C. Faloutsos, “Think before You Discard: Accurate Triangle Counting in Graph Streams with Deletions”, ECML/PKDD 2018
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Triangles in a Graph•A triangle is 3 nodes connected to each other
• The count of triangles is an important primitive◦ Applications: ▪community detection, spam detection, link prediction
◦ Structure measures:▪transitivity ratio, clustering coefficients, trussness
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1 2
3
4
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Remaining Challenge•Counting triangles in real-world graphs
◦ Large: not fitting in main memory
◦ Fully dynamic: both growing and shrinking
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Online social networks
Citation networks Call networks
Web
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Previous Work
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LargeGraph
Fully dynamic GraphAccurate
Growing Shrinking
MASCOT [LJK18]
Triest-IMPR [DERU17]
WRS [Shi17]
ESD [HS17]
Triest-FD [DERU17]
- Given: a large and fully-dynamic graph- To estimate: the count of triangles accurately
ThinkD (Proposed)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Our Contribution: ThinkD
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•We develop ThinkD (Think before You Discard):
Fast & Accurate: outperforming competitors
Scalable: linear data scalability
Theoretically Sound: unbiased estimates
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Roadmap• T1. Structure Analysis (Part 1)
◦ T1.1 Triangle Counting
▪Handling Deletions (§6)
◦ Problem Definition <<
◦ Proposed Method: ThinkD
◦ Experiments▪…
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
•…
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Fully Dynamic Graph Stream•Our model for a large and fully-dynamic graph
•Discrete time 𝑡, starting from 1 and ever increasing
•At each time 𝑡, a change in the input graph arrives◦ change: either an insertion or deletion of an edge
𝑎 𝑏
Time 𝑡 1
Change(given)
+(𝑎, 𝑏)
Graph(unmate-rialized)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Fully Dynamic Graph Stream
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•Our model for a large and fully-dynamic graph
•Discrete time 𝑡, starting from 1 and ever increasing
•At each time 𝑡, a change in the input graph arrives◦ change: either an insertion or deletion of an edge
𝑐
𝑎 𝑏 𝑎 𝑏
Time 𝑡 1 2
Change(given)
+(𝑎, 𝑏) +(𝑎, 𝑐)
Graph(unmate-rialized)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Fully Dynamic Graph Stream
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•Our model for a large and fully-dynamic graph
•Discrete time 𝑡, starting from 1 and ever increasing
•At each time 𝑡, a change in the input graph arrives◦ change: either an insertion or deletion of an edge
𝑐
𝑎 𝑏 𝑎 𝑏
Time 𝑡 1 2 3
Change(given)
+(𝑎, 𝑏) +(𝑎, 𝑐) +(𝑏, 𝑐)
Graph(unmate-rialized)
𝑐
𝑎 𝑏
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Fully Dynamic Graph Stream
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•Our model for a large and fully-dynamic graph
•Discrete time 𝑡, starting from 1 and ever increasing
•At each time 𝑡, a change in the input graph arrives◦ change: either an insertion or deletion of an edge
𝑐
𝑎 𝑏 𝑎 𝑏
Time 𝑡 1 2 3 4
Change(given)
+(𝑎, 𝑏) +(𝑎, 𝑐) +(𝑏, 𝑐) −(𝑎, 𝑏)
Graph(unmate-rialized)
𝑐
𝑎 𝑏
𝑐
𝑎 𝑏
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Fully Dynamic Graph Stream
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•Our model for a large and fully-dynamic graph
•Discrete time 𝑡, starting from 1 and ever increasing
•At each time 𝑡, a change in the input graph arrives◦ change: either an insertion or deletion of an edge
𝑐
𝑎 𝑏 𝑎 𝑏
Time 𝑡 1 2 3 4 5 …
Change(given)
+(𝑎, 𝑏) +(𝑎, 𝑐) +(𝑏, 𝑐) −(𝑎, 𝑏) +(𝑏, 𝑑) …
Graph(unmate-rialized)
…𝑐
𝑎 𝑏
𝑐
𝑎 𝑏
𝑐 𝑑
𝑎 𝑏
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Fully Dynamic Graph Stream
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•Our model for a large and fully-dynamic graph
•Discrete time 𝑡, starting from 1 and ever increasing
•At each time 𝑡, a change in the input graph arrives◦ change: either an insertion or deletion of an edge
𝑐
𝑎 𝑏 𝑎 𝑏
Time 𝑡 1 2 3 4 5 …
Change(given)
+(𝑎, 𝑏) +(𝑎, 𝑐) +(𝑏, 𝑐) −(𝑎, 𝑏) +(𝑏, 𝑑) …
Graph(unmate-rialized)
…𝑐
𝑎 𝑏
𝑐
𝑎 𝑏
𝑐 𝑑
𝑎 𝑏
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
Not Materialized
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Problem Definition
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•Given: ◦ a fully-dynamic graph stream (possibly infinite)
◦ memory space (finite)
• Estimate: the count of triangles
• To Minimize: estimation error
Time 𝑡 1 2 3 4 5 …
Changes +(𝑎, 𝑏) +(𝑎, 𝑐) +(𝑏, 𝑐) −(𝑎, 𝑏) +(𝑏, 𝑑) …
# Triangles…
Given(input)
Estimate(output)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Roadmap• T1. Structure Analysis (Part 1)
◦ T1.1 Triangle Counting▪Handling Deletions (§6)
◦ Problem Definition◦ Proposed Method: ThinkD <<◦ Experiments▪…
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
• Future Directions
•Conclusions
33/127Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Overview of ThinkD•Maintains and updates ∆
◦ Number of (non-deleted) triangles that it has observed
•How it processes an insertion:
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- arrive: an insertion of an edge arrives
- count: count new triangles and increase ∆- test: toss a coin - store: store the edge in memory
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
storearrive
No
(keep?)Yes
testcount:
increase ∆
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Overview of ThinkD (cont.)
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•Maintains and updates ∆◦ Number of (non-deleted) triangles that it has observed
•How it processes an deletion:
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
deletearriveYes
- arrive: a deletion of an edge arrives
- count: count deleted triangles and decrease ∆- test: test whether the edge is stored in memory- delete: delete the edge in memory
No
test(stored?)
count:
decrease ∆
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Why is ThinkD Accurate?
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• ThinkD (Think before You Discard):◦ every arrived change is used to update ∆
• Triest-FD [DERU17]:◦ some changes are discarded without being used to update ∆
store /delete
testarrivecount:
update ∆ Yes
No (discard)
store /delete
arriveYes
No (discard) → information loss!
count:update ∆
test
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Two Versions of ThinkD
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• ThinkD-FAST: simple and fast◦ independent Bernoulli trials with probability 𝑝
• ThinkD-ACC: accurate and parameter-free◦ random pairing [GLH08]
Q1: How to test in the test step
Q2: How to estimate the count of all triangles from ∆
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Unbiasedness of ThinkD-FAST•∆
𝒑𝟐: estimated count of all triangles
•∆: true count of all triangles
[ Theorem 1 ] At any time 𝒕,
• Proof and a variance of ∆/p2: see the thesis
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𝔼∆
𝒑𝟐= 𝜟
Unbiased estimate of 𝜟
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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ThinkD-ACC: More Accurate•Disadvantage of ThinkD-FAST:
◦ setting the parameter 𝑝 is not trivial
▪small 𝑝 → underutilize memory
→ inaccurate estimation
▪large 𝑝 → out-of-memory error
• ThinkD-ACC uses Random Pairing [RLH08] ◦ always utilizes memory as fully as possible
◦ gives more accurate estimation
39/127Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Scalability of ThinkD• Let 𝑘 be the size of memory
• For processing 𝑡 changes in the input stream,
[ Theorem 2 ] The time complexity of ThinkD-ACC is
[ Theorem 3 ] If 𝑝 = 𝑂𝑘
𝑡,
the time complexity ThinkD-FAST is
𝑂(𝑘 ⋅ 𝑡) linear in data size
𝑂(𝑘 ⋅ 𝑡)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Advantages of ThinkD
Fast & Accurate: outperforming competitors
Scalable: linear data scalability (Theorems 2 & 3)
Theoretically Sound: unbiased estimates (Theorem 1)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Roadmap• T1. Structure Analysis (Part 1)
◦ T1.1 Triangle Counting
▪Handling Deletions (§6)
◦ Problem Definition
◦ Proposed Method: ThinkD
◦ Experiments <<▪…
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
•…
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Experimental Settings•Competitors: Triest-FD [DERU17] & ESD [HS17]
◦ triangle counting in fully-dynamic graph streams
• Implementations:
•Datasets:◦ insertions (edges in graphs) + deletions (random 20%)
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Web(6M+)
Citation(16M+)
Social Networks(1.8B+ edges, …)
Synthetic(100B edges)
Trust(0.7M+)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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EXP1. Variance Analysis
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ThinkD is accurate with small varianceTriest-FD
ThinkD-ACC
ThinkD-FAST
True Count
Number of Processed Changes
- dataset:
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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EXP2. Scalability [THM 2 & 3]
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ThinkD-ACC
ThinkD-FAST
Number of Changes
- dataset:
100 billion changes
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
ThinkD is scalable
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EXP3. Space & Accuracy
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ThinkD outperforms its best competitors
ThinkD-FAST
ThinkD-ACC
Memory budget (ratio)
Estim
ation E
rror
(ratio)
Triest-FD
ESD
- dataset:
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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EXP4. Speed & Accuracy
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Running time (Sec)
Estim
ation E
rror
(ratio)
ThinkD-FAST
ThinkD-ACC
ESD
Triest-FD
- dataset:
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
ThinkD outperforms its best competitors
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Advantages of ThinkD
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Fast & Accurate: outperforming competitors
Scalable: linear data scalability
Theoretically Sound: unbiased estimates
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Summary of §6•We propose ThinkD (Think Before you Discard)
◦ for accurate triangle counting
◦ in large and fully-dynamic graphs
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ThinkDDownload
Fast & Accurate: outperforming competitors
Scalable: linear data scalability
Theoretically Sound: unbiased estimates
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 50/127
Part1.Structure Analysis
Part2. AnomalyDetection
Part3. BehaviorModeling
GraphsTriangle Count
(§§ 3-6) Anomalous Subgraph
(§ 9)
PurchaseBehavior
(§ 14)Summarization(§ 7)
Tensors Summarization(§ 8)
Dense Subtensors(§§ 10-13)
Progression(§ 15)
Organization of the Thesis (Recall)
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T1.2 Summarization
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 51/127
T1-2𝑎, 𝑏 𝑐, 𝑑, 𝑒 𝑓, 𝑔
𝑎
𝑏
𝑐
𝑑
𝑒
𝑓
𝑔
“Given a web-scale graph or tensor, how can we succinctly represent it?”
Input graph Summary graph
• §7: Summarizing Graphs
• §8: Summarizing Tensors (via Tucker Decomposition)◦ External-memory algorithm with 1,000× improved scalability
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Roadmap• T1. Structure Analysis (Part 1)
◦ …◦ T1-2. Summarization (§§ 7-8)▪Summarizing Graphs (§ 7)
◦ Problem Definition <<◦ Proposed Method: SWeG◦ Experiments▪…
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
•…
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 52/127K. Shin, A. Ghoting, M. Kim, and H. Raghavan, “SWeG: Lossless and Lossy
Summarization of Web-Scale Graphs”, WWW 2019
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Graph Summarization: Example
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− (𝑎, 𝑑)𝑎
𝑏
𝑐
𝑑
𝑒
𝑓
𝑔 𝑎, 𝑏
𝑐
𝑑
𝑒
𝑓
𝑔
𝑎, 𝑏 𝑐, 𝑑, 𝑒
𝑓
𝑔
− (𝑎, 𝑑)− (𝑐, 𝑒)+ 𝑑, 𝑔𝑎, 𝑏 𝑐, 𝑑, 𝑒 𝑓, 𝑔
− (𝑎, 𝑑)− (𝑐, 𝑒)+ 𝑑, 𝑔
Input Graph (w/ 9 edges)
Output (w/ 6 edges)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Graph Summarization [NRS08]•Given: an input graph
• Find:◦ a summary graph
◦ positive and negative residual graphs
• To Minimize: the edge count (≈ description length)
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Summary Graph
Residual Graph (Positive)
Residual Graph (Negative)
𝑎
𝑏
𝑑
𝑒
𝑓
𝑔 𝑎, 𝑏 𝑐, 𝑑, 𝑒 𝑓, 𝑔− (𝑎, 𝑑)− (𝑐, 𝑒)
+ 𝑑, 𝑔
Input Graph
𝑐
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Restoration: Example
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𝑎, 𝑏 𝑐, 𝑑, 𝑒 𝑓, 𝑔
− (𝑎, 𝑑)− (𝑐, 𝑒)+ 𝑑, 𝑔 𝑎, 𝑏 𝑐, 𝑑, 𝑒
𝑓
𝑔
− (𝑎, 𝑑)− (𝑐, 𝑒)+ 𝑑, 𝑔
𝑎, 𝑏
𝑐
𝑑
𝑒
𝑓
𝑔− (𝑎, 𝑑)
𝑎
𝑏
𝑐
𝑑
𝑒
𝑓
𝑔
Restored Graph (w/ 9 edges)
Summarized Graph (w/ 6 edges)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Why Graph Summarization?• Summarization:
◦ the summary graph is easy to visualize and interpret
•Compression:◦ support efficient neighbor queries
◦ applicable to lossy compression
◦ combinable with other graph compression techniques
▪the outputs are also graphs
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Summary Graph
Residual Graph (Positive)
Residual Graph (Negative)𝑎, 𝑏 𝑐, 𝑑, 𝑒 𝑓, 𝑔
− (𝑎, 𝑑)− (𝑐, 𝑒)
+ 𝑑, 𝑔
discussed in the thesis
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Challenge: Scalability!
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Maximum Size of Graphs
Compression Performance
Good
Bad
VoG [KKVF14]Greedy [NSR08]
millions 10 millions
Randomized [NSR08]
SAGS [KNL15]
billions
10,000 ×
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
SWeG
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Our Contribution: SWeG
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Fast with Concise Outputs
Memory Efficient
Scalable
•We develop SWeG (Summarizing Web-scale Graphs):
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Roadmap• T1. Structure Analysis (Part 1)
◦ …◦ T1-2. Summarization (§§ 7-8)▪Summarizing Graphs (§ 7)
◦ Problem Definition ◦ Proposed Method: SWeG <<◦ Experiments▪…
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
•…
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Terminologies
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Summary Graph 𝑺
{𝑎, 𝑏}= 𝐴
𝑐, 𝑑, 𝑒= 𝐵
𝑓, 𝑔= 𝐶
− (𝑎, 𝑑)− (𝑐, 𝑒)
+ 𝑑, 𝑔
super node
Residual Graph 𝑹 PositiveResidual Graph 𝑹+
Negative Residual Graph 𝑹−
𝑺𝒂𝒗𝒊𝒏𝒈 𝑨,𝑩 := 1 −𝐶𝑜𝑠𝑡(𝐴 ∪ 𝐵)
𝐶𝑜𝑠𝑡 𝐴 + 𝐶𝑜𝑠𝑡(𝐵)
Encoding cost when 𝐴 and 𝐵
are merged
Encoding cost of 𝐴 Encoding cost of 𝐵
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Overview of SWeG• Inputs: - input graph 𝑮
- number of iterations 𝑻
•Outputs: - summary graph 𝑺
- residual graph 𝑹 (or 𝑹+ and 𝑹−)
•Procedure:
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• S0: Initializing Step• repeat 𝑇 times
• S1-1: Dividing Step• S1-2: Merging Step
• S2: Compressing Step (optional)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Overview: Initializing Step
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𝐴 = {𝑎}
𝐵 = {𝑏}
𝐷 = {𝑑}
𝐸 = {𝑒}
𝐹 = {𝑓}
𝐺 = {𝑔}
𝐶 = {𝑐}
Summary Graph 𝑺 = 𝑮 Residual Graph 𝑹 = ∅
• S0: Initializing Step <<• repeat 𝑇 times
• S1-1: Dividing Step• S1-2: Merging Step
• S2: Compressing Step (optional)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Overview: Dividing Step
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• S0: Initializing Step• repeat 𝑇 times
• S1-1: Dividing Step <<• S1-2: Merging Step
• S2: Compressing Step (optional)
𝐴 = {𝑎}
𝐵 = {𝑏}
𝐷 = {𝑑}𝐶 = {𝑐}
𝐸 = {𝑒}
𝐹 = {𝑓}
𝐺 = {𝑔}
•Divides super nodes into groups◦ MinHashing (used), EigenSpoke, Min-Cut, etc.
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Overview: Merging Step•Merge some supernodes within each group if 𝑆𝑎𝑣𝑖𝑛𝑔 > 𝜃(𝑡)
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𝐴 = {𝑎, 𝑏}𝐷 = {𝑑, 𝑒}
𝐶 = {𝑐}𝐹 = {𝑓, 𝑔}
• S0: Initializing Step • repeat 𝑇 times
• S1-1: Dividing Step• S1-2: Merging Step <<
• S2: Compressing Step (optional)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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• S0: Initializing Step • repeat 𝑇 times
• S1-1: Dividing Step• S1-2: Merging Step
• S2: Compressing Step (optional)
𝐴 = {𝑎, 𝑏}
𝐷 = {𝑑, 𝑒}
𝐹 = {𝑓, 𝑔}
𝐶 = {𝑐}
Summary Graph 𝑺 Residual Graph 𝑹
− (𝑎, 𝑑)
+ 𝑑, 𝑔
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
Overview: Merging Step (cont.)
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Overview: Dividing Step
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• S0: Initializing Step • repeat 𝑇 times
• S1-1: Dividing Step <<• S1-2: Merging Step
• S2: Compressing Step (optional)
𝐴 = {𝑎, 𝑏} 𝐷 = {𝑑, 𝑒}𝐶 = {𝑐}𝐹 = {𝑓, 𝑔}
•Divides super nodes into groups
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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𝐴 = {𝑎, 𝑏} 𝐶 = {𝑐, 𝑑, 𝑒}𝐹 = {𝑓, 𝑔}
• S0: Initializing Step • repeat 𝑇 times
• S1-1: Dividing Step• S1-2: Merging Step <<
• S2: Compressing Step (optional)
•Merge some supernodes within each group if 𝑆𝑎𝑣𝑖𝑛𝑔 > 𝜃(𝑡)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
Overview: Merging Step
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𝐴 = {𝑎, 𝑏}
𝐵 =𝑐, 𝑑, 𝑒
𝐹 = 𝑓, 𝑔− (𝑎, 𝑑)− (𝑐, 𝑒)
+ 𝑑, 𝑔
Summary Graph 𝑺 Residual Graph 𝑹
• S0: Initializing Step• repeat 𝑇 times
• S1-1: Dividing Step• S1-2: Merging Step
• S2: Compressing Step (optional)
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
Overview: Merging Step (cont.)
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• S0: Initializing Step • repeat 𝑇 times
• S1-1: Dividing Step• S1-2: Merging Step <<
• S2: Compressing Step (optional)
•Merge some supernodes within each group if 𝑆𝑎𝑣𝑖𝑛𝑔 > 𝜃(𝑡)
•Decreasing 𝜃(𝑡) = 1 + 𝑡 −1
◦ exploration of other groups
◦ exploitation within each group
◦~ 30% better compression than 𝜃(𝑡) = 0
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
Overview: Merging Step (cont.)
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Overview: Compressing Step•Compress each output graph (𝑺, 𝑹+ and 𝑹−)
•Use any off-the-shelf graph-compression algorithm◦ Boldi-Vigna [BV04]
◦ VNMiner [BC08]
◦ Graph Bisection [DKKO+16]
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• S0: Initializing Step• repeat 𝑇 times
• S1-1: Dividing Step• S1-2: Merging Step
• S2: Compressing Step (optional) <<
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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No need to load the entire graph in memory!
•Map stage: compute min hashes in parallel
• Shuffle stage: divide super nodes using min hashes
•Reduce stage: process groups independently in parallel
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Parallel & Distributed Processing
𝐴 = {𝑎}
𝐵 = {𝑏}𝐶 = {𝑐}
MinHash = 𝟏
Merge!
𝐷 = {𝑑}
𝐸 = {𝑒}
MinHash = 𝟐
𝐹 = {𝑓}
𝐺 = {𝑔}
MinHash = 𝟑
Merge! Merge!
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Roadmap• T1. Structure Analysis (Part 1)
◦ ...◦ T1-2. Summarization (§§ 7-8)▪Summarizing Graphs (§ 7)
◦ Problem Definition ◦ Proposed Method: SWeG◦ Experiments <<▪…
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
•…
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 72/127
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Experimental Settings
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• 13 real-world graphs (10K - 20B edges)
•Graph summarization algorithms: ◦ Greedy [NRS08], Randomized [NSR08], SAGS [KNL15]
• Implementations: &
Social Collaboration Citation Web …
…
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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EXP1. Speed and Compression
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SWeG outperforms its competitors
SWeG
- dataset:
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Advantages of SWeG (Recall)
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Fast with Concise Outputs
Memory Efficient
Scalable
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Memory Usage
Input Graph
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EXP2. Memory EfficiencySWeG loads ≤0.1−4% of edges
in main memory at once
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
294X1209X
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Advantages of SWeG (Recall)
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Fast with Concise Outputs
Memory Efficient
Scalable
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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About 20 iterations are enough
EXP3. Effect of Iterations
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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EXP4. Data Scalability
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SWeG is linear in the number of edges
SWeG(Hadoop)
SWeG(Single machine)
≥ 𝟐𝟎 billion edges
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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EXP5. Machine Scalability
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SWeG(Hadoop)
SWeG(Single machine)
SWeG scales up
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
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Advantages of SWeG (Recall)
81/127Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
Fast with Concise Outputs
Memory Efficient
Scalable
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Summary of §7•We propose SWeG (Summarizing Web Graphs)
◦ for summarizing large-scale graphs
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Fast with Concise Outputs
Memory Efficient
Scalable
Graph / TensorPart 1 / Part 2 / Part 3 §4 / §5 / §6 / §7
SWeG(Hadoop)
SWeG
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Triangle counting algorithms [ICDM17, PKDD18, PAKDD18]
Summarization algorithms [WSDM17, WWW19]
Patent on SWeG: filed by LinkedIn Inc.
Open-source software: downloaded 82 times
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Contributions and Impact (Part 1)
Mining Large Dynamic Graphs and Tensors (by Kijung Shin)
github.com/kijungs
SWeG
ThinkD
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Part1.Structure Analysis
Part2. AnomalyDetection
Part3. BehaviorModeling
GraphsTriangle Count
(§§ 3-6) Anomalous Subgraph
(§ 9)
PurchaseBehavior
(§ 14)Summarization(§ 7)
Tensors Summarization(§ 8)
Dense Subtensors(§§ 10-13)
Progression(§ 15)
Organization of the Thesis (Recall)
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T2. Anomaly Detection (Part 2)“How can we detect anomalies or fraudsters
in large dynamic graphs (or tensors)?”
Hint: fraudsterstend to form dense subgraphs
??
benigndense
subgraphs
Graph / Tensor Part 1 / Part 2 / Part 3
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T2-1. Utilizing Patterns• T2-1. Patterns and Anomalies in Dense Subgraphs (§ 9)
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“What are patterns in dense subgraphs?”“What are anomalies deviating from the patterns?”
K. Shin, T. Eliassi-Rad, C. Faloutsos, “Patterns and Anomalies in k-Cores of Real-world Graphs with Applications”, KAIS 2018 (formerly, ICDM 2016)
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T2-2. Utilizing Side Information
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Item
s
Accounts
Graph / Tensor Part 1 / Part 2 / Part 3 §11 / §12 / §13 / §14
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“How can we detect dense subtensors in large dynamic data?”
• T2-2. Detecting Dense Subtensors (§§ 11-13)◦ In-memory Algorithm (§ 11)
◦ Distribute Algorithm for Web-scale Tensors (§ 12)
◦ Incremental Algorithms for Dynamic Tensors (§ 13)
T2-2. Utilizing Side Information
Graph / Tensor Part 1 / Part 2 / Part 3 §11 / §12 / §13 / §14
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Contributions and Impact (Part 2)Patterns in dense subgraphs [ICDM16]
◦ Award: best paper candidate at ICDM 2016
◦ Class:
Algorithms for dense subtensors [PKDD16, WSDM17, KDD17]
◦ Real-world usage:
Open-source software: downloaded 257 times
89/127Mining Large Dynamic Graphs and Tensors (by Kijung Shin)
github.com/kijungs
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Part1.Structure Analysis
Part2. AnomalyDetection
Part3. BehaviorModeling
GraphsTriangle Count
(§§ 3-6) Anomalous Subgraph
(§ 9)
PurchaseBehavior
(§ 14)Summarization(§ 7)
Tensors Summarization(§ 8)
Dense Subtensors(§§ 10-13)
Progression(§ 15)
Organization of the Thesis (Recall)
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“How can we model the behavior of individualsin graph and tensor data?”
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T3. Behavior Modeling (Part 3)
Social Network Behavior Log on Social Media
• T3-1. Modeling Purchase Behavior in a Social Network (§14)
• T3-2. Modeling Progression of Users of Social Media (§15)
“How do users evolve over time on social media?”
Graph Tensor Part 1 / Part 2 / Part 3 §14 / §15
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Roadmap• T1. Structure Analysis (Part 1)
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)◦ T3-1. Modeling Purchases (§14) <<◦ …
• Future Directions
•Conclusions
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 92/127
K Shin, E Lee, D Eswaran, AD Procaccia, “Why You Should Charge Your Friends for Borrowing Your Stuff”, IJCAI 2017
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Sharable Goods: Question
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“What do they have in common?”
Portable crib IKEA toolkit DVDs
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Sharable Goods: Properties
•Used occasionally
• Share with friends
•Do not share with strangers
94/127Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Motivation: Social Inefficiency
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Efficiencyof Purchase
Likelihoodof Purchase
High(share with many)
Low(share with few)
can be Low(likely to borrow)
can be High(likely to buy)
Popular Lonely
Q1 “How large can social inefficiency be?”Q2 “How to nudge people towards efficiency?”
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Roadmap• T1. Structure Analysis (Part 1)
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)◦ T3-1. Modeling Purchases (§14)
▪Toy Example <<
▪Game-theoretic Model
▪Best Rental-fee Search
◦ …
• Future Directions
•Conclusions
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 96/127
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Social Network•Consider a social network, which is a graph
◦ Nodes: people
◦ Edges: friendship
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Alice
Carol
Bob
“How many people should buy an IKEA toolkit for everyone to use it?”
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Socially Optimal Decision• The answer is at least 𝟐
• Socially optimal: ◦ everyone uses a toolkit
◦ with minimum purchases (or with minimum cost)
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Bob
“Does everyone want to stick to their current decisions?”
Alice
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Individually Optimal Decision• The answer is No
• Individually optimal: ◦ everyone best responses to others’ decisions
• Socially inefficient (suboptimal):◦ 4 purchases happen when 2 are optimal
99/127
AliceBob
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Social Inefficiency• Individually optimal outcome with 6 purchases
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Dan
Carol
“How can we prevent this social inefficiency?”
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Moving toward Social Optimum
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“How can we make people stick with this socially optimal outcome?”
•Recall the socially optimal outcome
BobAlice
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Imposing Rental Fee•Renters pay rental fee for getting permanent access
•Rental fee is half the price of a toolkit
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BobAlice
“Does everyone want to stick to their current decisions?”
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Socially & Individually Optimal• The answer is Yes
◦ Alice & Bob: are paid more than the price
◦ The others: renting is cheaper than buying
• Individually optimal
• Socially optimal with minimum (2) purchases
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BobAlice
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Socially & Individually Optimal• The answer is Yes
◦ Alice & Bob: are paid more than the price
◦ The others: renting is cheaper than buying
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BobAlice
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
Q1 “How do rental fees affect social inefficiency?”Q2 “What are the ‘socially optimal’ rental fees?”
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Roadmap• T1. Structure Analysis (Part 1)
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)◦ T3-1. Modeling Purchases (§14)
▪Toy Example
▪Game-theoretic Model <<
▪Best Rental-fee Algorithm
◦ …
• Future Directions
•Conclusions
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Formal Game-theoretic Model•Players: nodes in a social network
• Strategies:◦ buy a good / rent a good from a friend with a good
•Nash Equilibrium (NE): individually optimal outcome
• Social Optimum: socially optimal outcome
• Inefficiency of an NE:
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# purchases in a social optimum
# purchases in the NE
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Inefficiency without Rental Fee• [THM 1] Existence of NEs
◦ In every social network, there exists an NE.
• [THM 2] Inefficiency without Rental Fee◦ There exists a social network with 𝒏 nodes
◦ where all NEs have 𝜣(𝒏) inefficiency.
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… …
𝑛 − 1
2
Input graph
… …
Social optimum (𝟐 owners)
Best NE(𝒏/𝟐 owners)
… …
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Inefficiency with Rental Fee• [THM 3] Inefficiency with Rental Fee
◦ In every social network,
◦ if 𝑝𝑟𝑖𝑐𝑒
3< 𝑟𝑒𝑛𝑡𝑎𝑙 𝑓𝑒𝑒 < 𝑝𝑟𝑖𝑐𝑒,
◦ then, there exists a socially optimal NE,
◦ otherwise …
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… …
𝑛 − 1
2
Input graph… …
Social optimum (𝟐 owners)
→ NE with proper rental fee
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Roadmap• T1. Structure Analysis (Part 1)
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)◦ T3-1. Modeling Purchases (§14) <<
▪Toy Example
▪Game-theoretic Model
▪Best Rental-fee Search <<
◦ …
• Future Directions
•Conclusions
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Finding Best Rental Fee•Given:
◦ a social network
◦ a sharable good with price 𝑝
• Find: a range of rental fees
• To Minimize: inefficiencies of NEs
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,Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Searching NEs (SGG-Nash)• Linear-time algorithm for searching NEs
•Gives different NEs depending on initial strategies
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randomly initialize strategies
repeat until an NE is reached◦ for each node
▪optimize its strategy while fixing the others’
[THM 4] ConvergenceIn every social network, an NE is reached within 𝟑 iterations.
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Scalability of SGG-Nash• SGG-Nash is linear in the number of edges
112/127Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
- Dataset:
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Best Rental Fee in Real Graphs•Datasets:
• Inefficiency is minimized consistently when
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1.5
2.0
2.5
3.0
0.0 0.5 1.0Ave
rage
Ine
ffic
iency
Rental Fee (Relative to Price)
2
3
4
0.0 0.5 1.0Ave
rage
Ine
ffic
iency
Rental Fee (Relative to Price)
𝑝𝑟𝑖𝑐𝑒/3 < 𝑟𝑒𝑛𝑡𝑎𝑙 𝑓𝑒𝑒 < 𝑝𝑟𝑖𝑐𝑒/2
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
Theoretically best [THM3]
Theoretically best [THM3]
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1.5
2.0
2.5
3.0
0.0 0.5 1.0Ave
rage
Ine
ffic
iency
Summary of §14•Game-theoretic Model: Sharable good game
• Theoretical Analysis:◦ Existence of NEs
◦ Inefficiency of NEs
•Algorithm: for linear-time NE search
• Suggestion: “socially optimal” rental fees
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Rental Fee (Relative to Price)
Graph TensorPart 1 / Part 2 / Part 3 §14 / §15
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Tools for purchase modeling [IJCAI17]
◦ Suggest ‘socially optimal’ rental fees
◦ Media:
Tools for progression modeling [WWW18]
◦ Scale to datasets with a trillion records
◦ Real-world usage:
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Contributions and Impact (Part 3)
Graph Tensor Part 1 / Part 2 / Part 3 §14 / §15
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Part1.Structure Analysis
Part2. AnomalyDetection
Part3. BehaviorModeling
GraphsTriangle Count
(§§ 3-6) Anomalous Subgraph
(§ 9)
PurchaseBehavior
(§ 14)Summarization(§ 7)
Tensors Summarization(§ 8)
Dense Subtensors(§§ 10-13)
Progression(§ 15)
Organization of the Thesis (Recall)
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Roadmap• T1. Structure Analysis (Part 1)
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)
• Future Directions <<
•Conclusions
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Model & Theory
Scalable Algorithms for Big Data Mining
D3
Vision: Algorithms for “Big Data”
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D1: Distributed Graph Stream Processing“How to analyze large dynamic graphs
on a cluster of machines?”
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Current
SourcesDestination
Short Term
More Graph Mining Tasks
Mid Term
Programming Model
(Generalization)
Long Term
System / Platform
Triangle Counting
(§5)
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D2: Detecting Adversarial Anomalies
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Anomalies / Fraudsters Detection System
improve
avoid
Profits of Anomalies
Cost to AvoidAlgorithms
Algorithms Costly
to Avoid
Algorithms for Static
Anomalies
Current Short Term Mid Term Long Term
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D3: Co-Evolution of Beliefs and Graphs
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“How to model the co-evolution of nodes’ beliefs and edges?” Change of Beliefs
OR
Change of Edges
Game Theory/ Nash
Equilibrium
PredictionAlgorithms
Reducing Polarization
Regression Model
[PKDD18]
Current Short Term Mid Term Long Term
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Roadmap• T1. Structure Analysis (Part 1)
• T2. Anomaly Detection (Part 2)
• T3. Behavior Modeling (Part 3)◦ T3-1. Modeling Purchases (§14) <<
◦ T3-2. Modeling Progression (§ 15) (Skip)
• Future Directions
•Conclusions <<
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Conclusion•Goal: “To fully understand and utilize …………. large dynamic graphs and tensors”
•Contributions: developing scalable algorithms for
• Impact:
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T1. StructureAnalysis (Part 1)
T2. Anomaly Detection (Part 2)
T3. Behavior Modeling (Part 3)
1 23
4
github.com/kijungs
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[1] K Shin, B Hooi, and C Faloutsos, “M-Zoom: Fast Dense-Block Detection in Tensors with Quality Guarantees”, ECML/PKDD 2016 (§11)
[2] K Shin, T Eliassi-Rad, and C Faloutsos, “CoreScope: Graph Mining Using k-Core Analysis - Patterns, Anomalies and Algorithms”, ICDM 2016 (§9)
[3] K Shin, “Mining Large Dynamic Graphs and Tensors for Accurate Triangle Counting in Real Graph Streams”, ICDM 2017 (§4)
[4] K Shin, B Hooi, J Kim, and C Faloutsos, “D-Cube: Dense-Block Detection in Terabyte-Scale Tensors”, WSDM 2017 (§12)
[5] K Shin, E Lee, D Eswaran, and AD. Procaccia, “Why You Should Charge Your Friends for Borrowing Your Stuff”, IJCAI 2017 (§14)
[6] K Shin, B Hooi, J Kim, and C Faloutsos, “DenseAlert: Incremental Dense-Subtensor Detection in Tensor Streams”, KDD 2017 (§13)
[7] J Oh, K Shin, EE Papalexakis, C Faloutsos, and Hwanjo Yu, “S-HOT: Scalable High-Order Tucker Decomposition”, WSDM 2017 (§8)
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References
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References (cont.)
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[8] K Shin, B Hooi, and C Faloutsos, “Fast, Accurate and Flexible Algorithms for Dense Subtensor Mining”, TKDD 2018 (§11)
[9] K Shin, M Shafiei, M Kim, A Jain, and H Raghavan, “Discovering Progression Stages in Trillion-Scale Behavior Logs” WWW 2018 (§14)
[10] K Shin, T Eliassi-Rad, and C Faloutsos, “Patterns and Anomalies in k-Cores of Real-world Graphs with Applications”, KAIS 2018 (§9)
[11] K Shin, M Hammoud, E Lee, J Oh, and C Faloutsos. “Tri-fly: Distributed estimation of global and local triangle counts in graph streams” PAKDD 2018 (§5)
[12] K Shin, J Kim, B Hooi, and C Faloutsos, “Think before You Discard: Accurate Triangle Counting in Graph Streams with Deletions”, ECML/PKDD 2018 (§6)
[13] K Shin, A Ghoting, M Kim and H Raghavan, “SWeG: Lossless and Lossy Summarization of Web-Scale Graphs”, WWW 2019 (§7)
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Thank You!• Sponsors:
•Admins:
•Collaborators:
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Thank You!
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•Homepage (Software & Datasets): http://www.cs.cmu.edu/~kijungs/defense/
ThinkD
-ACC