gSpan: Graph-based substructure pattern mining

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gSpan: Graph-based substructure pattern mining
Authors: Xifeng Yan and Jiawei Han Presented by: Colin Luther

Copyright note: This presentation was originally provided by Prof. Xifeng Yan upon request from a student. Citation: Xifeng Yan and Jiawei Han. gSpan: graph- based substructure pattern mining. In IEEE International Conference on Data Mining (ICDM), 2002

Background Problem Definition Authors Contribution
Outlines Background Problem Definition Authors Contribution Concepts behind gSpan Experimental Result Conclusion

Background Frequent Subgraph Mining is an extension to existing frequent pattern mining algorithms A major challenge is to count how many instances of a pattern are in the dataset Counting instances might be easy for sets, but subtle for graphs Recall the graph isomorphism problem

Two Isomorphic graph (a) and (b) with their mapping function (c)
Background G1=(V1,E1,L1) G2=(V2,E2,L2) f(V1.1) = V2.2 f(V1.2) = V2.5 f(V1.3) = V2.3 f(V1.4) = V2.4 f(V1.5) = V2.1 Y 5 V 3 X 3 1 W 1 V 4 W U 2 Y 2 U 5 4 X (a) (b) (c) Two Isomorphic graph (a) and (b) with their mapping function (c) Two graphs are isomorphic if one can find a mapping of nodes of the first graph to the second graph such that labels on nodes and edges are preserved.

Problem: Finding Frequent Subgraphs
Problem setting: similar to finding frequent itemsets for association rule discovery Input: Database of graph transactions Undirected simple graph (no multiples edges) Each graph transaction has labeled edges/vertices. Transactions may not be connected Minimum support thresholds Output: Frequent subgraphs that satisfy the support threshold, where each frequent subgraph is connected.

Finding Frequent Subgraphs
Xifeng Yan

Authors Contribution Representing graphs as strings (like TreeMiner)
No candidate generation! “It combines the growing and checking of frequent subgraphs into one procedure, thus accelerates the mining process.” Really fast, still a standard baseline system that most rivals compare their systems to.

Concepts behind gSpan The idea is to produces a Depth-First Search (DFS) codes for each edge in graphs Edges are sorted according to lexicographic order of codes Yan and Han proved that graph isomororphism can be tested for two graphs annotated with DFS codes Starting with small graph patterns containing 1-edge, patterns are expanded systemically by the DFS search Employ anti-monotonic property of graph frequency

Anti-Monotonicity of graph frequency
The frequency of a super-pattern is less than or equal to the frequency of a sub-pattern. Copyright SIGMOD’08

Lexicographic Ordering in Graph
It can tell us the order of two graphs. The design can help us build a similar hierarchy. The design should guarantee easy-growing from one level to the lower level and easy-rolling-up from low level to higher level. It may be difficult to have such design that no two nodes in this tree are same for graph case. It can tell us whether the graph has been discovered. And more, the most important, if a graph has been discovered, all its children nodes in the hierarchy must have been discovered.

Lexicographic Ordering in Graph
1-edge ... 2-edge ... ... ... ... 3-edge ... ... ...

DFS code and Minimum DFS code
Depth First Tree and Forward/Backward Edge Set

DFS code and Minimum DFS code
We use a 5-tuple (vi, vj, l(vi), l(vj), l(vi,vj)) to represent an edge. (it may be redudant, but much easier to understand.) Turn a graph into a sequence whose basic element is 5-tuple. Form the sequence in such an order: to extend one new node, add the forward edge that connect one node in the old graph with this new node. Add all backward edge that connect this new node to other nodes in the old graph repeat this procedure.

DFS code Y a e0: (0,1,x,y,a) X X a a a e2: (2,0,x,x,a) X b
e1: (1,2,y,x,b) Y d b Z d e5: (1,4,x,z,d) b e4: (3,1,x,y,b) Z X b c Z c e3: (2,3,x,z,c) Z v0 v3 v1 v4 v2

Minimum DFS code Each Graph may have lots of DFS code (why?):
one smallest lexicographic one is its Minimum DFS Code

Graph Parent and its Children
Given a DFS code c0=(e0,e1,…,en) if c1=(e0,e1,…,en,ex) if c0<c1, then c0 is c1’s parent, c1 is c0’s child. ? X Y Z a b c

DFS Code Tree 1-edge ... 2-edge ... ... ... ... 3-edge ... ... ...

Theorem 1. Given two graph G0 and G1, G0 is isomorphic to G1 iff min_dfs_code(G0)=min_dfs_code(G1). 2. DFS Code Tree covers all graphs although some tree nodes may represent the same graph 3. Given a node in DFS Code Tree, if its DFS code is not its minimum DFS code, prune this node and its all descendants won’t change. “Covering”.

Algorithm

Algorithm

Experimental Result

Experimental Result

Conclusion No Candidate Generation and False Test
Space Saving from Depth First Search Good Performance: using “memory Pool” and one major counting improvement, it seems the performance will be improved 5 times more. (but need more testing).

Exam Questions Q1) What two major costs from Apriori-like, frequent substructure mining algorithms did gSpan aim to reduce/avoid? Answer: 1) The creation of size k+1 candidate subgraphs from size k frequent subgraphs is more complicated and costly the standard Apriori large itemset generation. 2) Pruning false positives is an expensive process. Subgraph isomorphism problem is NP-Complete.

Exam Questions (cont.) Q2) Which DFS tree does the DFS code below belong to? Answer: tree (c)

Exam Questions Q3) 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.

Questions?