1. Simulation based approach Shang Zechao 1010161920
Graph Matching
Simulation based approach
Shang Zechao

2. Introduction What is graph matching?
When the one graph matches with another?

3. Introduction (cont.) Graph: G=(V, E). GQ = (VQ, EQ)
Can be easily extended with labels.
Exact matching: isomorphism
Find a bijection function f between V and VQ
(u, v) in E iff (f(u), f(v)) in EQ

4. Introduction (cont.) Graph isomorphism Sub-graph isomorphism Too hard!
GI class
Sub-graph isomorphism
NP-Complete
Too hard!

5. Simulation based approach [Henzinger95]
Find a relation S: V x VQ
(u, u’) in S if
u and u’ has same labels
for all children v’ of u’, there exists v
V is child of u
(v, v’) in S

6. Simulation based approach
The major difference between graph simulation and graph isomorphism
Isomorphism requires an bijection (one to one) function
Graph simulation based on relation (many to many)
Simulation is in polynomial time

7. An Example [Fan10] Drug dealer network B: Boss S: Secretary
AM: Assistant manager
FW: Field worker

8. An Example (cont.) In real world S and AM is same
AM maps to multiple worker

9. Bounded Simulation [Fan10]
Each edge in pattern graph has label
Either a positive integer K
Or * (infinite)
The length of path connects these two nodes

10. The Example (cont.)
AM should be able to reach FW within 3 hops.

11. Matching Algorithm
Similar with the EffcientSimilarity algorithm in [Henzinger95].
Pre-compute the distance matrix between all pairs of node in G.
Complexity O(|V||E| + |Ep||V|2 + |Vp||V|)

12. Strong Simulation [Ma12]
Recall the condition that two nodes match:
Have same label
Children could be matched by simulation
Two issues
Parent information is not captured
Matching size is not limited

13. An Example [Ma12] Bio can match to Bio1, Bio2, Bio3, Bio4
Actually only Bio4 makes sense

14. Strong Simulation two nodes match if:
Have same label
Children could be matched by simulation
Parent could be matched by simulation
The matched sub-graph should have same diameter as pattern graph

15. An Example (cont.)
Bio only matches to Bio4 in strong simulation

16. Comparison of different approaches
children topology
parents topology
connectivity
cycle info
simulation Y N
with parent topology
with diameter constrain
isomorphism

17. Comparison of different approaches
locality
bounded matches
bisimulation
bounded cycle
simulation N Y
with parent topology
with diameter constrain
isomorphism

18. But Bounded cycle problem is intractable
NP-hard
Bisimilar problem is intractable
coNP-hard

19. References
[Henzinger95] M. R. Henzinger, T. A. Henzinger, and P. W. Kopke Computing simulations on finite and infinite graphs. In Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS '95). IEEE Computer Society, Washington, DC, USA, 453-.
[Fan10] Wenfei Fan, Jianzhong Li, Shuai Ma, Nan Tang, Yinghui Wu, and Yunpeng Wu Graph pattern matching: from intractable to polynomial time. Proc. VLDB Endow. 3, 1-2 (September 2010),
[Ma12] Shuai Ma, Yang Cao, Wenfei Fan, Jinpeng Huai , Tianyu Wo Capturing Topology in Graph Pattern Matching. PVLDB. To appear.