1. Mining in Graphs and Complex Structures01
Background 02
gSpan 03
Improvement
Presented by: Yi He 04
Application

2. 1 2 3 4 01 Background Why we need Graphs Mining
What is Frequent Subgraphs
Background 3
Approaches of Graphs Mining 4
Two Challenges

3. Background: Why Graph Mining?
Graphs are prevalent
Chemical compounds
Protein structures, biological networks
Program control flow, traffic flow, and workflow analysis
Social network analysis
Graph is a general model
Trees, lattices, sequences, and items are degenerated graphs
Diversity of graphs
Directed vs. undirected
Labeled vs. unlabeled (edges & vertices)
Weighted, with angles & geometry (topological vs. 2-D/3-D)
Complexity of algorithms:
many problems are of high complexity! 01
Background
Graph better than 1000 words
100 years of solitude by Garcia Markquez

4. 01 Background Aspirin Yeast protein interaction network Internet
Social network

5. Frequent subgraphs: How to find?
A (sub)graph is frequent if its support (occurrence frequency) in a given dataset is no less than a minimum support threshold (minsup). 01
Background
hydroxide-ion

6. Frequent Subgraph: Mining Approaches 01 Apriori-based approach
AGM/AcGM: Inokuchi, et al. (PKDD’00)
FSG: Kuramochi and Karypis (ICDM’01)
PATH: Vanetik and Gudes (ICDM’02, ICDM’04)
FFSM: Huan, et al. (ICDM’03)
Pattern growth-based approach
MoFa, Borgelt and Berthold (ICDM’02)
gSpan: Yan and Han (ICDM’02)
Gaston: Nijssen and Kok (KDD’04)
Background
Search order: apriori vs. pattern growth
Elimination of duplicate subgraphs: passive vs. active
We do simple job, mining in different graphs, find frequent transaction, rather than the whole structure

7. Frequent Subgraph: Challenges 01 Background
Pattern growth = generating longer and longer
How to order and generate efficiently by trunk some at early stage.
How to identify the duplicated ones
How to reduce exponential time cost induced by candidate generation.
How to reduce exponential time cost induced by isomorphism test.

8. Frequent subgraphs: Isomorphism Test01
Background
Kernel = NP-complete problem = no algorithm can solve it yet
Reference point X in graph (a)
Frequent subgraphs contains information

9. 02 Introduction DFS Code DFS Lexicographic Order VS Minimum DFS Code
Depth First Search
DFS Code Tree
& Rightmost Path

10. Depth First Search: 02
Introduction

11. 02 Introduction DFS Code DFS Lexicographic Order VS Minimum DFS Code
DFS Code Tree
& Rightmost Path
Depth First Search
& Rightmost Path

12. DFS Code: 02
Introduction

13. DFS code: Edge Order 02
Introduction

14. 02 Introduction DFS Code DFS Lexicographic Order VS Minimum DFS Code
DFS Code Tree
& Rightmost Path
Depth First Search
& Rightmost Path

15. DFS Lexicographic Ordering:
→ Different search starting point.
→Mutiple DFS Code sets 02
Introduction

16. α is a code set of at, β is a code set of bt.
DFS Lexicographic Ordering 02
Introduction
α is a code set of at, β is a code set of bt.
B=backward edges
F= forward edges

17. 02
Introduction

18. 02 Introduction DFS Code DFS Lexicographic Order VS Minimum DFS Code
DFS Code Tree
& Rightmost Path
Depth First Search
& Rightmost Path

19. Righmost Path for Extension02
When building DFS codes, must expand all back edges first!
Introduction
shortest path between V0 and Vn using forward edges

20. Righmost Path (completeness)? 02 X Y Z a b c
Introduction
shortest path between V0 and Vn using forward edges

21. gSpan Algorithm: 02 Introduction Mining Result Dataset
Since they are same, we select the one has minimum DFS.

22. DFS Code Tree 02 Introduction
Minsup = 3 02
Introduction
Since they are same, we select the one has priori consequencial DFS code .
Iff the min(G)=min(G’), G is isomorphic to G’by given two graphs G and G’.

23. Depth First Search: 02
Convert Graphs into Strings
Introduction

24. Experimental Result: 02 Introduction Size of frequent kernel I
Possible label N
D L fixed

25. 03
Improvement
gPrune
Closest
Path
gSpan
FSG
2001
2002
2003
2007

26. Closed Frequent Graphs:
Motivation: Handling graph pattern explosion problem.
Closed frequent graph:
A frequent graph G is closed if there exists no super-graph of G that carries the same support as G
If some of G’s subgraphs have the same support, it is unnecessary to output these subgraphs (non-closed graphs)
Lossless compression: still ensures that the mining result is complete
Improvement
Always have the same support for G and G’
Mined all the frequent graphs and cut them = not efficient
gSpan = 2^100 complexisity

27. CLOSEGRAPH (Yan & Han, KDD’03)
Improvement

28. CLOSEGRAPH:
Performance
Improvement

29. Constraint-Based Graph Pattern Mining
gPrune:
Constraint-Based Graph Pattern Mining
Improvement
Main Idea: Push constraints deeply into the mining process to reduces search space.
(Constraints can be classified into different categories,
Different categories require different pushing strategies)
There are often various kinds of constraints (with domain knowledge) specified for mining graph pattern P, e.g.,
max_degree(P) ≥ 10
diameter(P) ≥δ

30. Pattern Pruning vs. Data Pruning
gPrune:
Pattern Pruning vs. Data Pruning
Improvement
P Pruning = generate the sub pattern, if not satisfied and keep grow no way to satisfiy= cut
D Pruning = Divided the data set into small set = cut off the set have no way to satisfy the constrain

31. Pattern Pruning vs. Data Pruning
gPrune:
Pattern Pruning vs. Data Pruning
Improvement

32. Pruning Pattern Search Space
gPrune:
Pruning Pattern Search Space
Improvement
Density ratio = available / Possible edges
In big graph, at least one small one will meet the requirement of density ratio.

33. gPrune: Pruning Data Space Improvement
Separable = rest + exist < 10 then trunk
Why middle?
Find frequent part = useful part
=cut not useful componants=
Know not sufficient

34. gPrune: Pruning Data Space Improvement
Always look back to the dataset, as soon as constraint cannot be satisfied= cut

35. gPrune: Graph Constraints-- A General Picture
Improvement
Not efficient all the time, at least one knife is useful.

36. An traffic mining approach:
Application
Traffic old = Euclidean space distance
New= high crime, construction, traffic jam, high crash rate.
Suggest path contains as much as frequent path segment
*Road hierarchy-based partitioning
*Speed rule mining
Driving Pattern mining
Adaptive pre computation
Road upgrading
Adaptive fastest path algorithms

37. Exam Questions:
Q1) What Does Gspan Compare When Testing For Isomorphism Between Two Graphs, And Why?
Answer: Gspan Compares The Minimum DFS Codes Of The Two Graphs. Given Two Graphs G And G’, G Is Isomorphic To G’ If Min(g)=min(g’). This Theorem Allows For A Simple String Comparison Of More Complicated Graphs. If Two Nodes Contain The Same Graph But Different Minimum DFS Codes, We Can Prune The Sub-branch Of The Rightmost Of The Two Nodes. This Greatly Decreases The Problem Size.
Application

38. Q2) Which pattern Does The DFS Code Below Belong To?
Exam Questions:
Application
Q2) Which pattern Does The DFS Code Below Belong To?
Answer: tree (c)

39. Exam Questions:
Application
Q3) What is the main idea for closedGraph mining?
Answer: To reduce the search space, by finding the “early termination”. If G and G’ are frequent, G is an subgraph of G’. If in any part of the graph in the dataset where G occurs, G’ also occurs, that is G and G’ share the same support, then we stop here and do not need grow G, since none of G’s children will be closed except those of G’.

40. THANK YOU!
Presented by Yi He