1. NED: An Inter-Graph Node Metric Based On Edit Distance
Haohan Zhu (Facebook, Inc)
Xianrui Meng (Apple, Inc)
George Kollios (Boston University)

2. Node Similarity Applications
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Node Similarity Applications
Transfer Learning
Communication Networks
Protein–Protein Interaction Networks (PPI)
Graph De-anonymization
Social Networks
Label-Free
Inter-Graph
Metric
From
From S. Rajagopala, et al., The binary protein-protein interaction landscape of Escherichia coli, Nature Biotechnology 32, 285–290 (2014)
From

3. Existing Node Similarity Measures
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Existing Node Similarity Measures

4. NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Node Similarity: NED
Use the neighborhood topological structure to describe a node.
Use Neighborhood Trees rather than Sub-Graphs
Computability - Let 𝛿(𝑢, 𝑣) be a distance function between a pair of nodes 𝑢 and 𝑣. The neighborhood subgraphs of 𝑢 and 𝑣 are isomorphic, iff 𝛿(𝑢, 𝑣) = 0. Then 𝛿 is not polynomial-computable. (As hard as graph isomorphism)
Effectiveness - Non-identity is not good for top-k similar nodes ranking and k-nearest-neighbor similar nodes finding.

5. 𝛿 𝑁𝐸𝐷 (𝑢,𝑣) = 𝛿 𝑇𝐸𝐷 ∗ (𝑇(𝑢,𝑘),𝑇(𝑣,𝑘))
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Tree Edit Distance
The similarity between two nodes is measured by the edit distance between neighborhood adjacent-trees
𝛿 𝑁𝐸𝐷 (𝑢,𝑣) = 𝛿 𝑇𝐸𝐷 ∗ (𝑇(𝑢,𝑘),𝑇(𝑣,𝑘))
The computation of tree edit distance on unordered unlabeled trees belongs to NP-Complete, even MaxSNP-Hard.

6. Modified Tree Edit Distance: TED*
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Modified Tree Edit Distance: TED*
Modified Tree Edit Distance:
No operation can change the depth of any existing node in the tree
Edit Operation in TED* :
Insert a leaf node
Delete a leaf node
Move a node at the same level
Advantage of TED* ?
Metric: The same as TED
Efficient: Polynomial time computable

7. Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆
Iterative Bottom-Up Algorithm

8. Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L
Level 3 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Padding
Padding Lv 3 = 1

9. Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L
Level 3 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Canonization

10. Bipartite Graph Construction
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L
Level 3 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Bipartite Graph Construction

11. Bipartite Graph Matching
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L
Level 3 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Bipartite Graph Matching
Matching Lv 3 = 0

12. Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L
Level 3 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Re-Canonization

13. Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F
Level 2 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Padding
Padding Lv 2 = 0

14. Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D 2 3 E 1 F 𝛿𝛿 2 2 𝜀 𝜁𝜁 1 G H I J K
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D 2 3 E 1 F
Level 2 𝛿𝛿 2 2 𝜀 𝜁𝜁 1 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Canonization

15. Bipartite Graph Construction
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D 2 3 E 1 F
Level 2 2 𝛿𝛿 𝜀 2 𝜁𝜁 1 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Bipartite Graph Construction D 𝛿𝛿 1 1 E 𝜀 1 1 1 F 2 𝜁𝜁

16. Bipartite Graph Matching
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D 2 3 E 1 F
Level 2 𝛿𝛿 2 𝜀 2 1 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Bipartite Graph Matching
Matching Lv 2 = 0 D 𝛿𝛿 1 1 E 𝜀 1 1 1 F 2 𝜁𝜁

17. Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D 2 3 E 1 F 𝛿𝛿 2 2 𝜀 3 𝜁𝜁 1 G H I J
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D 2 3 E 1 F
Level 2 𝛿𝛿 2 2 𝜀 3 𝜁𝜁 1 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Re-Canonization

18. Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D 2 3 E 1 F 𝛿𝛿 2 𝜀 3 𝜁𝜁 1 G H I J K
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C
Level 1 𝛽𝛽 𝛾𝛾 D 2 3 E 1 F 𝛿𝛿 2 𝜀 3 𝜁𝜁 1 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Padding
Padding Lv 1 = 0

19. Computation of TED* A 𝛼𝛼 B 5 6 C 𝛽𝛽 7 𝛾𝛾 4 D 2 3 E 1 F 𝛿𝛿 2 3 𝜀 1 𝜁𝜁 G
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B 5 6 C
Level 1 𝛽𝛽 7 𝛾𝛾 4 D 2 3 E 1 F 𝛿𝛿 2 3 𝜀 1 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Canonization

20. Bipartite Graph Construction
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 5 B 6 C
Level 1 7 𝛽𝛽 𝛾𝛾 4 D 2 3 E 1 F 𝛿𝛿 2 3 𝜀 1 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Bipartite Graph Construction B 𝛽𝛽 1 2 2 C 𝛾𝛾 1

21. Bipartite Graph Matching
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 5 B 6 C
Level 1 7 𝛽𝛽 𝛾𝛾 4 D 2 3 E 1 F 𝛿𝛿 2 3 𝜀 1 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Bipartite Graph Matching
Matching Lv 1 = 1 B 𝛽𝛽 1 2 2 C 𝛾𝛾 1

22. Computation of TED* A 𝛼𝛼 7 B 5 6 4 C 7 𝛽𝛽 𝛾𝛾 4 D 2 3 E 1 F 2 𝛿𝛿 3 𝜀 1
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 7 B 5 6 4 C
Level 1 7 𝛽𝛽 𝛾𝛾 4 D 2 3 E 1 F 2 𝛿𝛿 3 𝜀 1 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Re-Canonization

23. Computation of TED* A 𝛼𝛼 B 7 4 C 7 𝛽𝛽 𝛾𝛾 4 D 2 3 E 1 F 𝛿𝛿 2 3 𝜀 𝜁𝜁 1 G
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED*
Level 0 A 𝛼𝛼 B 7 4 C 7 𝛽𝛽 𝛾𝛾 4 D 2 3 E 1 F 𝛿𝛿 2 3 𝜀 𝜁𝜁 1 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆 𝜇𝜇
Padding
Padding Lv 0 = 0

24. 𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑢,𝑘),𝑇(𝑣,𝑘) = 𝑖=0 𝑘 𝑃 𝑖 + 𝑀 𝑖
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED* A 𝛼𝛼 B C 𝛽𝛽 𝛾𝛾 D E F 𝛿𝛿 𝜀 𝜁𝜁 G H I J K L 𝜂𝜂 𝜃𝜃 𝜄 𝜅𝜅 𝜆𝜆
𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑢,𝑘),𝑇(𝑣,𝑘) = 𝑖=0 𝑘 𝑃 𝑖 + 𝑀 𝑖
𝛿 𝑇𝐸𝐷 ∗ 𝑇 𝐴,3 , 𝑇(𝛼, 3) =2

25. 𝑚( 𝐺 𝑖 2 )= 𝑀𝑖𝑛 ∀𝑥𝜖 𝐿 𝑖 (𝑢) 𝑤(𝑥, 𝑓 𝑖 (𝑥))
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
Computation of TED*
Canonization
Padding
𝑃 𝑖 = 𝐿 𝑖 (𝑢) − 𝐿 𝑖 (𝑣)
Matching
𝑀 𝑖 =(𝑚( 𝐺 𝑖 2 )− 𝑃 𝑖+1 )/2
Bipartite Graph
Bipartite Matching
𝑚( 𝐺 𝑖 2 )= 𝑀𝑖𝑛 ∀𝑥𝜖 𝐿 𝑖 (𝑢) 𝑤(𝑥, 𝑓 𝑖 (𝑥))

26. NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
TED* vs TED & GED

27. TED* is a metric, so is NED
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
TED* is a metric, so is NED
TED* is a metric for trees
𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑢,𝑘),𝑇(𝑣,𝑘) = 𝑖=0 𝑘 𝑃 𝑖 + 𝑀 𝑖 = 𝑖=1 𝑘 𝑃 𝑖 + 𝑖=0 𝑘−1 𝑚( 𝐺 𝑖 2 )/2
Metric properties:
𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑢,𝑘),𝑇(𝑣,𝑘) ≥0
𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑢,𝑘),𝑇(𝑣,𝑘) = 𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑣,𝑘),𝑇(𝑢,𝑘)
𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑢,𝑘),𝑇(𝑣,𝑘) =0, iff 𝑇 𝑢,𝑘 ⋍𝑇(𝑣,𝑘)
𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑢,𝑘),𝑇(𝑣,𝑘) ≤ 𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑢,𝑘),𝑇(𝑤,𝑘) + 𝛿 𝑇𝐸𝐷 ∗ 𝑇(𝑤,𝑘),𝑇(𝑣,𝑘)
NED is a metric for inter-graph nodes

28. TED* vs TED on Efficiency
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
TED* vs TED on Efficiency

29. NED Case Study: De-anonymization
NED: An Inter-Graph Node Metric Based On Edit Distance
08/30/2017
NED Case Study: De-anonymization

30. Thank You !