SWeG: Lossless and Lossy Summarization of Web-Scale Graphs

SWeG: Lossless and
Lossy Summarization of
Web-Scale Graphs
Kijung Shin Amol Ghoting Myunghwan Kim Hema Raghavan
SWeG: Lossless and Lossy Summarization of Web-Scale Graphs (by Kijung Shin)

Graphs are Everywhere
SWeG: Lossless and Lossy Summarization of Web-Scale Graphs (by Kijung Shin) 2/42

Graphs are Everywhere (cont.)

Real-world Graphs are Huge
2B+ active users
500M+ products
300M+ customers
4B+ Web pages
600M+ users
20B+ connections
× 30+
How can we analyze and utilize
such large graph data?

Limitations of Existing Tools
•Graph algorithms in textbooks
◦ Assume graphs fit in main memory
(i.e., random-access memory)
•Tools for out-of-core graph processing
◦ Not for every graph algorithm
◦ Requiring engineering for each algorithm
◦ Inappropriate for real-time applications
1000+
new graph
algorithms
per year!

Solution: Graph Compression
Textbook
algorithms
Any new
algorithms
Real-time
algorithms
•Compressing large graphs so that
◦ Compressed data fit in main memory
◦ Algorithms can be performed on compressed
data without changes

Roadmap
• Problem Definition <<
• Proposed Method: SWeG
• Experimental Results
• Conclusions

Graph Summarization: Example
• Graph summarization is a graph compression technique
− (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, 𝑔
𝑎, 𝑏 𝑐, 𝑑, 𝑒 𝑓, 𝑔
− (𝑎, 𝑑)
− (𝑐, 𝑒)
+ 𝑑, 𝑔
Input Graph (w/ 9 edges)
Output (w/ 6 edges)

Graph Summarization: Definition
• Given: an input graph
• Find:
◦ a summary graph
◦ positive and negative residual graphs
• To Minimize: the edge count (≈ description length)
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

Restoration: Example
11/42
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)

Why Graph Summarization?
• Neighbor queries can be rapid and efficient
§Given a seed node, return its neighbors
§Key building block of most graph algorithms
• Easily extended to lossy compression
• Easily combined with other graph-compression methods
◦ the outputs are also graphs
12/42SWeG: Lossless and Lossy Summarization of Web-Scale Graphs (by Kijung Shin)
Summary Graph
Residual Graph (Positive)
Residual Graph (Negative)𝑎, 𝑏 𝑐, 𝑑, 𝑒 𝑓, 𝑔 − (𝑎, 𝑑)
− (𝑐, 𝑒)
+ 𝑑, 𝑔
discussed
in the paper

Challenge: Scalability!
13/42
Maximum Size of Input Graphs
Compression
Performance
Good
Bad
VoG [KKVF14]
Greedy [NSR08]
millions 10 millions
Randomized [NSR08]
SAGS [KNL15]
billions
10,000×
SWeG
(Proposed)

Our Contribution: SWeG
14/42
Fast with Concise Outputs
Memory Efficient
Scalable
• We develop SWeG (SWeG: Lossless and Lossy
Summarization of Web-Scale Graphs):

Roadmap
• Problem Definition
• Proposed Method: SWeG <<
• Conclusions

Main Idea behind SWeG
• Graph summarization ≈ node clustering
◦ Finding sets of similar nodes to be merged into super nodes
• Previous heuristics are greedy algorithms
• Why are even greedy algorithms slow?
◦ Too many node pairs are considered: 𝑂(𝑛3)
• How can we reduce the number of node pairs to be
considered without missing similar node pairs?
𝑎
𝑏
𝑐
𝑑
𝑒
𝑓
𝑔
𝑎
𝑏
𝑐
𝑑
𝑒
𝑓

Main Idea behind SWeG (cont.)
• Step 1: Coarse clustering (Grouping)
◦ Fast and careless
• Step 2: Fine clustering (Merging)
◦ Greedy algorithm
◦ Only node pairs within each group are considered
Repeated
𝑎
𝑏
𝑐
𝑑
𝑒
𝑓
𝑔
𝑎
𝑏
𝑐
𝑑
𝑒
𝑓

Terminologies
18/42
Summary Graph 𝑺
{𝑎, 𝑏}
= 𝐴
𝑐, 𝑑, 𝑒
= 𝐵
𝑓, 𝑔
= 𝐶
− (𝑎, 𝑑)
− (𝑐, 𝑒)
+ 𝑑, 𝑔
super node
Residual Graph 𝑹 Positive
Residual
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

Details of SWeG
• Inputs: - input graph 𝑮
- number of iterations 𝑻
• Outputs: - summary graph 𝑺
- residual graph 𝑹 (or 𝑹<
and 𝑹=
)
• Procedure:
19/42
• S0: Initializing Step
• repeat 𝑇 times
• S1-1: Dividing Step
• S1-2: Merging Step
• S2: Compressing Step (optional)

SWeG: Initializing Step
20/42
𝐴 = {𝑎}
𝐵 = {𝑏}
𝐷 = {𝑑}
𝐸 = {𝑒}
𝐹 = {𝑓}
𝐺 = {𝑔}
𝐶 = {𝑐}
Summary Graph 𝑺 = 𝑮 Residual Graph 𝑹 = ∅
• S0: Initializing Step <<

SWeG: Dividing Step
21/42
• S1-1: Dividing Step <<
𝐴 = {𝑎}
𝐵 = {𝑏}
𝐷 = {𝑑}
𝐶 = {𝑐}
𝐸 = {𝑒}
𝐹 = {𝑓}
𝐺 = {𝑔}
• Divides super nodes into groups
◦ MinHashing: rapid and out-of-core

SWeG: Merging Step
• Merge some supernodes within each group if 𝑆𝑎𝑣𝑖𝑛𝑔 > 𝜃(W)
22/42
𝐴 = {𝑎, 𝑏}
𝐷 = {𝑑, 𝑒}
𝐶 = {𝑐}
𝐹 = {𝑓, 𝑔}
• S1-2: Merging Step <<

23/42
𝐴 = {𝑎, 𝑏}
𝐷 = {𝑑, 𝑒}
𝐹 = {𝑓, 𝑔}
𝐶 = {𝑐}
Summary Graph 𝑺 Residual Graph 𝑹
− (𝑎, 𝑑)
+ 𝑑, 𝑔
SWeG: Merging Step (cont.)

SWeG: Dividing Step
24/42
• S1-1: Dividing Step <<
𝐴 = {𝑎, 𝑏} 𝐷 = {𝑑, 𝑒}
𝐶 = {𝑐}𝐹 = {𝑓, 𝑔}
• Divides super nodes into groups

25/42
𝐴 = {𝑎, 𝑏} 𝐶 = {𝑐, 𝑑, 𝑒}
𝐹 = {𝑓, 𝑔}
SWeG: Merging Step

26/42
𝐴 = {𝑎, 𝑏}
𝐵 =
𝑐, 𝑑, 𝑒
𝐹 = 𝑓, 𝑔
− (𝑎, 𝑑)
− (𝑐, 𝑒)
+ 𝑑, 𝑔
Summary Graph 𝑺 Residual Graph 𝑹

27/42
• Decreasing 𝜃(W)
= 1 + 𝑡 =X
◦ exploration of other groups
◦ exploitation within each group
◦ ~ 30% better compression than 𝜃(W)
= 0
Details

SWeG: Compressing Step
• Compress each output graph (𝑺, 𝑹<
and 𝑹=
)
• Use any off-the-shelf graph-compression algorithm
◦ Boldi-Vigna [BV04] / VNMiner [BC08] / BFS [AD09] /
Graph Bisection [DKKO+16]
•SWeG+: SWeG with the compressing step
28/42
• S2: Compressing Step (optional) <<

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
29/42
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!
Details

Roadmap
• Experimental Results <<
• Conclusions

Experimental Settings
31/42
• 13 real-world graphs (10K - 20B edges)
• Graph summarization algorithms:
◦ Greedy [NRS08], Randomized [NSR08], SAGS [KNL15]
• Implementations: &
Social Collaboration Citation Web …
…

EXP1. Speed and Compression
32/42
SWeG outperforms its competitors
SWeG
- dataset:
𝟑 𝟕 𝟎
−
𝟒, 𝟒 𝟗 𝟎×
faster

Advantages of SWeG
33/42
Memory Efficient
Scalable

Memory
Usage
Input
Graph
34/42
EXP2. Memory Efficiency
SWeG loads ≤0.1−4% of edges
in main memory at once
294X1209X

Advantages of SWeG
35/42
Memory Efficient
Scalable

36/42
About 20 iterations are enough
EXP3. Effect of Iterations

EXP4. Data Scalability
37/42
SWeG is linear in the number of edges
SWeG
(Hadoop)
SWeG
(Single machine)
≥ 𝟐𝟎 billion edges

EXP5. Machine Scalability
38/42
SWeG
(Hadoop)
SWeG
(Single machine)
SWeG scales up
ideal
ideal

Advantages of SWeG
39/42
Memory Efficient
Scalable

EXP6. Further Compression
SWeG+ achieves ~0.7 bit / link for Web graphs
BV BFS BP SWeG+
3.4X
- dataset:
𝟏. 𝟐
−
𝟑. 𝟒×
additional com
pression

Roadmap
• Conclusions <<

Conclusions
• We propose SWeG (Summarizing Web Graphs)
◦ for summarizing large-scale graphs
42/42
Memory Efficient
Scalable
SWeG
(Hadoop)
SWeG

SWeG: Lossless and Lossy Summarization of Web-Scale Graphs

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