1. Frequent Subgraph Pattern Miningon 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 graphse.g. [Leskovec et al. KDD’05] Patterns of graphse.g., [Yan et al. ICDM’02] Uncertainties of graphs Privacy of graphse.g., [Zou et al. VLDB’09] Evolution of graphse.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 0.147 0.639 0.698 NTG1 SMT3 RAD59 RPC40 The data are taken from the STRING Database (http://string-db.org).

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. Outline • Background • Problem Definition • Algorithm • Experimental Results • Conclusions

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

9. 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?

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

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

12. A B B 0.8 0.5 A y x x y 0.1 0.8 x y 0.7 0.6 0.7 z B B B B Uncertain graph G1 Uncertain graph G2 exist in this form A B B A y x y x y B B B B Implicated graph of G1 Implicated graph of G2 Uncertain Graph Databases 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)) = 4.032 * 10-3

13. implicating implicating implicating …… d1 d2 dn Expected Support D uncertain graph DB 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 Sis

14. 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.

15. Outline • Background • Problem Definition • Algorithm • Experimental Results • Conclusions

16. 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 0 1 (1-e) minsup minsup

17. 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.

18. A B B 0.8 0.5 A y x x y 0.1 0.8 x y 0.7 0.6 0.7 z B B B B Uncertain graph G2 Uncertain graph G1 Discard Arbitrary Output expected support 0 1 (1-e) minsup minsup 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.

19. Discard Arbitrary Output expected support 0 1 (1-e) minsup minsup 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.

20. expected support 0 1 (1-e) minsup minsup Method for Objective II • Step 1: Approximate esup(S) by an interval [l, u] such that esup(S)∈[l, u]. • Step 2: Decide whether Scan be output or not by testing the following conditions. Output Discard Shrink

21. 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:

22. B B A 0.8 0.5 A y x (x1) (x4) x y x y (x2) (x3) 0.6 0.7 B B B B pattern S uncertain graph Gi Approximate Pr(S occurs in Gi) by [li, ui] Step 1: Find all embeddings of S in Gi. 4 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].

23. Outline • Background • Problem Definition • Algorithm • Experimental Results • Conclusions

24. Experimental Results • Data • The STRING Database (http://string-db.org)

25. Time Efficiency

26. Approximation Quality

27. Scalability

28. 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.

29. Thank you