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Repairing Entities using Star Constraints in Multi-relational Graphs - - PowerPoint PPT Presentation

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Repairing Entities using Star Constraints in Multi-relational Graphs Peng Lin 1 Qi Song 1 Yinghui Wu 2,3 Jiaxing Pi 4 1 2 4 3 Erroneous entities: how to capture? Multi-relational graphs: a labeled graph with attributes on nodes

slide-1
SLIDE 1

Repairing Entities using Star Constraints in Multi-relational Graphs

Peng Lin1 Qi Song1 Yinghui Wu2,3 Jiaxing Pi4

1 2 3 4

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SLIDE 2

Erroneous entities: how to capture?

Graph G: a football database

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Stadium

name: ATC

  • wner: AHP

city: LDN

Facility

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“ Β§ Multi-relational graphs: a labeled graph with attributes on nodes

1

slide-3
SLIDE 3

Erroneous entities: how to capture?

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Stadium

name: ATC

  • wner: AHP

city: LDN

Facility

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“

Graph G: a football database

Β§ Multi-relational graphs: a labeled graph with attributes on nodes Β§ Entity errors: incorrect node attributes

1

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SLIDE 4

Erroneous entities: how to capture?

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Stadium

name: ATC

  • wner: AHP

city: LDN

Facility

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“

Graph G: a football database

Β§ Multi-relational graphs: a labeled graph with attributes on nodes Β§ Entity errors: incorrect node attributes Β§ Semantics: relevant paths from a center node

1

β€œFor stadium and facility relevant to player (π’˜πŸ) from Premier League, if they have the same

  • wner, then they should locate at the same city.”
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SLIDE 5

Regular path queries

Β§ Regular expressions: 𝑆 = π‘š π‘š&' 𝑆 % 𝑆|𝑆 βˆͺ 𝑆

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Facility

name: ATC

  • wner: AHP

city: LDN

Stadium

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“

Graph G: a football database 2

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SLIDE 6

Regular path queries

Β§ Regular expressions: 𝑆 = π‘š π‘š&' 𝑆 % 𝑆|𝑆 βˆͺ 𝑆

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Facility

name: ATC

  • wner: AHP

city: LDN

Stadium

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“

Graph G: a football database Β§ Paths from Player to Stadium Β§ 𝑆! = (playsFor , operates) βˆͺ (coachedBy , worksAt) 2

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SLIDE 7

Regular path queries

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Facility

name: ATC

  • wner: AHP

city: LDN

Stadium

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

Β§ Paths from Player to Facility Β§ 𝑆" = (playsFor , operates) βˆͺ (teammate#! , trainsAt) Β§ Paths from Player to Stadium Β§ 𝑆! = (playsFor , operates) βˆͺ (coachedBy , worksAt)

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“

Graph G: a football database

Β§ Regular expressions: 𝑆 = π‘š π‘š&' 𝑆 % 𝑆|𝑆 βˆͺ 𝑆

2

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SLIDE 8

Contributions

Graph 𝐻, StarFDs Ξ£ (𝐻 does not satisfy Ξ£) Repair 𝐻’ (𝐻’ satisfies Ξ£)

StarRepair framework

Error detection

3

Repair

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SLIDE 9

Contributions

StarFDs: star functional dependencies new constraints for graphs

Graph 𝐻, StarFDs Ξ£ (𝐻 does not satisfy Ξ£) Repair 𝐻’ (𝐻’ satisfies Ξ£)

Entity repair problem: minimum editing cost, NP-hard and APX-hard Feasible framework with provable guarantees whenever possible StarRepair framework

Error detection

3

Repair

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SLIDE 10

Contributions

StarFDs: star functional dependencies new constraints for graphs

Graph 𝐻, StarFDs Ξ£ (𝐻 does not satisfy Ξ£) Repair 𝐻’ (𝐻’ satisfies Ξ£)

Entity repair problem: minimum editing cost, NP-hard and APX-hard Feasible framework with provable guarantees whenever possible StarRepair framework

Error detection Is approximable? Approximation solution Is optimal repairable?

3

Repair Optimal solution

Yes

Heuristic solution

No Yes No

Repair workflow

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SLIDE 11

Star constraints

Β§ StarFDs: πœ’ = (𝑄(𝑣(), π‘Œ β†’ 𝑍) Β§ Star pattern 𝑄(𝑣(): Β§ Value constraints: π‘Œ β†’ 𝑍

4

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SLIDE 12

Star constraints

Β§ StarFDs: πœ’ = (𝑄(𝑣(), π‘Œ β†’ 𝑍) Β§ Star pattern 𝑄(𝑣():

  • A two-level tree with center node 𝑣(
  • Each branch is a regular expression

Player Stadium Facility π‘ΊπŸ π‘ΊπŸ‘

π’—πŸ π’—πŸ π’—πŸ‘

𝑆# = (playsFor 0 operates) βˆͺ (teammate$% 0 trainsAt) 𝑆% = (playsFor 0 operates) βˆͺ (coachedBy 0 worksAt)

Β§ Value constraints: π‘Œ β†’ 𝑍

4

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SLIDE 13

Star constraints

Β§ StarFDs: πœ’ = (𝑄(𝑣(), π‘Œ β†’ 𝑍) Β§ Star pattern 𝑄(𝑣():

  • A two-level tree with center node 𝑣(
  • Each branch is a regular expression

Player Stadium Facility π‘ΊπŸ π‘ΊπŸ‘

π’—πŸ π’—πŸ π’—πŸ‘

𝑆# = (playsFor 0 operates) βˆͺ (teammate$% 0 trainsAt) 𝑆% = (playsFor 0 operates) βˆͺ (coachedBy 0 worksAt)

Β§ Value constraints: π‘Œ β†’ 𝑍

  • π‘Œ and 𝑍 are two sets of literals
  • Literals: 𝑣. 𝐡 = 𝑑, or 𝑣. 𝐡 = 𝑣). 𝐡′

π‘Œ : 𝑣$. league = EPL, 𝑣!. owner = 𝑣". owner 𝑍 : 𝑣!. city = 𝑣". city 4

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SLIDE 14

Star constraints

Β§ Matching semantics: maximum set matched by star pattern

5

Player Stadium Facility π‘ΊπŸ π‘ΊπŸ‘

π’—πŸ π’—πŸ π’—πŸ‘

Star pattern 𝑄(𝑣$)

π‘Œ : 𝑣&. league = EPL, 𝑣%. owner = 𝑣#. owner 𝑍 : 𝑣%. city = 𝑣#. city

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SLIDE 15

Star constraints

Β§ Matching semantics: maximum set matched by star pattern

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Facility

name: ATC

  • wner: AHP

city: LDN

Stadium

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“ π’—πŸ matches π’˜πŸ π’—πŸ matches π’˜πŸ and π’˜πŸ“ π’—πŸ‘ matches π’˜πŸ‘ and π’˜πŸ’

5

Player Stadium Facility π‘ΊπŸ π‘ΊπŸ‘

π’—πŸ π’—πŸ π’—πŸ‘

Star pattern 𝑄(𝑣$)

π‘Œ : 𝑣&. league = EPL, 𝑣%. owner = 𝑣#. owner 𝑍 : 𝑣%. city = 𝑣#. city

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SLIDE 16

Star constraints

Β§ Matching semantics: maximum set matched by star pattern Β§ Inconsistencies 𝑱: matches that π‘Œ holds but 𝑍 does not hold

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Facility

name: ATC

  • wner: AHP

city: LDN

Stadium

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“

Player Stadium Facility π‘ΊπŸ π‘ΊπŸ‘

π’—πŸ π’—πŸ π’—πŸ‘

Star pattern 𝑄(𝑣$)

π‘Œ : 𝑣&. league = EPL, 𝑣%. owner = 𝑣#. owner 𝑍 : 𝑣%. city = 𝑣#. city

5

π’—πŸ matches π’˜πŸ π’—πŸ matches π’˜πŸ and π’˜πŸ“ π’—πŸ‘ matches π’˜πŸ‘ and π’˜πŸ’

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SLIDE 17

Summary of results

Β§ Notations 6

Problem Description Hardness Solution Satisfiability Input: Ξ£ decide whether there exists 𝐻 that satisfies Ξ£ NP-complete Implication Input: Ξ£ and πœ’ decide whether for all 𝐻 satisfy Ξ£, they satisfy πœ’ coNP-hard Error detection (validation) Input: 𝐻 and Ξ£ Output: all inconsistencies 𝑱 PTIME Evaluate regular path queries and validate values

  • time complexity: 𝑃( Ξ£ V + |π‘Š|( π‘Š + |𝐹|))

Repair Input: Ξ£ and 𝐻 that does not satisfy Ξ£ Ouput: 𝐻′ that satisfies Ξ£ with least repair cost NP-hard APX-hard Approximable cases (PTIME checkable)

  • time complexity 𝑃( 𝑱 Ξ£ ! + 𝑱 ( 𝑱 Ξ£ ! + |𝑱| Ξ£ ))
  • approximation ratio: 𝑱 Ξ£ !

Optimal cases

  • time complexity 𝑃( 𝑱 Ξ£ ))

Heuristic cases

  • time complexity 𝑃( 𝑱 Ξ£ ! + 𝑱 ( 𝑱 Ξ£ ! + |𝑱| Ξ£ ))
  • bounded repairable: cost ≀ 𝑱

𝐻: graph π‘Š: nodes 𝐹: edges πœ’: a single StarFD Ξ£: a set of StarFDs 𝑱: all inconsistencies.

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SLIDE 18

Updates and repairs

7

Β§ Updates 𝑃: operators 𝑝 = (𝑀. 𝐡, 𝑏, 𝑑) with editing cost cost 𝑃 = βˆ‘(∈+ cost 𝑝 Β§ Repair 𝑃: applying 𝑃 to 𝐻, such that obtain 𝐻′ that satisfies Ξ£

slide-19
SLIDE 19

Updates and repairs

Player

name: VanPersie

Club

name: MU

Club

name: AFC

Player

name: Rooney

Coach

name: Wenger

Stadium

name: OT

  • wner: MUP

city: MAN

Facility

name: AON

  • wner: MUP

city: LD Facility

name: ATC

  • wner: AHP

city: LDN

Stadium

name: EM

  • wner: AHP

city: BZ

  • perates

playsFor playsFor worksAt teammate trainsAt

  • perates

coachedBy trainsAt

π’˜πŸ π’˜πŸ π’˜πŸ‘ π’˜πŸ’ π’˜πŸ“

7 Two repairs:

𝑃! = {(𝑀".city, LD, MAN), (𝑀#.city, BZ, LDN)} 𝑃" = {(𝑀".owner, MUP, CFG), (𝑀#.owner, EM, ENIC)}

Β§ Updates 𝑃: operators 𝑝 = (𝑀. 𝐡, 𝑏, 𝑑) with editing cost cost 𝑃 = βˆ‘(∈+ cost 𝑝 Β§ Repair 𝑃: applying 𝑃 to 𝐻, such that obtain 𝐻′ that satisfies Ξ£

slide-20
SLIDE 20

Entity repair problem

Β§ Input: StarFDs Ξ£, and graph 𝐻 does not satisfy Ξ£ Β§ Output: a repair 𝑃, such that

  • btain 𝐻’ that satisfies Ξ£
  • cost(𝑃) ≀ cost(𝑃’) for any 𝑃’

8

slide-21
SLIDE 21

Entity repair problem

Β§ Input: StarFDs Ξ£, and graph 𝐻 does not satisfy Ξ£ Β§ Output: a repair 𝑃, such that

  • btain 𝐻’ that satisfies Ξ£
  • cost(𝑃) ≀ cost(𝑃’) for any 𝑃’

Β§ Solution overview

  • Connected components (CCs): inconsistencies connected at shared node attributes
  • Isolated CCs: no new inconsistency is introduced when a CC is repaired

8 𝐽! 𝐽" 𝐽%

𝑀". 𝐡" 𝑀!. 𝐡!

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SLIDE 22

Entity repair problem

Β§ Input: StarFDs Ξ£, and graph 𝐻 does not satisfy Ξ£ Β§ Output: a repair 𝑃, such that

  • btain 𝐻’ that satisfies Ξ£
  • cost(𝑃) ≀ cost(𝑃’) for any 𝑃’

Β§ Solution overview

  • Connected components (CCs): inconsistencies connected at shared node attributes
  • Isolated CCs: no new inconsistency is introduced when a CC is repaired

8

Is approximable? Approximation solution Is optimal repairable? Optimal solution

Yes

Heuristic solution

No Yes No

Repair workflow 𝐽! 𝐽" 𝐽%

𝑀". 𝐡! 𝑀!. 𝐡!

Isolated CCs have approximate solutions

slide-23
SLIDE 23

Optimal case

Β§ Updates π’‘π‘š: flip the condition of a literal π‘š in π‘Œ βˆͺ 𝑍 Β§ Optimal solution: hyper star structure

  • Select the π’‘βˆ— with least cost in center
  • Select one 𝒑 with least cost in each petal, and induce 𝑷
  • If cost(π’‘βˆ—) ≀ cost(𝑷), return π’‘βˆ—; otherwise, return 𝑷

𝐽! 𝐽" 𝐽% 𝒑"

'

𝒑(

'

Optimal case

9

Is approximable? Approximation solution Is optimal repairable? Optimal solution

Yes

Heuristic solution

No Yes No

Repair workflow 𝒑!

'

𝒑)

'

𝒑*

'

𝒑%

'

Example:

  • π’‘βˆ—= 𝒑!

'

  • 𝑷 = 𝒑%

' βˆͺ 𝒑( ' βˆͺ 𝒑* '

  • Return π’‘βˆ— that has less cost
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SLIDE 24

Approximable case

Β§ Updates π’‘π‘š: flip the condition of a literal π‘š in π‘Œ βˆͺ 𝑍 Β§ Approximation solution:

  • Hypergraph vertex cover without forbidden pairs
  • Forbidden pairs

10

Is approximable? Approximation solution Is optimal repairable? Optimal solution

Yes

Heuristic solution

No Yes No

Repair workflow Example:

  • Return 𝑷 = 𝒑"

' βˆͺ 𝒑( '

  • 𝒑*

' is pruned

𝐽! 𝐽%

Approximable case

𝒑"

'

𝒑(

'

𝒑!

'

𝒑)

'

𝒑*

'

𝒑%

'

𝒑'

( = {(𝑀#.owner, MUP, CFG), (𝑀).owner, EM, ENIC)}

𝒑*

( = {(𝑀#.owner, MUP, FSG), (𝑀).owner, EM, ENIC)}

𝐽"

slide-25
SLIDE 25

Heuristic case

Β§ Updates π’‘π‘š: flip the condition of a literal π‘š in π‘Œ βˆͺ 𝑍 Β§ Heuristic solution (for non-isolated CC):

  • Select CC introducing fewest inconsistencies
  • Invoke approximation/optimal solution
  • Re-detect inconsistencies
  • Repeat until incur a cost bound

11

Is approximable? Approximation solution Is optimal repairable? Optimal solution

Yes

Heuristic solution

No Yes No

Repair workflow

Heuristic case

Repair CC1 consisting of 𝐽!, 𝐽", and 𝐽%

CC1

𝐽" 𝐽! 𝐽%

slide-26
SLIDE 26

Heuristic case

Β§ Updates π’‘π‘š: flip the condition of a literal π‘š in π‘Œ βˆͺ 𝑍 Β§ Heuristic solution (for non-isolated CC):

  • Select CC introducing fewest inconsistencies
  • Invoke approximation/optimal solution
  • Re-detect inconsistencies
  • Repeat until incur a cost bound

12

Is approximable? Approximation solution Is optimal repairable? Optimal solution

Yes

Heuristic solution

No Yes No

Repair workflow 𝐽(

Heuristic case

𝐽)

CC2 CC3

Two new inconsistencies 𝐽) and 𝐽(

slide-27
SLIDE 27

Experiment settings

Β§ Datasets Β§ Error generation: adopt silver standard and an error generation benchmark (Arocena et al. 2015) Β§ StarFD generation: discovered from silver standard (first star patterns and then value constraints) Β§ Algorithms:

  • StarRepair: use bidirectional search for regular path queries with incremental error detection
  • biBFSRepair: use bidirectional search without incremental error detection
  • SubIsoRepair: use subgraph isomorphism as matching semantics with incremental error detection

Data Description # of nodes # of edges

  • avg. # of attributes per node

Yago Knowledge graph 2.1M 4.0M 3 DBPedia Knowledge graph 2.2M 7.4M 4 Yelp Business reviews 1.5M 1.6M 5 IMDb Movie network 5.9M 3.2M 3 13

slide-28
SLIDE 28

Experiment results

Β§ StarFD repairs: efficiency and effectiveness Β§ Case study

0.1 1 10 100 1000

YAGO Yelp DBP IMDb

Time (seconds)

StarRepair biBFSRepair SubIsoRepair

StarRepair outperforms biBFSRepair and SubIsoRepair by 3.4 and 7.1 times respectively 14

slide-29
SLIDE 29

Experiment results

Β§ StarFD repairs: efficiency and effectiveness Β§ Case study

0.1 1 10 100 1000

YAGO Yelp DBP IMDb

Time (seconds)

StarRepair biBFSRepair SubIsoRepair

StarRepair outperforms biBFSRepair and SubIsoRepair by 3.4 and 7.1 times respectively StarRepair outperforms SubIsoRepair by 10% in f-score (9% in precision and 14% in recall)

0.2 0.4 0.6 0.8 1

YAGO Yelp DBP IMDb

f-score

StarRepair SubIsoRepair

14

slide-30
SLIDE 30

Experiment results

Β§ StarFD repairs: efficiency and effectiveness Β§ Case study

0.1 1 10 100 1000

YAGO Yelp DBP IMDb

Time (seconds)

StarRepair biBFSRepair SubIsoRepair

StarRepair outperforms biBFSRepair and SubIsoRepair by 3.4 and 7.1 times respectively StarRepair outperforms SubIsoRepair by 10% in f-score (9% in precision and 14% in recall)

0.2 0.4 0.6 0.8 1

YAGO Yelp DBP IMDb

f-score

StarRepair SubIsoRepair

StarFD: If a person 𝑣, is a politician or president of U.S., and is married to another person 𝑣!, then 𝑣!’s child is 𝑣,’s child. We found more than 100 such errors in Yago.

Person Country Person

𝑆!=marriedTo 𝑆"= presidentOf βˆͺ politicianOf

π’—πŸ π’—πŸ π’—πŸ‘

Person

name: G.W. Bush child: B. Obama

Country

name: U.S.

Person

name: Laura Bush child: Barbara Bush

marriedTo presidentOf

π’˜πŸ π’˜πŸ π’˜πŸ‘

14

slide-31
SLIDE 31

Compare with GFDs (Fan et al. 2016)

Β§ StarFDs: star functional dependencies Β§ Definition: πœ’ = (𝑄(𝑣(), π‘Œ β†’ 𝑍) Β§ GFDs: graph functional dependencies Β§ Definition: πœ’ = (𝑄, π‘Œ β†’ 𝑍)

Problem StarFDs GFDs Semantic star patterns with regex queries subgraph isomorphism Satisfiability NP-complete coNP-complete Implication coNP-hard NP-complete Error detection (validation) PTIME coNP-complete 15

slide-32
SLIDE 32

Repairing Entities using Star Constraints in Multi-relational Graphs

Β§ Notations 6

Problem Description Hardness Solution Satisfiability Input: Ξ£ decide whether there exists 𝐻 that satisfies Ξ£ NP-complete Implication Input: Ξ£ and πœ’ decide whether for all 𝐻 satisfy Ξ£, they satisfy πœ’ coNP-hard Error detection (validation) Input: 𝐻 and Ξ£ Output: all inconsistencies 𝑱 PTIME Evaluate regular path queries and validate values

  • time complexity: 𝑃( Ξ£ V + |π‘Š|( π‘Š + |𝐹|))

Repair Input: Ξ£ and 𝐻 that does not satisfy Ξ£ Ouput: 𝐻′ that satisfies Ξ£ with least repair cost NP-hard APX-hard Approximable cases (PTIME checkable)

  • time complexity 𝑃( 𝑱 Ξ£ ! + 𝑱 ( 𝑱 Ξ£ ! + |𝑱| Ξ£ ))
  • approximation ratio: 𝑱 Ξ£ !

Optimal cases

  • time complexity 𝑃( 𝑱 Ξ£ ))

Heuristic cases

  • time complexity 𝑃( 𝑱 Ξ£ ! + 𝑱 ( 𝑱 Ξ£ ! + |𝑱| Ξ£ ))
  • bounded repairable: cost ≀ 𝑱

𝐻: graph π‘Š: nodes 𝐹: edges πœ’: a single StarFD Ξ£: a set of StarFDs 𝑱: all inconsistencies.

slide-33
SLIDE 33

Thank you!

Kronos: Lightweight Knowledge-based Event Analysis in Cyber-Physical Data Streams To appear in Demo Session