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"SSumM: Sparse Summarization of Massive Graphs", KDD 2020

K
KyuhanLee4

The document discusses the challenges associated with analyzing massive graphs, emphasizing the computational costs due to their size. It introduces graph summarization as a solution, highlighting its advantages such as elasticity, analyzability, and compatibility with existing tools. Various graph compression techniques are mentioned, with a focus on how summarization can effectively reduce graph size while retaining crucial information.

Data & Analytics
Related topics:
Graph Mining • Data Mining Insights
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SSumM : Sparse Summarization of Massive Graphs Kyuhan Lee* Hyeonsoo Jo* Jihoon Ko Sungsu Lim Kijung Shin
Graphs are Everywhere Citation networksSubway networks Internet topologies Introduction Algorithms Experiments ConclusionProblem Designed by Freepik from Flaticon
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
Difficulties in Analyzing Massive graphs Computational cost (number of nodes & edges) Introduction Algorithms Experiments ConclusionProblem
Difficulties in Analyzing Massive graphs > Cannot fit Introduction Algorithms Experiments ConclusionProblem Input Graph
Solution: Graph Summarization > Introduction Algorithms Experiments ConclusionProblem Can fit Summary Graph
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
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
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
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
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
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
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
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
Road Map • Introduction • Problem << • Proposed Algorithm: SSumM • Experimental Results • Conclusions
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
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
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
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
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
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
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
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
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
Road Map • Introduction • Problem • Proposed Algorithm: SSumM << • Experimental Results • Conclusions
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
Main Idea: Combining Two Strategies Introduction Algorithms Experiments ConclusionProblem Node Grouping Sparsification
Main Idea: Combining Two Strategies Introduction Algorithms Experiments ConclusionProblem Node Grouping Sparsification
Main Idea: Combining Two Strategies Introduction Algorithms Experiments ConclusionProblem Node Grouping Sparsification
Main Idea: Combining Two Strategies Introduction Algorithms Experiments ConclusionProblem Node Grouping Sparsification
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?
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?
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?
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
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
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
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
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}
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}
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)
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)
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
Road Map • Introduction • Problem • Proposed Algorithm: SSumM • Experimental Results << • Conclusions
• 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
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
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
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
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
Introduction Algorithms Experiments ConclusionProblem SSumM is Scalable
Introduction Algorithms Experiments ConclusionProblem SSumM Converges Fast
Road Map • Introduction • Problem • Proposed Algorithm: SSumM • Experimental Results • Conclusions <<
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
SSumM : Sparse Summarization of Massive Graphs Kyuhan Lee* Hyeonsoo Jo* Jihoon Ko Sungsu Lim Kijung Shin

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"SSumM: Sparse Summarization of Massive Graphs", KDD 2020

  • 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