1. SSumM : Sparse Summarization
of Massive Graphs
Kyuhan Lee* Hyeonsoo Jo* Jihoon Ko Sungsu Lim Kijung Shin

2. Graphs are Everywhere
Citation networksSubway networks Internet topologies
Introduction Algorithms Experiments ConclusionProblem
Designed by Freepik from Flaticon

3. Massive Graphs Appeared
Social networks
2.49 Billion active users
Purchase histories
0.5 Billion products
World Wide Web
5.49 Billion web pages
Introduction Algorithms Experiments ConclusionProblem

4. Difficulties in Analyzing Massive graphs
Computational cost
(number of nodes & edges)
Introduction Algorithms Experiments ConclusionProblem

5. Difficulties in Analyzing Massive graphs
>
Cannot fit
Introduction Algorithms Experiments ConclusionProblem
Input Graph

6. Solution: Graph Summarization
>
Introduction Algorithms Experiments ConclusionProblem
Can fit
Summary Graph

7. Advantages of Graph Summarization
Introduction Algorithms Experiments ConclusionProblem
• Many graph compression techniques are available
• TheWebGraph Framework [BV04]
• BFS encoding [AD09]
• SlashBurn [KF11]
• VoG [KKVF14]
• Graph summarization stands out because
• Elastic: reduce size of outputs as much as we want
• Analyzable: existing graph analysis and tools can be applied
• Combinable for Additional Compression: can be further compressed

8. Example of Graph Summarization
1
2
3 4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
Adjacency Matrix
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
Introduction Algorithms Experiments ConclusionProblem
Input Graph

9. Example of Graph Summarization
1
2
3 4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
Adjacency Matrix
Input Graph
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
𝑽𝑽𝟏𝟏={1,2}
𝑽𝑽𝟕𝟕={7,8,9}
𝑽𝑽𝟑𝟑={3,4,5,6}
Summary Graph
Introduction Algorithms Experiments ConclusionProblem
Subnode
Subedge
Supernode

10. Example of Graph Summarization
1
2
3 4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
Adjacency Matrix
Input Graph
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
𝑽𝑽𝟏𝟏={1,2}
𝑽𝑽𝟕𝟕={7,8,9}
𝑽𝑽𝟑𝟑={3,4,5,6}
Summary Graph
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟑𝟑} =3
Introduction Algorithms Experiments ConclusionProblem
Subnode
Subedge
Superedge
Supernode

11. Example of Graph Summarization
1
2
3 4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
Adjacency Matrix
Input Graph
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
𝑽𝑽𝟏𝟏={1,2}
𝑽𝑽𝟕𝟕={7,8,9}
𝑽𝑽𝟑𝟑={3,4,5,6}
Summary Graph
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟕𝟕} =2
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟑𝟑} =3
𝝎𝝎 {𝑽𝑽𝟑𝟑, 𝑽𝑽𝟕𝟕} =2
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟏𝟏} =1
𝝎𝝎 {𝑽𝑽𝟕𝟕, 𝑽𝑽𝟕𝟕} =3
𝝎𝝎 {𝑽𝑽𝟑𝟑, 𝑽𝑽𝟑𝟑} =5
Introduction Algorithms Experiments ConclusionProblem

12. Example of Graph Summarization
1
2
3 4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
Adjacency Matrix
Input Graph
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
𝑽𝑽𝟏𝟏={1,2}
𝑽𝑽𝟕𝟕={7,8,9}
𝑽𝑽𝟑𝟑={3,4,5,6}
Summary Graph
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟕𝟕} =2
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟑𝟑} =3𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟏𝟏} =1
𝝎𝝎 {𝑽𝑽𝟕𝟕, 𝑽𝑽𝟕𝟕} =3
𝝎𝝎 {𝑽𝑽𝟑𝟑, 𝑽𝑽𝟑𝟑} =5
Introduction Algorithms Experiments ConclusionProblem

13. Example of Graph Summarization
Introduction Algorithms Experiments ConclusionProblem
1 2 3 4 5 6 7 8 9
1 0 1 3/8 3/8 3/8 3/8 1/3 1/3 1/3
2 1 0 3/8 3/8 3/8 3/8 1/3 1/3 1/3
3 3/8 3/8 0 5/6 5/6 5/6 0 0 0
4 3/8 3/8 5/6 0 5/6 5/6 0 0 0
5 3/8 3/8 5/6 5/6 0 5/6 0 0 0
6 3/8 3/8 5/6 5/6 5/6 0 0 0 0
7 1/3 1/3 0 0 0 0 0 1 1
8 1/3 1/3 0 0 0 0 1 0 1
9 1/3 1/3 0 0 0 0 1 1 0
Reconstructed Adjacency Matrix
𝑽𝑽𝟏𝟏={1,2}
𝑽𝑽𝟕𝟕={7,8,9}
𝑽𝑽𝟑𝟑={3,4,5,6}
Summary Graph
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟕𝟕} =2
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟑𝟑} =3𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟏𝟏} =1
𝝎𝝎 {𝑽𝑽𝟕𝟕, 𝑽𝑽𝟕𝟕} =3
𝝎𝝎 {𝑽𝑽𝟑𝟑, 𝑽𝑽𝟑𝟑} =5

14. Example of Graph Summarization
𝑽𝑽𝟏𝟏={1,2}
𝑽𝑽𝟕𝟕={7,8,9}
𝑽𝑽𝟑𝟑={3,4,5,6}
Summary Graph
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟕𝟕} =2
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟑𝟑} =3𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟏𝟏} =1
𝝎𝝎 {𝑽𝑽𝟕𝟕, 𝑽𝑽𝟕𝟕} =3
𝝎𝝎 {𝑽𝑽𝟑𝟑, 𝑽𝑽𝟑𝟑} =5
Introduction Algorithms Experiments ConclusionProblem
1 2 3 4 5 6 7 8 9
1 0 1 3/8 3/8 3/8 3/8 1/3 1/3 1/3
2 1 0 3/8 3/8 3/8 3/8 1/3 1/3 1/3
3 3/8 3/8 0 5/6 5/6 5/6 0 0 0
4 3/8 3/8 5/6 0 5/6 5/6 0 0 0
5 3/8 3/8 5/6 5/6 0 5/6 0 0 0
6 3/8 3/8 5/6 5/6 5/6 0 0 0 0
7 1/3 1/3 0 0 0 0 0 1 1
8 1/3 1/3 0 0 0 0 1 0 1
9 1/3 1/3 0 0 0 0 1 1 0
Reconstructed Adjacency Matrix
𝝎𝝎 {𝑽𝑽𝟏𝟏, 𝑽𝑽𝟑𝟑} = 3
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 8

15. Road Map
• Introduction
• Problem <<
• Proposed Algorithm: SSumM
• Experimental Results
• Conclusions

16. Problem Definition: Graph Summarization
𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮:
a graph 𝑮𝑮 and the target number of node 𝑲𝑲
𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭:
a summary graph �𝑮𝑮
𝑻𝑻𝑻𝑻 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴:
the difference between graph 𝑮𝑮and the restored graph �𝑮𝑮
𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝒕𝒕𝒕𝒕:
the number of supernodes in �𝑮𝑮 ≤ 𝑲𝑲
Introduction Algorithms Experiments ConclusionProblem

17. Problem Definition: Graph Summarization
𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮:
a graph 𝑮𝑮 and the target number of node 𝑲𝑲
𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭:
a summary graph �𝑮𝑮
𝑻𝑻𝑻𝑻 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴:
the difference between graph 𝑮𝑮and the restored graph �𝑮𝑮
𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝒕𝒕𝒕𝒕:
the number of supernodes in �𝑮𝑮 ≤ 𝑲𝑲
Shouldn’t we
consider sizes?
Introduction Algorithms Experiments ConclusionProblem

18. Problem Definition: Graph Summarization
𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮:
a graph 𝑮𝑮 and the desired size 𝑲𝑲 (in bits)
𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭:
a summary graph �𝑮𝑮
𝑻𝑻𝑻𝑻 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴:
the difference with graph graph 𝑮𝑮and the restored graph �𝑮𝑮
𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝒕𝒕𝒕𝒕:
size of �𝑮𝑮 in bits ≤ 𝑲𝑲
Introduction Algorithms Experiments ConclusionProblem

19. Details: Size in Bits of a Graph
𝐸𝐸 : set of edges
𝑉𝑉 : set of nodes
Encoded using log2|𝑉𝑉| bits
Introduction Algorithms Experiments ConclusionProblem
𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝒐𝒐𝒐𝒐 𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈: 2 𝐸𝐸 log2 𝑉𝑉
Input graph 𝑮𝑮

20. Details: Size in Bits of a Summary Graph
4
5
1
4
1
5
Introduction Algorithms Experiments ConclusionProblem
𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝒐𝒐𝒐𝒐 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈: 𝑃𝑃 2 log2 𝑆𝑆 + log2 𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑉𝑉 log2 𝑆𝑆
𝑆𝑆 : set of supernodes
𝑃𝑃 : set of superedges
𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 : maximum superedge weight
Summary graph �𝑮𝑮

21. Details: Size in Bits of a Summary Graph
4
5
1
4
1
5
Introduction Algorithms Experiments ConclusionProblem
𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝒐𝒐𝒐𝒐 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈: 𝑃𝑃 2 log2 𝑆𝑆 + log2 𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑉𝑉 log2 𝑆𝑆
𝑆𝑆 : set of supernodes
𝑃𝑃 : set of superedges
𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 : maximum superedge weight
Summary graph �𝑮𝑮

22. Details: Size in Bits of a Summary Graph
4
5
1
4
1
5
Introduction Algorithms Experiments ConclusionProblem
𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝒐𝒐𝒐𝒐 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈: 𝑃𝑃 2 log2 𝑆𝑆 + log2 𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑉𝑉 log2 𝑆𝑆
𝑆𝑆 : set of supernodes
𝑃𝑃 : set of superedges
𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 : maximum superedge weight
Summary graph �𝑮𝑮

23. Details: Size in Bits of a Summary Graph
4
5
1
4
1
5
Introduction Algorithms Experiments ConclusionProblem
𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝒐𝒐𝒐𝒐 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈: 𝑃𝑃 2 log2 𝑆𝑆 + log2 𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑉𝑉 log2 𝑆𝑆
𝑆𝑆 : set of supernodes
𝑃𝑃 : set of superedges
𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 : maximum superedge weight
Summary graph �𝑮𝑮

24. Details: Error Measurement
Introduction Algorithms Experiments ConclusionProblem
1 2 3 4 5 6 7 8 9
1 0 1 3/8 3/8 3/8 3/8 1/3 1/3 1/3
2 1 0 3/8 3/8 3/8 3/8 1/3 1/3 1/3
3 3/8 3/8 0 5/6 5/6 5/6 0 0 0
4 3/8 3/8 5/6 0 5/6 5/6 0 0 0
5 3/8 3/8 5/6 5/6 0 5/6 0 0 0
6 3/8 3/8 5/6 5/6 5/6 0 0 0 0
7 1/3 1/3 0 0 0 0 0 1 1
8 1/3 1/3 0 0 0 0 1 0 1
9 1/3 1/3 0 0 0 0 1 1 0
1 2 3 4 5 6 7 8 9
1 0 1 0 0 0 1 0 0 1
2 1 0 1 0 0 1 1 0 0
3 0 1 0 1 1 1 0 0 1
4 0 0 1 0 1 0 0 0 0
5 0 0 1 1 0 1 1 0 0
6 1 1 1 0 1 0 0 0 0
7 0 1 0 0 1 0 0 1 1
8 0 0 0 0 0 0 1 0 1
9 1 0 1 0 0 0 1 1 0
Reconstructed Adjacency Matrix �𝑨𝑨Reconstructed Adjacency Matrix 𝑨𝑨
𝑅𝑅𝐸𝐸𝑝𝑝(𝑨𝑨, �𝑨𝑨) = �
𝑖𝑖=1
𝑉𝑉
�
𝑗𝑗=1
𝑉𝑉
𝐴𝐴 𝑖𝑖, 𝑗𝑗 − ̂𝐴𝐴 𝑖𝑖, 𝑗𝑗
𝑝𝑝
𝟏𝟏
𝒑𝒑

25. Road Map
• Introduction
• Problem
• Proposed Algorithm: SSumM <<
• Experimental Results
• Conclusions

26. Main ideas of SSumM
Introduction Algorithms Experiments ConclusionProblem
Combines node grouping and edge sparsification
Prunes search space
Balances error and size of the summary graph using MDL principle
• Practical graph summarization problem
◦ Given: a graph 𝑮𝑮
◦ Find: a summary graph �𝑮𝑮
◦ To minimize: the difference between 𝑮𝑮and the restored graph �𝑮𝑮
◦ Subject to: 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑜𝑜𝑜𝑜 �𝑮𝑮 in bits ≤ 𝑲𝑲

27. Main Idea: Combining Two Strategies
Introduction Algorithms Experiments ConclusionProblem
Node Grouping Sparsification

28. Main Idea: Combining Two Strategies
Introduction Algorithms Experiments ConclusionProblem
Node Grouping Sparsification

29. Main Idea: Combining Two Strategies
Introduction Algorithms Experiments ConclusionProblem
Node Grouping Sparsification

30. Main Idea: Combining Two Strategies
Introduction Algorithms Experiments ConclusionProblem
Node Grouping Sparsification

31. Main Idea: MDL Principle
Introduction Algorithms Experiments ConclusionProblem
1
2
3
4
5
6
Merge (5, 6)
1
2
3
4
Merge (1, 2)
3
4
5
6
5
6
1
2
{1,2}
{5,6}Merge (1, {5,6})
Merge (1, 2)
Merge (1, 3)
Merge (1, 3)
How to choose a next action?

32. Main Idea: MDL Principle
Introduction Algorithms Experiments ConclusionProblem
1
2
3
4
5
6
Merge (5, 6)
1
2
3
4
Merge (1, 2)
3
4
5
6
5
6
1
2
{1,2}
{5,6}Merge (1, {5,6})
Merge (1, 2)
Merge (1, 3)
Merge (1, 3)
Graph Summarization is
A Search Problem
How to choose a next action?

33. Main Idea: MDL Principle
Introduction Algorithms Experiments ConclusionProblem
1
2
3
4
5
6
Merge (5, 6)
1
2
3
4
Merge (1, 2)
3
4
5
6
5
6
1
2
{1,2}
{5,6}
Summary graph size + Information loss
Merge (1, {5,6})
Merge (1, 2)
Merge (1, 3)
Merge (1, 3)
Graph Summarization is
A Search Problem
How to choose a next action?

34. Main Idea: MDL Principle
Introduction Algorithms Experiments ConclusionProblem
1
2
3
4
5
6
Merge (5, 6)
1
2
3
4
Merge (1, 2)
3
4
5
6
5
6
1
2
{1,2}
{5,6}
Summary graph size + Information loss
MDL Principle
Merge (1, {5,6})
Merge (1, 2)
Merge (1, 3)
Merge (1, 3)
Graph Summarization is
A Search Problem
How to choose a next action?
arg min 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝑮𝑮 + 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶(𝑮𝑮|�𝑮𝑮)
�𝑮𝑮
# 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓𝑓𝑓 �𝑮𝑮 # 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓𝑓𝑓 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑮𝑮
𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 �𝑮𝑮

35. Overview: SSumM
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
Introduction Algorithms Experiments ConclusionProblem
Procedure
• Given:
◦ (1) An input graph 𝑮𝑮, (2) the desired size 𝑲𝑲, (3) the number 𝑻𝑻 of iterations
• Outputs:
◦ Summary graph �𝑮𝑮

36. Input graph 𝑮𝑮 Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase <<
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
𝑎𝑎
𝑏𝑏𝑐𝑐
𝑑𝑑
𝑒𝑒
𝑓𝑓
𝑔𝑔
ℎ
𝑖𝑖
𝐴𝐴 = {𝑎𝑎}
𝐵𝐵 = {𝑏𝑏}C = {𝑐𝑐}
𝐷𝐷 = {𝑑𝑑}
𝐸𝐸 = {𝑒𝑒}
𝐹𝐹 = {𝑓𝑓}
𝐺𝐺 = {𝑔𝑔}
𝐻𝐻 = {ℎ}
𝐼𝐼 = {𝑖𝑖}
Initialization Phase

37. Candidate Generation Phase
𝐵𝐵 = {𝑏𝑏}
𝐶𝐶 = {𝑐𝑐}
D = {𝑑𝑑}
𝐸𝐸 = {𝑒𝑒}
𝐴𝐴 = {𝑎𝑎}
𝐹𝐹 = {𝑓𝑓}
𝐺𝐺 = {𝑔𝑔}
𝐻𝐻 = {ℎ}
𝐼𝐼 = {𝑖𝑖}
Input graph 𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase <<
 Merge and sparsification phase
 Further sparsification phase
𝑎𝑎
𝑏𝑏𝑐𝑐
𝑑𝑑
𝑒𝑒
𝑓𝑓
𝑔𝑔
ℎ
𝑖𝑖

38. Merging and Sparsification Phase
For each candidate set 𝑪𝑪 Among possible candidate pairs
Introduction Algorithms Experiments ConclusionProblem
(A, B) (B, C) (C, D) (D, E)
(A, C) (B, D) (C, E)
(A, D) (B, E)
(A, E)
𝐵𝐵 = {𝑏𝑏}
𝐶𝐶 = {𝑐𝑐}
D = {𝑑𝑑}
𝐴𝐴 = {𝑎𝑎}
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
𝐸𝐸 = {𝑒𝑒}

39. Merging and Sparsification Phase
For each candidate set 𝑪𝑪 Among possible candidate pairs
Introduction Algorithms Experiments ConclusionProblem
(A, B) (B, C) (C, D) (D, E)
(A, C) (B, D) (C, E)
(A, D) (B, E)
(A, E)
𝐵𝐵 = {𝑏𝑏}
𝐶𝐶 = {𝑐𝑐}
D = {𝑑𝑑}
𝐴𝐴 = {𝑎𝑎}
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
Sample log2 |𝑪𝑪| pairs𝐸𝐸 = {𝑒𝑒}

40. Merging and Sparsification Phase
Select the pair with
the greatest (relative) reduction
in the cost function
𝒊𝒊𝒊𝒊 reduction(C, D) > 𝜽𝜽:
merge(C, D)
𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆
sample log2 |𝑪𝑪| pairs again
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
(A, B) (A, D) (C, D)

41. Merging and Sparsification Phase
Select the pair with
the greatest (relative) reduction
in the cost function
𝒊𝒊𝒊𝒊 reduction(C, D) > 𝜽𝜽:
merge(C, D)
𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆
sample log2 |𝑪𝑪| pairs again
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
(A, B) (A, D) (C, D)

42. Merging and Sparsification Phase (cont.)
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}𝐶𝐶 = {𝑐𝑐}
𝐷𝐷 = {𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

43. Merging and Sparsification Phase (cont.)
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

44. Merging and Sparsification Phase (cont.)
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}
C = {𝑐𝑐, 𝑑𝑑}

45. Merging and Sparsification Phase (cont.)
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}
C = {𝑐𝑐, 𝑑𝑑}

46. Merging and Sparsification Phase (cont.)
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
Sparsify or not according
to total description cost
{𝑎𝑎}
{𝑏𝑏}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}
C = {𝑐𝑐, 𝑑𝑑}

47. Merging and Sparsification Phase (cont.)
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
Sparsify or not according
to total description cost
{𝑎𝑎}
{𝑏𝑏}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}
C = {𝑐𝑐, 𝑑𝑑}

48. Merging and Sparsification Phase (cont.)
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase <<
 Further sparsification phase
Sparsify or not according
to total description cost
{𝑎𝑎}
{𝑏𝑏}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}
C = {𝑐𝑐, 𝑑𝑑}

49. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

50. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

51. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

52. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓} {ℎ}
{𝑔𝑔, 𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

53. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓} {ℎ}
{𝑔𝑔, 𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

54. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓} {ℎ}
{𝑔𝑔, 𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

55. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓} {𝑔𝑔, ℎ, 𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

56. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓} {𝑔𝑔, ℎ, 𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

57. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔, ℎ, 𝑖𝑖}
Summary graph �𝑮𝑮
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓} {𝑔𝑔, ℎ, 𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}

58. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑓𝑓}
{𝑔𝑔, ℎ, 𝑖𝑖}
Summary graph �𝑮𝑮
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓} {𝑔𝑔, ℎ, 𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}
{𝑎𝑎, 𝑒𝑒}

59. Repetition
Summary graph �𝑮𝑮
Introduction Algorithms Experiments ConclusionProblem
Summary graph �𝑮𝑮
Different
candidate sets
and decreasing
threshold 𝜃𝜃
over iteration
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝒕𝒕++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑓𝑓}
{𝑔𝑔, ℎ, 𝑖𝑖}
Summary graph �𝑮𝑮
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓} {𝑔𝑔, ℎ, 𝑖𝑖}
{𝑎𝑎}
{𝑏𝑏}
{𝑐𝑐, 𝑑𝑑}
{𝑒𝑒}
{𝑓𝑓}
{𝑔𝑔}
{ℎ}
{𝑖𝑖}
{𝑎𝑎, 𝑒𝑒}

60. Introduction Algorithms Experiments ConclusionProblem
Further Sparsification Phase
Summary graph �𝑮𝑮
𝐴𝐴 𝐴𝐴
𝐵𝐵 𝐴𝐴
𝐹𝐹 𝐴𝐴
𝐺𝐺 𝐺𝐺
Superedges sorted by ∆𝑅𝑅𝐸𝐸𝑝𝑝
𝐶𝐶 𝐶𝐶
Size of �𝑮𝑮 in bits ≤ 𝑲𝑲
Procedure
 Initialization phase
 𝑡𝑡 = 1
 While 𝑡𝑡++ ≤ 𝑻𝑻 and 𝑲𝑲 < size of �𝑮𝑮 in bits
 Candidate generation phase
 Merge and sparsification phase
 Further sparsification phase <<
C = {𝑐𝑐, 𝑑𝑑}
𝐴𝐴 = {𝑎𝑎, 𝑒𝑒}
𝐵𝐵 = {𝑏𝑏}
𝐹𝐹 = {𝑓𝑓}
𝐺𝐺 = {𝑔𝑔, ℎ, 𝑖𝑖}

61. Road Map
• Introduction
• Problem
• Proposed Algorithm: SSumM
• Experimental Results <<
• Conclusions

62. • 10 datasets from 6 domains (up to 0.8B edges)
• Three competitors for graph summarization
◦ k-Gs [LT10]
◦ S2L [RSB17]
◦ SAA-Gs [BAZK18]
Introduction Algorithms Experiments ConclusionProblem
Social Internet Email Co-purchase Collaboration Hyperlinks
Experiments Settings

63. Email-Enron Caida Ego-Facebook
Web-UK-05 Web-UK-02 LiveJournal
DBLP Amazon-0302Skitter
k-GsSSumM S2LSAA-Gs SAA-Gs (linear sample)
Introduction Algorithms Experiments ConclusionProblem
o.o.t. >12hours
o.o.m. >64GB
SSumM Gives Concise and Accurate Summary

64. Email-Enron Caida Ego-Facebook
Web-UK-05 Web-UK-02 LiveJournal
DBLP Amazon-0302Skitter
k-GsSSumM S2LSAA-Gs SAA-Gs (linear sample)
Introduction Algorithms Experiments ConclusionProblem
o.o.t. >12hours
o.o.m. >64GB
SSumM Gives Concise and Accurate Summary

65. Email-Enron Caida Ego-Facebook
Web-UK-05 Web-UK-02 LiveJournal
Amazon-0601 DBLPSkitter
k-GsSSumM S2LSAA-Gs SAA-Gs (linear sample)
Introduction Algorithms Experiments ConclusionProblem
SSumM is Fast

66. Email-Enron Caida Ego-Facebook
Web-UK-05 Web-UK-02 LiveJournal
Amazon-0601 DBLPSkitter
k-GsSSumM S2LSAA-Gs SAA-Gs (linear sample)
Introduction Algorithms Experiments ConclusionProblem
SSumM is Fast

67. Introduction Algorithms Experiments ConclusionProblem
SSumM is Scalable

68. Introduction Algorithms Experiments ConclusionProblem
SSumM Converges Fast

69. Road Map
• Introduction
• Problem
• Proposed Algorithm: SSumM
• Experimental Results
• Conclusions <<

70. Code available at https://github.com/KyuhanLee/SSumM
Concise & Accurate Fast Scalable
Introduction Algorithms Experiments ConclusionProblem
Practical Problem Formulation
Extensive Experiments on 10 real world graphs
Scalable and Effective Algorithm Design
Conclusions

71. SSumM : Sparse Summarization
of Massive Graphs
Kyuhan Lee* Hyeonsoo Jo* Jihoon Ko Sungsu Lim Kijung Shin