1. Frequent Subgraph Pattern Mining on Uncertain Graph Data
Zhaonian Zou, Jianzhong Li, Hong Gao, Shuo Zhang
Harbin Institute of Technology, China
CIKM’09, Hong Kong
Nov 4, 2009

2. Outline Background Problem Definition Algorithm Experimental Results
Conclusions

3. Background
Graph mining has played an important role in a range of real world applications.
medicines: structures of molecules
bioinformatics: biological networks
technologies: WWW
social science: social networks
many others

4. Directions of Graph Mining
Models of graphs e.g. [Leskovec et al. KDD’05]
Patterns of graphs e.g., [Yan et al. ICDM’02]
Uncertainties of graphs
Privacy of graphs e.g., [Zou et al. VLDB’09]
Evolution of graphs e.g., [Faloutsos et al. SIGMOD’07]

5. Uncertainties of Graphs: Example I
Protein-Protein Interaction (PPI) Networks
Vertices: proteins
Edges: interactions between proteins
Uncertainties: probabilities of interactions really existing
TIF34
0.375
0.639
0.867
0.651
0.651
FET3
0.698
0.147
0.639
NTG1
SMT3
RAD59
RPC40
The data are taken from the STRING Database (

6. Uncertainties of Graphs: Example II
Topologies of wireless sensor networks (WSNs)
Vertices: sensor nodes
Edges: wireless links between sensor nodes
Uncertainties: probabilities of wireless links functioning at any given time
0.75
0.95
0.88
0.92
0.69

7. The Goal of This Paper Models of graphs e.g. [Leskovec et al. KDD’05]
Patterns of graphs e.g., [Yan et al. ICDM’02]
Uncertainties of graphs
Privacy of graphs e.g., [Zou et al. VLDB’09]
Evolution of graphs e.g., [Faloutsos et al. SIGMOD’07]

8. Outline Background Problem Definition Algorithm Experimental Results
Conclusions

9. Preliminaries B A x y z graph G2 B A x y graph G1 Graph Database B x y
Subgraph Pattern
support = 1.0
support = 0.5
The support of S = the number of graphs containing S the total number of graphs

10. Frequent Subgraph Pattern Mining Problem
Input: a graph database D, and a support threshold minsup
Output: all subgraph patterns with support no less than minsup
FSP mining on biological networks (e.g., PPI networks) is an important tool for discovering functional modules [Koyutürk et al. Bioinformatics 04, Turanalp et al. BMC Bioinformatics 08].
PPI networks are subject to uncertainties.
How do we define support?

11. Model of Uncertain GraphsB A x y
exist in this form
Implicated Graph B A x y
0.5
0.6
0.7
0.8
(1 – 0.5) * 0.6 * 0.7 * 0.8 = 0.168 B A x y
exist in this form
Uncertain Graph
0.5 * (1 – 0.6) * 0.7 * 0.8 = 0.112

12. Model of Uncertain Graphs (Cont’d)
Theorem: An uncertain graph represents a probability distribution over all its implicated graphs.

13. Uncertain Graph Databasesx y
0.5
0.6
0.7
0.8 z
0.1
Uncertain graph G1
Uncertain graph G2 B A x y
exist in this form
Implicated graph of G1
Implicated graph of G2
Theorem: An uncertain graph DB represents a probability distribution over all its implicated graph DBs.
Totally, 24 * 23 = 128 implicated graph databases.
Implicated Graph Database
((1 – 0.5) * 0.6 * 0.7 * 0.8) * (0.8 * 0.1 * (1 – 0.7)) = * 10-3

14. …… Expected Support D uncertain graph DB d1 d2 dn implicating
p1 = Pr(D implicates d1)
p2 = Pr(D implicates d2)
pn = Pr(D implicates dn)
s1 = support of S in d1
s2 = support of S in d2
sn = support of S in dn
The expected support of S is

15. FSP Mining Problem on Uncertain Graphs
Input: an uncertain graph database D, and an expected support threshold minsup
Output: all subgraph patterns with expected support no less than minsup
It is #P-hard to count the number of frequent subgraph patterns.
Reduction from the problem of counting the number of satisfying truth assignments of a monotone k-CNF formula.
The FSP mining problem on uncertain graphs is NP-hard.

16. Outline Background Problem Definition Algorithm Experimental Results
Conclusions

17. Approximation Method
It is #P-hard to compute the expected support of a subgraph pattern.
We develop an approximation method to find an approximate set of frequent subgraph patterns.
Let e (0 < e < 1) be a relative error tolerance.
Discard
Arbitrary
Output
expected support
(1-e) minsup
minsup 1

18. Objective I
Difficulty I: # of frequent subgraph patterns is exponentially large.
Objective I: Examine subgraph patterns as efficiently as possible to find all frequent ones.

19. Method for Objectives I
Step 1: Build a search tree T of subgraph patterns.
Step 2: Examine subgraph patterns in T in depth-first order
If S is infrequent, then all its descendents can be pruned. B A x y
0.5
0.6
0.7
0.8 z
0.1
Uncertain graph G1
Uncertain graph G2
expected support
minsup
(1-e) minsup
Output
Discard
Arbitrary 1

20. Objective II
Difficulty II: It is #P-hard to compute the expected support esup(S) of a subgraph pattern S.
Objective II: Make the following judgments without computing esup(S) exactly.
If esup(S) is surely not in the green region, then discard.
If esup(S) is probable to be in the green region and surely not in the red region, then output.
expected support
minsup
(1-e) minsup
Output
Discard
Arbitrary 1

21. Method for Objective II
Step 1: Approximate esup(S) by an interval [l, u] such that esup(S)∈[l, u].
Step 2: Decide whether S can be output or not by testing the following conditions.
expected support
minsup
(1-e) minsup 1
Output
Discard
Shrink

22. Approximating esup(S) by [l,u]
A subgraph pattern S occurs in an uncertain graph G if S is contained in at least one implicated graph of G.
Algorithm Approximate esup(S) by [l,u]
Step 1: For each uncertain graph Gi in D, approximate Pr(S occurs in Gi) by an interval [li, ui] of width at most
e*minsup.
Step 2:

23. Approximate Pr(S occurs in Gi) by [li, ui]
0.5
0.6
0.7
0.8
uncertain graph Gi
pattern S
(x1)
(x2)
(x4)
(x3)
Step 1: Find all embeddings of S in Gi embeddings
Step 2: Assign boolean variables to the edges in the embeddings. Pr(x1) = 0.5, Pr(x2) = 0.6, Pr(x3) = 0.7, Pr(x4) = 0.8.
Step 3: Construct a conjunctive formula for each embedding. C1 = (x1 ^ x2), C2 = (x1 ^ x4), C3 = (x2 ^ x3), C4 = (x3 ^ x4).
Step 4: Construct a DNF formula. F = C1 V C2 V C3 V C4.
Step 5: Estimate Pr(F = TRUE) by p using Karp & Luby’s Markov-Chain Monte-Carlo method with absolute error e*minsup/2 and confidence d (d ∈[0,1]).
Step 6: [li, ui] = [p - e*minsup/2, p + e*minsup/2].

24. Outline Background Problem Definition Algorithm Experimental Results
Conclusions

25. Experimental Results
Data
The STRING Database (

26. Time Efficiency

27. Approximation Quality

28. Scalability

29. Conclusions A new model of uncertain graph data has been proposed.
The frequent subgraph pattern mining problem on uncertain graph data has been formalized.
The computational complexity of the problem has been formally proved to be NP-hard.
An approximate mining algorithm has been proposed.
The proposed algorithm has high efficiency, high approximation quality, and high scalability.

30. Thank you