1. by Dayu Yuan The Pennsylvania State University
Graph Feature Mining for Indexing and Classification A Thesis Proposal in Department of Computer Science and Engineering
by Dayu Yuan
The Pennsylvania State University
© Dayu Yuan
4/26/2017

2. Welcome Committee Chairs: CSE Department Faculty Members:
Dr. Prasenjit Mitra
Dr. C. Lee Giles
CSE Department Faculty Members:
Dr. Jessy Barlow
Dr. Daniel Kifer
Outside Member:
Dr. Zan Huang
© Dayu Yuan
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3. Outline 1. Motivation & Introduction
2. Graph index design for subgraph search (WebDB 2011)
3. Graph feature mining for subgraph search (In submission)
4. Future work
© Dayu Yuan
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4. Motivation: Graphs are prevalent: Chemistry: Chemical Molecule
Biology: Protein Structure
Computer Aided Design:
Image Processing & Computer Vision:
Social Network:
Mine and Manage Graph Data
Above figures all come from internet
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5. Our focus Data: Graph Database: a collection of graphs
Graph Scale: hundreds of nodes & edges
Chemical Molecules
Small Social Communities
Mechanical Parts
Graph Type: labeled, connected, undirected graph
(can be extend to other types)
Give the intuitve definition of subgraph features a b c c d c d c c c d c d a a a a a d b a b d b d b a b g5 g1 g2 g3 g4
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6. Graph Feature/Pattern
Nothing but Subgraphs/Subtrees/Random walk paths
Exponential number of subgraphs in a graph database
Impossible to enumerate
Frequent subgraphs are popular c c d c d c c a
Features b b b a b d P1 P2 P3 P4
Graph Database a b c c d c d c c c d c d a a a a a d b a b d b d b a b g5 g1 g2 g3 g4
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7. Graph Feature/Patternb c P1 b c d a P2 c b d P3 b c d P4
P1: g1, g2, g3, g4 P2: g1, g4
P3: g2, g3, g4, g5 P4: g1, g3
Features
Graph Database
P1 and P3 will enlarge a b c c d c d c c c d c d a a a a a d b a b d b d b a b g5 g1 g2 g3 g4
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8. Graph Features in Subgraph Search
In a graph database D = {g1,g2,...gn}, given a query graph q, the subgraph search algorithm returns all database graphs containing q as a subgraph. a c b d q1
Graph Database: D
G2 and G4 will enlarge a b c d g5 c d g4 b a a b c d g1 a c b d g2 a c g3 b d
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9. Graph Features in Subgraph Search
Filtering + Verification (gIndex 04):
If a graph g contains the query q, then g has to contain all q’s subgraphs.
Features
P1: g1, g2, g3, g4 a c b d q1 a b c P1 b c d a P2 c b d P3 b c d P4
P2: g1, g4
P3: g2, g3, g4, g5
P4: g1, g3
Graph Database: D a b c d g5 c d g4 b a a b c d g1 a c b d g2 a c g3 b d
© Dayu Yuan
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10. Graph Features in Supergraph Search
In a graph database D = {g1,g2,...gn}, given a query graph q, the supergraph search algorithm returns all database graphs have q as a supergraph. c d q2 b a
Graph Database: D a b c d g5 c d g4 b a a b c d g1 a c b d g2 a c g3 b d
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11. Graph Features in Supergraph Search
Filter + Verification (cIndex 07):
For any subgraph features that are not contained in the query, their supporting sets can be filtered.
Features
P1: g1, g2, g3, g4 c d q2 b a a b c P1 b c d a P2 c b d P3 b c d P4
P2: g1, g4
P3: g2, g3, g4, g5
P4: g1, g3
Graph Database: D a b c d g5 c d g4 b a a b c d g1 a c b d g2 a c g3 b d
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12. Graph Features in Graph Classification
Protein activity prediction
Drug toxicity prediction
Image classification
Graph Kernel:
Hard to interpret the results and the rules
P1: g1, g2, g3, g4
P2: g1, g4
P3: g2, g3, g4, g5
P4: g1, g3
Graph Database: D
Conclusion in the next slides a b c d g5 c d g4 b a a b c d g1 a c b d g2 a c g3 b d
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13. Graph Feature Mining Motivation Challenges:
Graph query operations (subgraph/supergraph search) need features to build the index
Graph learning needs features to explicitly represent graphs as vectors
Challenges:
Indexing: limited memory
Classification: curse of dimensionality
Exponential number of subgraphs
Too many frequent subgraphs, most of them are redundant & not discriminative/informative
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14. Research in Graph Feature Mining
1. Mine Frequent Subgraphs:
(1) Not computational efficient
(2) Curse of Dimensionality
(3) Most frequent subgraphs are redundant or not discrimiantive.
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15. Research in Graph Feature Mining
2. Batch mode discriminative & frequent subgraph mining
First enumerate all frequent subgraphs [Bottleneck]
Mine discriminative & frequent subgraphs out of frequent subgraphs
Challenge:
How to set the minimum support in step (A)? Small? Big?
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16. Research in Graph Feature Mining
3. Direct Feature Mining:
A. Search for a feature f optimizing an objective function
B. Find K features: run above algorithm iteratively (forward feature selection)
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17. Proposal: Our initial study on Subgraph Search Future work plan
Graph Index Structure Design
Graph Feature Mining
Future work plan
Supergraph Search
Graph Index Structure Design (initial study)
Graph Classification
Graph Descriptor Mining for Classification
Irredundant Iterative Feature Mining
Conclusion:
Usefulness
Challenges of graph feature minig
History
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18. Outline 1. Motivation & Introduction
2. Graph index design for subgraph search (WebDB 2011)
Background of Subgraph Search:
Preliminary & Problem Definition
Filter + Verification [Feature Based Index Approach]
3. Graph feature mining for subgraph search (In submission)
Direct feature mining for sub search
4. Future work
© Dayu Yuan
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19. Subgraph Search: Definition
Problem Definition:
In a graph database D = {g1,g2,...gn}, given a query graph q, the subgraph search algorithm returns all database graphs containing q as a subgraph.
Solution:
Brute force: For each query q, scan the dataset, find D(q)
Filter + Verification:
Given a query q, find a candidate set C(q), then verify each graph in C(q) to obtain D(q) q
C(q)
D(q) D
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20. Subgraph Search: Solutions
Filter + Verification:
Rule:
If a graph g contains the query q, then g has to contain all q’s subgraphs.
Inverted Index: <Key, Value> pair
Left: subgraph features (small segment of subgraphs),
Right: Posting List (IDs of all db graphs containing the “key” subgraph)
Given a query q, containing subgraph features p1 and p3. Then the candidate graphs are {g1, g2} \intersect {g1, g2, g3}, which is {g1, g2}. And g3 is filtered out.
For g1 and g2, we need to run subgraph isomorphism test to exam whether they are the true answer.
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21. Subgraph Search: Related Work
Total Query Processing Time:
(1) filtering cost: D to C(q)
Cost of the search for subgraph features contained in the query
Cost of loading the postings file, cost of intersecting the postings
(2) verification cost: C(q) to D(q)
subgraph isomorphism tests
NP-complete, dominates overall cost
Related work:
Reduce the verification cost by mining subgraph features
Disadvantages:
(1) “Batch mode” feature mining
(2) Different index structure designed for different features
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22. Outline 1. Motivation & Introduction
2. Graph index design for subgraph search (WebDB 2011)
Background of Subgraph Search:
Lindex: A general index structure for subsearch
Effective (filtering power)
Efficient (response time)
Compact (memory consumption)
Experiment Results
3. Graph feature mining for subgraph search (In submission)
Direct feature mining for sub search
4. Future work
© Dayu Yuan
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23. Lindex: A general index structure
Lattice-like Index:
Organize indexing features with a lattice
Partition the value set (supporting set/ posting list)
Contributions:
Orthogonal to related work (feature mining)
Applicable to all subgraph/subtree features. Lindex decouples feature mining and index structure design
Compact, Effective and Efficient
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24. Lindex: Effective in Filtering
Definition (maxSub, minSuper).
S is all indexing features
(1) sg2 and sg4 are maxSub of q
(2) sg5 is
minsup of q
back
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25. Lindex: Effective in Filtering
Strategy One: Minimal Supergraph Filtering
Given a query q and Lindex L(D,S), the candidate set on which an algorithm should check for subgraph isomorphism is
(1) sg2 and sg4 are maxSub of q
(2) sg5 is
minsup of q
(3)
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26. Lindex: Effective in Filtering
Strategy Two: Postings Partition
Direct & Indirect Value Set.
Direct Set: such that sg can extend to g, without being isomorphic to any other features
Indirect Set:
Index
Why “b” is in the direct value set of “sg1”, but “a” is not?
Data Based Graphs
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27. Lindex: Effective in Filtering
Given a query q and Lindex L(D,S), the candidate set on which an algorithm should check for subgraph isomorphism is b c
In Fig. 1, let us say that the query is the graph a. The maximal subgraphs of a are determined to be sg1 and sg2 and the minimal supergraph of a is sg4. The algo- rithm takes the intersection of the direct value sets of sg1 and sg2. The lists shown in the figure beside each node depicts its value-set. The first list is the direct value-set and the second list is the indirect value-set. The algorithm takes the intersection of the direct value-sets of the maxi- mal subgraphs and obtains the candidate set {a, c} ∩ a = a for verification. The union of the value-sets of the minimal supergraph sg4 is {c} and that is directly included in the answer and deducted from the candidate set. Finally, the set {a} is taken and its element a is verified to contain the subgraph of the query and is added to the answer set. The answer set {a, c} is output.
Query “a”
Graphs need to be verified
Traditional Model
Strategy(1)
Strategy(1 + 2)
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28. Lindex: Compact Space Saving (Extension Labeling)
Each Edge in a graph is represented as:
<ID(u), ID(v), Label(u), Label(edge(u, v)), Label(v)>
the label of the graph sg2 is
< 1,2,6,1,7 >,< 1,3,6,2,6 >
the label of its chosen parent sg1 is
< 1,2,6,1,7 >
Then subgraph g2 can be stored
as just < 1, 3, 6, 2, 6 >
< 1,2,6,1,7 >
< 1, 3, 6, 2, 6 >
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29. Lindex: Empirical Evaluation of Memory
Index\Feature
DFG
∆TCFG
MimR
Tree+∆
DFT
Feature Count
7599/6238
9873/5712
5000
6172/38
7500/6172
Gindex
1359
1534
1348
1339
FGindex
1826
SwiftIndex
860
Lindex
677
841
772
676
671
Unit in KB
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30. Lindex: Efficient in Maxsub Feature Search
the label of the graph sg2 is
< 1,2,6,1,7 >,< 1,3,6,2,6 >
the label of its chosen parent sg1 is
< 1,2,6,1,7 >
Node1 of sg1 mapped to Node1 of sg2
< 1,2,6,1,7 >
< 1, 3, 6, 2, 6 >
instead of constructing canonical labels for each subgraph of q and comparing them with the existing labels in the index to check if a indexing feature matches
while traversing a graph lattice, mappings constructed to check that a graph sg1 is contained in q can be extended to check whether a supergraph of sg1 in the lattice, sg2, is contained in q by incrementally expanding the mappings from sg1 to q.
© Dayu Yuan
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31. Lindex: Efficient in Minsup Feature Search
The set of minimal supergraph of a query q in the Lindex is a subset of the intersection of the set of descendants of each subgraph node of q in the partial lattice.
< 1,2,6,1,7 >
< 1, 3, 6, 2, 6 >
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32. Lindex: Experiments Exp on AIDS Dataset: 40,000 Graphs © Dayu Yuan
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33. Lindex: Experiments Exp on AIDS Dataset: 40,000 Graphs © Dayu Yuan
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34. Lindex: Experiments Exp on AIDS Dataset: 40,000 Graphs © Dayu Yuan
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35. Outline 1. Motivation & Introduction
2. Graph index design for subgraph search (WebDB 2011)
3. Graph feature mining for subgraph search (In submission)
Direct feature mining for sub search
Problem Definition & Objective Function
Branch & Bound
Heuristic-based search space exploration:
Partition of the search space
Experiment Results
4. Future work
© Dayu Yuan
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36. Feature Mining: Motivation
All previous feature selection algorithms for “subgraph search problem” follow “batch mode”
Assume stable database
Bottleneck (frequent subgraph enumeration)
Hard to tune the setting of parameters (minimum support, etc)
Our Contributions:
First direct feature mining algorithm for the subgraph search problem
Effective in index updating
Choose high quality features
© Dayu Yuan
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37. Feature Mining: Problem Definition
Iterative Index Updating:
Given database D, current index I with features P0
(1) Remove the Least Useful Feature
(2) Add a New Feature
(3) Goes to (1) until it converge
P1: g1, g2, g3, g4 q1 P0
P2: g2, g3, g4, g5 c d a b c p1 c b d p2 b c d a p3 a b d p4
P3: g1, g4 a
P4: g2, g3 b a
Graph Database: D a b c d g5 c d g4 b a a b c d g1 a c b d g2 a c g3 b d
© Dayu Yuan
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38. Feature Mining: Problem Definition
Previous work:
Given a graph database D, find a set of subgraph (subtree) features, minimizing the response time over training queries Q.
Our work:
Given a graph database D, an already built index I with feature set P0, search for a new feature p, such that the new feature set {P0 + p} minimizes the response time
© Dayu Yuan
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39. Feature Mining: Problem Definition
Iterative Index Updating:
Given database D, current index I with features P0
(1) Remove the Least Useful Feature
Find a feature p in P0
(2) Add a New Feature
Find a new feature p
(3) Goes to (1)
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40. Feature Mining: More on the Object Function
(1) Pros and Cons of using the query logs
The objective function of previous algorithms (i.e. Gindex, FGindex) depends on queries too. [Implicitly]
(2) Feature selected are “discriminative”
Previous work:
the discriminative power of ‘sg’ is measured w.r.t to sub(sg) or sup(sg), where sub(sg) denotes all subgraphs of ‘sg’, and sup(sg) denotes all supergraph of ‘sg’.
Our objective function:
discriminative power is measure w.r.t P0
(3) Computation Issue (next slides):
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41. Feature Mining: More on the Object Function (cont.)
MinSup Queries(p, Q) Q
Example
Computing D(p) for each enumerated feature ‘p’ is expensive
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42. Feature Mining: Estimate The Objective Function
The objective function of a new subgraph feature p, has an easy to compute upper bound and lower bound:
Inexpensive to compute
Two approaches to estimate
(1) Lazy calculation: don’t have to calculate gain(p, P0) when
Upp(p, P0) < gain(p*, P0)
Low(p, P0) > gain(p*, P0)
(2)
Omit Prof
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43. Feature Mining: Challenges
(1) Exponential search space for the new index subgraph feature “p”.
(2) Objective function is neither monotonic nor anti- monotonic. [Apriori rule can not be used]
(3) Traditional heuristic-based graph feature mining algorithms (e.g. LeapSearch) do not work. (They rely only on “frequencies”)
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44. Feature Mining: Branch and Bound
Exhaustive Search according to DFS Tree
A graph(pattern) can be canonically labeled as a string, the DFS tree is a pre-fix tree of the labels of graphs. n1
For each branch, e.g., branch starting from n5, find an branch upper bound > gain value of all nodes on that branch. n2 n7 n5 n6 n3 n4
The objective function is neither monotonic or anti-monotonic
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45. Feature Mining: Branch and Bound
Thm
For a feature p, an upper bound exists such that for all p’ that are supergraph of p, gain(p’, P0) <= BUpp(p, P0) n1 n2 n7 n5 n6 n3 n4
Omit Prof
© Dayu Yuan
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46. Feature Mining: Heuristic based search space partition
Problem:
The search always starts from the same root and search according to the same order
Observation
The new graph pattern p must be a super graph of some patterns in P0, i.e., p ⊃ p2 in Figure 4
1) A great proportion of the queries are supergraphs of root, otherwise there will be few queries using p ⊃ r for filtering
2) The average size of the set of candidates for queries ⊃ r are large, which means improvement over those queries is important.
© Dayu Yuan
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47. Feature Mining: Heuristic based search space partition
Procedure:
(1)gain(p*)=0
(2)Sort all P0 according to sPoint(pi) function in decreasing order
(3) Start Iterating
For i=1to|P| do
If branch upper bound of BUpp(ri) < gain(p∗) then break
Else Find the minimal supergraph queries minSup(r, Q)
p*(r) = Branch & Bound Search (minSup(r, Q), p∗)
If gain(p*(r)) > gain(p∗), update p∗ = p∗(r)
Discussion:
(1) Candidate features are enumerated as descendents of the “root”. (Partition of the search space)
(3) The “root” is visited according to sPoint(r) score; quick to find a close to optimal feature.
(2) Candidate features are ‘frequent’ on D(r), not all D
Smaller minimum support
(4) Top k feature selection
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48. Feature Mining: Experiment
The AIDS dataset D (40K chemical molecules),
Index0: Gindex with minsupport 0.05
IndexDF: Gindex with minsupport 0.02
[1175 new feature are added]
Index QG/BB/TK (Index updated based on Index0)
BB: branch and bound
QG: search space partitioned
TK: top k feature returned in on iteration
Achieving the same candidate set size decrease
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49. © Dayu Yuan
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50. Feature Mining: Experiment
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51. Feature Mining: Experiment
2 Dataset: D1 & D2 (80% same)
DF(D1): Gindex on Dataset D1
DF(D2): Gindex on Dataaset D2
Index QG/BB/TK (Index updated based on DF(D1))
BB: branch and bound
QG: search space partitioned
TK: top k feature returned in on iteration
Exp1: D2 = D1 + 20% New
Exp2: D2 = 80%D1 + 20%New
Iterative until the objective value is stable
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52. Feature Mining: Experiments
DF VS. iterative methods
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53. Feature Mining: Experiments
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54. Feature Mining: Experiments
TCFG VS. iterative methods
MimR VS. iterative methods
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55. Outline 1. Motivation & Introduction
2. Graph index design for subgraph search (WebDB 2011)
3. Graph feature mining for subgraph search (In submission)
4. Future work
Supergraph Search
Index structure
Feature Selection for Supergraph Search
Graph feature mining for classification
Time line
Conclusion So Far: solving the subgraph search problem
1. Lindex: index structure general enough to support any features
Compact
Effective
Efficient
2. Direct feature mining
Third generation algorithm (no frequent feature enumeration bottleneck)
Effective in updating the index to accommodate changes
Runs much faster than building the index from scratch
Feature selected can filter more false positives than features selected from scratch.
© Dayu Yuan
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56. Future work: supergraph search
Exclusive Logic Filtering:
For any subgraph features that are not contained in the query, their supporting sets can be filtered.
Problem Definition:
In a graph database D = {g1,g2,...gn}, given a query graph q, the supergraph search algorithm returns all database graphs have q as a supergraph.
In Memory Model:
Too many disk operations if the postings are stored on disk
© Dayu Yuan
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57. Future work: supergraph search
On Disk Model & Feature Selection
Features are organized in a lattice (same as Lindex)
Each feature is associated with one value set (disjoint value set)
The value set of feature f
Query processing model:
Sg1
Sg3
Sg2
Value Set
Sg1：［a］
Sg2：［c］
Sg3：［b］ b a c
Although, the above query processing model over poIndex does not involve explicit filtering step, it actually prunes some false graphs, such as the graph g in the Figure 5.1. But there are some false positive graphs that can not be filtered out by the poIndex, but can be pruned by using the explicit logic filtering. For example, graph g′ contains a feature f′ that is not contained in the query graph q, but when we assign the value sets, we assign g′ to the value set of feature f1 which is contained in q. g′ can be pruned out by applying the explicit logic filtering explicitly, but can not be filtered by the query processing model over poIndex. In order to minimize such g′ and improves the filtering power of poIndex, we need the support of a feature selection algorithm and also an value set assigning algorithm, which will be discussed in the next subsection in details. q
© Dayu Yuan
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58. Future work: supergraph search
On Disk Model & Feature Selection
Features are organized in a lattice (same as Lindex)
Each feature is associated with one value set (disjoint value set)
The value set of feature f
Query processing model:
Sg1
Sg3
Sg2
Value Set
Sg1：［］
Sg2：［a,c］
Sg3：［b］ b a c
Although, the above query processing model over poIndex does not involve explicit filtering step, it actually prunes some false graphs, such as the graph g in the Figure 5.1. But there are some false positive graphs that can not be filtered out by the poIndex, but can be pruned by using the explicit logic filtering. For example, graph g′ contains a feature f′ that is not contained in the query graph q, but when we assign the value sets, we assign g′ to the value set of feature f1 which is contained in q. g′ can be pruned out by applying the explicit logic filtering explicitly, but can not be filtered by the query processing model over poIndex. In order to minimize such g′ and improves the filtering power of poIndex, we need the support of a feature selection algorithm and also an value set assigning algorithm, which will be discussed in the next subsection in details. q
© Dayu Yuan
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59. Future work: supergraph search
On Disk Model & Feature Selection
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60. Future work: supergraph search
On Disk Model & Feature Selection
Tentative Solution to this problem:
Banking float maximization problem (similar to max cover problem). NP-hard problem. Polynomial solution exists with approximate ratio (1-1/e)
Scalability issues:
(a) Solve the problem with Map-reduce
(map-reduce solution exists for the max cover problem [63, 65])
(b) Direct Feature Mining
© Dayu Yuan
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61. Future work: graph classification
Feature vector for graph data:
Given a set of N graph features p1 , p2 , ..., pn , a graph g can be coded as a vector Xg=[x1, x2, ..., xn]T where xi is a {0, 1} valued variable and xi = 1 if and only if pi ⊆ g for all i.
Compared with Graph Kernel::
Interpretability
Graph Vectors based on feature p1, p2, p3 and p4
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62. Future work: graph classification
Previous Work:
(1) Class two algorithms: [batch mode]
(2) Class three algorithms: [direct mining]
Iterative mining: at each iteration, only feature is selected
Redundancy: a feature f with very low objective function value in iteration i, may be enumerated in the following iterations as well.
(3) Heuristic based feature mining:
The search space of subgraph features is explored randomly. Close to optimal features tend to be discovered quickly. (Much faster than exact search)
Why not use other descriptors instead of subgraphs as features?
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63. Future work: graph classification
Direction One: Descriptor Mining
Tentative Solution:
Use the information diffusion models to build a descriptor capturing the local context and topology information.
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64. Future work: graph classification
Direction One: Descriptor Mining
What is Information Diffusion? (Word of Mouth effect)
Different models
General Assumption
Nodes are either active or inactive; Active nodes may cause other to activate; active nodes never deactivate
Linear Threshold Model [69]
Independent Cascade Model [70]
Build a descriptor:
(1) For each node v, activate only v and trigger the information diffusion procedure.
(2) Collect all active nodes to build the descriptor.
© Dayu Yuan
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65. Information Diffusion Models
Linear Threshold Model [69]
A node v is influenced by each neighbor w with weight b(v, w)
Node v will be activated if
is a random variable in [0, 1]
Independent Cascade Model [70]
When node v is activated, it has a one and only one chance of activating its inactive neighbor w.
The activation attempt succeeds with probability pvw
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66. Future work: graph classification
Build a descriptor:
(1) For each node v, activate only v and trigger the information diffusion.
(2) Collect all active nodes to build the descriptor.
(3) P1 denotes the probability of a node with label `a’ becomes active
DesA
Probability
Label a
p1=1
Label b
p2=.8
Label c
p3=.9
Label d
p4=.4 ……
DesB
Probability
Label a
P1=0.5
Label b
p2=.4
Label c
p3=.3
Label d
p4=.7 ……
Des(v, l)
Node v, label l
© Dayu Yuan
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67. Future work: graph classification
Mining descriptors for graph classification
Convert to the probabilistic item set mining problem.
Future plan
(1) Try different information diffusion models to see their cons and pros on modeling the `local’ context and topology information.
(2) Explore discriminative probabilistic item-set mining algorithms for (binary) classification.
© Dayu Yuan
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68. Future work: graph classification
Direction Two: Irredundant Iterative Feature Mining
Embedding Fading approach:
For example, in Figure 5.2. After detection f is not significant w.r.t the objective function, we lower the weight of nodes and edges of f’s embedding on graph g1 and g2.
For the above example, if we assume that the dashed edge weight will drop from 1 to 0.5. Then, in the next iteration, the frequency of f decrease from 2 to 1. The frequency of f1 will not be affected much, since f1’s embedding does not overlap much with f.
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69. Future work: time line
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70. Thanks
Questions?
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71. Reference For a complete list of references, please refer to
© Dayu Yuan
4/26/2017

72. Backup Slides 1 A B 2 1 A B 2 1 A C 2 f1 A g1 D 3 D 3 A 4 3 B 1 A B 25 g2 q f2 A C 3 C 3
Features
Database Graph
Query Graph
© Dayu Yuan
4/26/2017