Data Mining-graph mining

1. Graph Mining
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2. Graph Mining
 Graphs
 Model sophisticated structures and their interactions
 Chemical Informatics
 Bioinformatics
 Computer Vision
 Video Indexing
 Text Retrieval
 Web Analysis
 Social Networks
 Mining frequent sub-graph patterns
 Characterization, Discrimination, Classification and Cluster
Analysis, building graph indices and similarity search
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3. Mining Frequent Subgraphs
 Graph g
 Vertex Set – V(g)
 Edge set – E(g)
 Label function maps a vertex / edge to a label
 Graph g is a sub-graph of another graph g’ if there exists a graph iso-
morphism from g to g’
 Support(g) or frequency(g) – number of graphs in D = {G1, G2,..Gn} where
g is a sub-graph
 Frequent graph – satisfies min_sup
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4. Discovery of Frequent Substructures
 Step 1: Generate frequent sub-structure candidates
 Step 2: Check for frequency of each candidate
 Involves sub-graph isomorphism test which is computationally
expensive
 Approaches
 Apriori –based approach
 Pattern Growth approach
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5. Apriori based Approach
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Start with graph of small size –
generate candidates with extra
vertex/edge or path
AprioriGraph
• Level wise mining method
• Size of new substructures is
increased by 1
• Generated by joining two similar
but slightly different frequent sub-
graphs
• Frequency is then checked
Candidate generation in graphs
is complex

6. Apriori Approach
 AGM (Apriori-based Graph Mining)
 Vertex based candidate generation – increases sub structure size by one
vertex at each step
 Two frequent k size graphs are joined only if they have the same (k-1)
subgraph (Size – number of vertices)
 New candidate has (k-1) sized component and the additional two
vertices
 Two different sub-structures can be formed
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7. Apriori Approach
 FSG (Frequent Sub-graph mining)
 Edge-based Candidate generation – increases by one-edge at a
time
 Two size k patterns are merged iff they share the same subgraph
having k-1 edges (core)
 New candidate – has core and the two additional edges
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8. Apriori Approach
 Edge disjoint path method
 Classify graphs by number of disjoint paths they have
 Two paths are edge-disjoint if they do not share any common edge
 A substructure pattern with k+1 disjoint paths is generated by joining
sub-structures with k disjoint paths
 Disadvantage of Apriori Approaches
 Overhead when joining two sub-structures
 Uses BFS strategy : level-wise candidate generation
 To check whether a k+1 graph is frequent – it must check all of its size-k sub graphs
 May consume more memory
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9. Pattern-Growth Approach
 Uses BFS as well as DFS
 A graph g can be extended by adding a new edge e. The newly
formed graph is denoted by g ♦x e.
 Edge e may or may not introduce a new vertex to g.
 If e introduces a new vertex, the new graph is denoted by g ♦xf e,
otherwise, g ♦xb e, where f or b indicates that the extension is in a forward
or backward direction.
 Pattern Growth Approach
 For each discovered graph g performs extensions recursively until all
frequent graphs with g are found
 Simple but inefficient
 Same graph is discovered multiple times – duplicate graph
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10. Pattern Growth
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11. gSpan Algorithm
 Reduces generation of duplicate graphs
 Does not extend duplicate graphs
 Uses Depth First Order
 A graph may have several DFS-trees
 Visiting order of vertices forms a linear order - Subscript
 In a DFS tree – starting vertex – root; last visited vertex – right-most vertex
 Path from v0 to vn – right most path
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Right most path: (b), (c) – (v0, v1, v3); (d) – (v0, v1, v2, v3)

12. gSpan Algorithm
 gSpan restricts the extension method
 A new edge e can be added
 between the right-most vertex and another vertex on the right-most path (backward
extension);
 or it can introduce a new vertex and connect to a vertex on the right-most path (forward
extension)
 Right-most extension, denoted by G ♦r e
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13. gSpan Algorithm
 Chooses any one DFS tree – base subscripting and
extends it
 Each subscripted graph is transformed into an edge sequence –
DFS code
 Select the subscript that generates minimum sequence
 Edge Order – maps edges in a subscripted graph into a sequence
 Sequence Order – builds an order among edge sequences
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Introduce backward edges:
Given a vertex v all of its backward edges should appear before
its forward edges (if any); If there are two backward edges (i,j1)
appears before (i,j2)
Order of forward edges: (0,1) (1,2) (1,3)
Complete sequence: (0,1) (1,2) (2,0) (1,3)

14. gSpan Algorithm
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DFS Lexicographic Ordering: Edge order, First Vertex label, Edge label, Second Vertex label
Here γ0 < γ1 < γ2
γ0 – Minimum DFS Code
Corresponding subscript – Base
Subscripting
gSpan – carries out right most
extension on the minimum
DFS code
gSpan – carries out right most
extension on the minimum
DFS code

15. gSpan Algorithm
 Root – Empty code
 Each node is a DFS code encoding a graph
 Each edge – rightmost extension from a (k-1) length DFS code to a
k-length DFS code
 If codes s and s’ encode the same graph – search space s’ can be safely
pruned
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16. gSpan Algorithm
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17. Mining Closed Frequent Substructures
 Helps to overcome the problem of pattern explosion
 A frequent graph G is closed if and only if there is no proper super graph G0
that has the same support as G.
 Closegraph Algorithm
 A frequent pattern G is maximal if and only if there is no frequent super-
pattern of G.
 Maximal pattern set is a subset of the closed pattern set.
 But cannot be used to reconstruct entire set of frequent patterns
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18. Mining Alternative Substructure Patterns
 Mining unlabeled or partially labeled graphs
 New empty label φ is assigned to vertices and edges that do not have labels
 Mining non-simple graphs
 A non simple graph may have a self-loop and multiple edges
 growing order - backward edges, self-loops, and forward edges
 To handle multiple edges - allow sharing of the same vertices in two neighboring
edges in a DFS code
 Mining directed graphs
 6-tuple (i; j; d; li; l(i; j) ; lj ); d = +1 / -1
 Mining disconnected graphs
 Graph / Pattern may be disconnected
 Disconnected Graph – Add virtual vertex
 Disconnected graph pattern – set of connected graphs
 Mining frequent subtrees
 Tree – Degenerate graph
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19. Constraint based Mining of Substructure
Patterns
 Element, set, or subgraph containment constraint
 user requires that the mined patterns contain a particular set of
subgraphs - Succinct constraint
 Geometric constraint
 A geometric constraint can be that the angle between each pair of
connected edges must be within a range – Anti-monotonic constraint
 Value-sum constraint
 the sum_of (positive) weights on the edges, must be within a range low
and high – (sum > low) Monotonic / Anti-monotonic (sum < high)
 Multiple categories of constraints may also be enforced
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20. Mining Approximate Frequent Substructures
 Approximate frequent substructures allow slight structural variations
 Several slightly different frequent substructures can be represented
using one approximate substructure
 SUBDUE – Substructure discovery system
 based on the Minimum Description Length (MDL) principle
 adopts a constrained beam search
 SUBDUE performs approximate matching
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21. Mining Coherent and Dense Sub structures
 A frequent substructure G is a coherent sub graph if the mutual information
between G and each of its own sub graphs is above some threshold
 Reduces number of patterns mined
 Application: coherent substructure mining selects a small subset of features that have high
distinguishing power between protein classes.
 Relational graph –each label is used only once
 Frequent highly connected or dense subgraph mining
 People with strong associations in OSNs
 Set of genes within the same functional module
 Cannot judge based on average degree or minimal degree
 Must ensure connectedness
 Example: Average degree: 3.25
Minimum degree 3
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22. Mining Dense Substructures
 Dense graphs defined in terms of Edge Connectivity
 Given a graph G, an edge cut is a set of edges Ec such that E(G) - Ec is
disconnected.
 A minimum cut is the smallest set in all edge cuts.
 The edge connectivity of G is the size of a minimum cut.
 A graph is dense if its edge connectivity is no less than a specified minimum cut
threshold
 Mining Dense substructures
 Pattern-growth approach called Close-Cut (Scalable)
 starts with a small frequent candidate graph and extends it until it finds the largest super graph with the
same support
 Pattern-reduction approach called Splat (High performance)
 directly intersects relational graphs to obtain highly connected graphs
 A pattern g discovered in a set is progressively intersected with subsequent components to give g’
 Some edges in g may be removed
 The size of candidate graphs is reduced by intersection and decomposition operations.
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23. Applications – Graph Indexing
 Indexing is essential for efficient search and query processing
 Traditional approaches are not feasible for graphs
 Indexing based on nodes / edges / sub-graphs
 Path based Indexing approach
 Enumerate all the paths in a database up to maxL length and index them
 Index is used to identify all graphs with the paths in query
 Not suitable for complex graph queries
 Structural information is lost when a query graph is broken apart
 Many false positives maybe returned
 gIndex – considers frequent and discriminative substructures as index features
 A frequent substructure is discriminative if its support cannot be approximated by the intersection of the
graph sets
 Achieves good performance at less cost
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24. Graph Indexing
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Only (c) is an exact match, but
others are also reported due to the
presence of sub-structures

25. Substructure Similarity Search
 Bioinformatics and Chem-informatics applications involve query
based search in massive complex structural data
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Form a set of sub-graph queries with one
or more edge deletions and then use
exact substructure search

26. Substructure Similarity Search
 Grafil (Graph Similarity Filtering)
 Feature based structural filtering
 Models each query graph as a set of features
 Edge deletions – feature misses
 Too many features – reduce performance
 Multi-filter composition strategy
 Feature Set - group of similar features
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27. Classification and Cluster Analysis using
Graph Patterns
 Graph Classification
 Mine frequent graph patterns
 Features that are frequent in one class but less in another – Discriminative
features – Model construction
 Can adjust frequency, connectivity thresholds
 SVM, NBM etc are used
 Cluster Analysis
 Cluster Similar graphs based on graph connectivity (minimal cuts)
 Hierarchical clusters based on support threshold
 Outliers can also be detected
 Inter-related process
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