Mining Frequent Subgraphs

COMP 790-90 Seminar

Spring 2007

OverviewIntroduction

Finding recurring subgraphs from graph databases.

gSpan

FFSM

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1L06

Labeled GraphWe define a labeled graph G as a five element tuple G = {V, E, V, E, } where

V is the set of vertices of G,

E V V is a set of undirected edges of G,

V (E) are set of vertex (edge) labels,

is the labeling function: V V and E E that maps vertices and edges to their labels.

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Frequent Subgraph Mining

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Input: A set GD of labeled undirected graphs

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Output: All frequent subgraphs (w. r. t. ) from GD.

Finding Frequent Subgraphs

Given a graph database GD = {G0,G1,…,Gn}, find all subgraphs appearing in at least graphs.

Isomorphic subgraphs are considered the same subgraph.

Apriori approachesGeneration of subgraph candidates is complicated and expensive.

Subgraph isomorphism is an NP-complete problem, so pruning is expensive.

gSpan

DFS without candidate generationRelabels graph representation to support DFS.

Discovers all frequent subgraphs without candidate generation or pruning.

DFS RepresentationMap each graph to a DFS code (sequence).

Lexicographically order the codes.

Construct a search tree based on the lexicographic order.

(a) (b) (c) (d)

Depth-First Search Tree

(a) (b)

(c) (d)

DFS Codes

Given ei = (i1,j1), e2 = (i2,j2): e1 < e2 if:

~i1 = i2 && j1 < j2

~i1 < j1 && j1 = i2

code(G,T) = edge sequence of ei < ei+1

edge (b) (c) (d)

0 (0,1,X,a,Y) (0,1,Y,a,X) (0,1,X,a,X)

1 (1,2,Y,b,X) (1,2,X,a,X) (1,2,X,a,Y)

2 (2,0,X,a,X) (2,0,X,b,Y) (2,0,Y,b,X)

3 (2,3,X,c,Z) (2,3,X,c,Z) (2,3,Y,b,Z)

4 (3,1,Z,b,Y) (3,0,Z,b,Y) (3,0,Z,c,X)

5 (1,4,Y,d,Z) (0,4,Y,d,Z) (2,4,Y,d,Z)

DFS Lexicographic Order

∂ = code(G∂,T∂) = (a0,a1,…,am)

ß = code(Gß,Tß) = (b0,b1,…,bn)

∂ ≤ ß iff (1) or (2):(1)(2)Minimum DFS code

The minimum DFS code min(G), in DFS lexicographic order, is the canonical label of graph G.

Graphs A and B are isomorphic if min(A) = min(B).

ak bk for 0 k m, n m

t, 0 t min m,n , ak bk for k t, at e bt

DFS Codes: Parents and Children

If ∂ = (a0,a1,…,am) and ß = (a0,a1,…,am,b):

ß is the child of ∂.

∂ is the parent of ß.

A valid DFS code requires that b grows from a vertex on the rightmost path.

DFS Code TreesOrganize DFS code nodes as parent-child.Pre-order traversal follows DFS lexicographic order.If s and s’ are the same graph with different DFS codes, s’ is not the minimum and can be pruned.

gSpanD is the set of all graphs.S is the result set.

Algorithm 1: GraphSet_Projection(D,S)1: sort labels in D by frequency2: remove infrequent vertices and edges3: relabel remaining vertices and edges4: S’ = all frequent 1-edge graphs in D5: sort S’ in DFS lexicographic order6: S = S’7: foreach edge e in S’ do8: s = graph defined by e9: s.D = subgraphs in D containing e10: Subgraph_Mining(D,S,s)11: D = D - e12: if |D| < minSup13: break

Subprocedure 1: Subgraph_Mining(D,S,s)1: if s != min(s)2: return3: S = S U {s}4: s’ = +1-edge children of s in s.D5: foreach child c of s’ do6: if support(c) ≥ minSup7:

Subgraph_Mining(Ds,S,c)

Runtime: SyntheticR

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Runtime: Chemical

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Support Threshold (%)

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Apriori (FSG) gSpan

gSpan AdvantagesLower memory requirements.Faster than naïve FSG by an order of magnitude.No candidate generation.Lexicographic ordering minimizes search tree.False positives pruning.

Any disadvantage?

FFSM: Fast Frequent Subgraph Mining -- An Overview:

How to solve graph isomorphism problem?

A Novel Graph Canonical Form: CAM

How to tackle subgraph isomorphism problem (NP-complete)?

Incrementally maintained embeddings

How to enumerate subgraphs:An Efficient Data Structure: CAM Tree

Two Operations: CAM-join, CAM-extension.

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Adjacency Matrix Every diagonal entry of adjacency matrix M corresponds to a distinct vertex in G and is filled with the label of this vertex. Every off-diagonal entry in the lower triangle

part of M1 corresponds to a pair of vertices in G and is filled with the label of the edge between the two vertices and zero if there is no edge.

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1for an undirected graph, the upper triangle is always a mirror of the lower triangle

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CodeA Code of n n adjacency matrix M is defined as sequence of lower triangular entries (including the diagonal entries) in the order:M1,1 M2,1 M2,2 … Mn,1 Mn,2 …Mn,n-1 Mn,n

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Code(M1): aybyxb0y0c00y0d > Code(M2): aybyxb00yd0y00c > Code(M3): bxby0d0y0cyy00aM2

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The Canonical Adjacency Matrix is the one produces the maximal code, using lexicographic order.

MP SubmatrixFor an m m matrix A, an n n matrix B is A’s maximal proper submatrix (MP Submatrix), iff N is obtained by removing the last none-zero entry from M.

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We define a CAM is connected iff the corresponding graph is connected. Theorem I: A CAM’s MP submatrix is CAMTheorem II: A connected CAM’s MP submatrix is connected

CAM Tree: Subgraphs

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CAM Tree: Frequent Subgraphs

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How to Enumerate Nodes in a CAM Tree?

Two operations to explore CAM tree:CAM-Join

CAM-Extension

Augmenting CAM tree with Suboptimal CAMsObjectives:

none false dismissal

no redundancy

Plus: We want to this efficiently!

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Suboptimal Tree

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c dWe define a Suboptimal CAM as a matrix that its MP submatrix is a CAM.

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SummaryTheorem:For a graph G, let CK-1 (Ck) be set of the suboptimal CAMs of all the size (K-1) (K) subgraphs of G (K ≥ 2). Every member of set CK can be enumerated unambiguously either by joining two members of set CK-1 or by extending a member in CK-1.

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Experimental StudyPredictive Toxicology Evaluation Competition (PTE)

Contains: 337 compounds

Each graph contains 27 nodes and 27 edges on average

NIH DTP Anti-Viral Screen Test (DTP CA/CM)Chemicals are classified to be Confirmed Active (CA), Confirmed Moderate Active (CM) and Confirmed Inactive (CI).

We formed a dataset contains CA (423) and CM (1083).

Each graph contains 25 nodes and 27 edges on average

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Performance (PTE)

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Performance (DTP CACM)

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Support Threshold (%) Support Threshold (%)