synopsys: foundationsfor...
TRANSCRIPT
Michael Rudolf1, Hannes Voigt1, Christof Bornhoevd2,and Wolfgang Lehner1
SynopSys: Foundations forMultidimensional Graph AnalyticsBusiness Intelligence for the Real-Time Enterprise (BIRTE 2014)1Database Technology Group, Technische Universität Dresden2SAP Labs, LLC, Palo Alto
September 1, 2014
Motivation: Big (Graph) Data
Peak Performance
Nov. 26, 2012: 26.5M items (306/sec)Nov. 23, 2013: 36.8M items (426/sec)
645M users135 K new every day
58M tweets & 2.1 G searches / day
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 2
Motivation: Big (Graph) Data
Peak Performance
Nov. 26, 2012: 26.5M items (306/sec)Nov. 23, 2013: 36.8M items (426/sec)
645M users135 K new every day
58M tweets & 2.1 G searches / day
Intensional vs. Extensional� Schema & integrity constraints
� Created at design time bydomain experts
ETL
Once & forever
� Collect lots of data �rst
� Try to deduce the intension
� �The Fourth Paradigm� [Mic09]
. . .
Time© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 2
The Property Graph Model
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� Provides directed, attributed multi-relational graphs
� Attributes on vertices and edges as key-value pairs(instance-level instead of class-level)
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 3
The Property Graph Model
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2
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� Provides directed, attributed multi-relational graphs
� Attributes on vertices and edges as key-value pairs(instance-level instead of class-level)
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 3
Agenda
Analytical Scenario: From Graphs to Cubes
Operations: Roll-up, Drill-down, Slice & Dice
Challenges: Unbalanced Hierarchies & OLAP Anomalies
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 4
Graph Cube
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1. Identify facts
2. Specify dimensions
3. De�ne measures
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 5
Graph Cube
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2
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�AppleiPhone 5�
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4�ConsumerElectronics�
5�Phones�7�Tablets�
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1. Identify facts
2. Specify dimensions
3. De�ne measures
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 5
Graph Cube
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64GB�Apple iPadMC707LL/A�
2
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32 GB
�AppleiPhone 5�
3white
16 GB�Apple
iPhone 4�
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4�ConsumerElectronics�
5�Phones�7�Tablets�
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8�Freddy�
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�Karl�DE
10�Mike�US
11�Steve�US
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135/5 stars
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delivered 24/02/14
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part ofpart of
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inauthors
authors
rates
rates
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in
likeslikes
records
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contains 1
contains 2
contains 1
1. Identify facts
2. Specify dimensions
3. De�ne measures
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 5
Graph Cube
1
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64GB�Apple iPadMC707LL/A�
2
black
32 GB
�AppleiPhone 5�
3white
16 GB�Apple
iPhone 4�
a
4�ConsumerElectronics�
5�Phones�7�Tablets�
b
8�Freddy�
FR
9
�Karl�DE
10�Mike�US
11�Steve�US
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125/5stars
135/5 stars
14
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delivered 24/02/14
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part ofpart of
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inauthors
authors
rates
rates
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likeslikes
records
records
contains 1
contains 2
contains 1
1. Identify facts
2. Specify dimensions
3. De�ne measures
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 5
Facts
Depending on the use case, a (base) fact can be
� a vertex attribute, an edge attribute, or
� the presence of an edge.
in general: a subgraph
Ô Use pattern matching Ô graphical speci�cation instead of DSL
Example
authorsrates Match reviews of products and
their authors (vertex typesindicated via color)
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 6
Facts
Depending on the use case, a (base) fact can be
� a vertex attribute, an edge attribute, or
� the presence of an edge.in general: a subgraph
Ô Use pattern matching Ô graphical speci�cation instead of DSL
Example
authorsrates Match reviews of products and
their authors (vertex typesindicated via color)
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 6
Facts
Depending on the use case, a (base) fact can be
� a vertex attribute, an edge attribute, or
� the presence of an edge.in general: a subgraph
Ô Use pattern matching Ô graphical speci�cation instead of DSL
Example
authorsrates Match reviews of products and
their authors (vertex typesindicated via color)
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 6
Facts
Depending on the use case, a (base) fact can be
� a vertex attribute, an edge attribute, or
� the presence of an edge.in general: a subgraph
Ô Use pattern matching Ô graphical speci�cation instead of DSL
Example
authorsrates Match reviews of products and
their authors (vertex typesindicated via color)
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 6
Dimensions
Dimensions can be
1. vertex or edge attributes
2. connectivity
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8�Freddy�
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�Karl�DE
10�Mike�US
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135/5 stars
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Structure in Dimensions� extrinsic: not contained in graph data,needs to be provided externally (e.g., GeoNames)
� intrinsic: embodied in graph data
explicit: captured as topological informationimplicit: has to be derived from attribute values
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 7
Dimensions
Dimensions can be
1. vertex or edge attributes
2. connectivity
1
black
64GB�Apple iPadMC707LL/A�
2
black
32 GB
�AppleiPhone 5�
3white
16 GB�Apple
iPhone 4�
a
4�ConsumerElectronics�
5�Phones�7�Tablets�
b
8�Freddy�
FR
9
�Karl�DE
10�Mike�US
11�Steve�US
c
125/5stars
135/5 stars
14
4/5 stars
d
e
f
15
delivered 24/02/14
16ordered24/02/14
g h
part ofpart of
in
inin
authors
authorsrates
rates
rates
likeslikes
records
records
contains 1
contains 2
contains 1
Structure in Dimensions� extrinsic: not contained in graph data,needs to be provided externally (e.g., GeoNames)
� intrinsic: embodied in graph data
explicit: captured as topological informationimplicit: has to be derived from attribute values
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 7
Intrinsic Dimensions
Explicit Dimensions� Can be speci�ed using path expressions� In general requires one path expression per level, e.g.
-[@type='belongsTo ']->[@type='state ']-[@type='partOf ']->[@type='country ']
Implicit Dimensions� Might require bucketization� In general requires one expression per level, e.g.
GetWeekOfYear(@ordered) and GetYear(@ordered)
alias @ attribute access of vertex or edge attribute
- [ edge predicate ] -> [ vertex predicate ] ( length )
paths (with optional recursion depth), optionally satisfying the predicates© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 8
Intrinsic Dimensions
Explicit Dimensions� Can be speci�ed using path expressions� In general requires one path expression per level, e.g.
-[@type='belongsTo ']->[@type='state ']-[@type='partOf ']->[@type='country ']
Implicit Dimensions� Might require bucketization� In general requires one expression per level, e.g.
GetWeekOfYear(@ordered) and GetYear(@ordered)
alias @ attribute access of vertex or edge attribute
- [ edge predicate ] -> [ vertex predicate ] ( length )
paths (with optional recursion depth), optionally satisfying the predicates© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 8
Intrinsic Dimensions
Explicit Dimensions� Can be speci�ed using path expressions� In general requires one path expression per level, e.g.
-[@type='belongsTo ']->[@type='state ']-[@type='partOf ']->[@type='country ']
Implicit Dimensions� Might require bucketization� In general requires one expression per level, e.g.
GetWeekOfYear(@ordered) and GetYear(@ordered)
alias @ attribute access of vertex or edge attribute
- [ edge predicate ] -> [ vertex predicate ] ( length )
paths (with optional recursion depth), optionally satisfying the predicates© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 8
Dimension Speci�cation
ExampleName Seed Pattern Levels
Nationality $c $c@nationality
Category $p
Product category:
$p-[@type='in']->
Product group:
$p-[@type='in']->-[@type='part-of']->
Product area:
$p-[@type='in']->-[@type='part-of']->(2)
Seed Pattern� Connects facts to dimensions� Is matched against facts
Ô Has to be a super pattern of the fact pattern (i.e., more general)
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 9
Dimension Speci�cation
ExampleName Seed Pattern Levels
Nationality $c $c@nationality
Category $p
Product category:
$p-[@type='in']->
Product group:
$p-[@type='in']->-[@type='part-of']->
Product area:
$p-[@type='in']->-[@type='part-of']->(2)
Seed Pattern� Connects facts to dimensions� Is matched against facts
Ô Has to be a super pattern of the fact pattern (i.e., more general)© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 9
Properties of Dimensions
MonotonyLevels should be ordered such thatthe number of items decreases.
Level Name # Elements
1 Region 125
2 Country 30
3 Continent 3
HierarchyLevels should form hierarchies.If two facts map to the sameelement in li, they should map tothe same element in li+1 as well.Ô Functional dependency
Fact Level 1 Level 2 Level 3
A Saxony Germany Europe
B Saxony Germany Europe
C Bavaria Germany Europe
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 10
Properties of Dimensions
MonotonyLevels should be ordered such thatthe number of items decreases.
Level Name # Elements
1 Region 125
2 Country 30
3 Continent 3
HierarchyLevels should form hierarchies.If two facts map to the sameelement in li, they should map tothe same element in li+1 as well.Ô Functional dependency
Fact Level 1 Level 2 Level 3
A Saxony Germany Europe
B Saxony Germany Europe
C Bavaria Germany Europe
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 10
Measures
A measure is a derived fact
� combining several facts
� computed by a speci�ed function(e.g., scalar, aggregation).
Ô Annotate the fact patternÔ Introduce representative vertex
Example� Average product rating byproduct category
� Minimum age of customersby nationality
$c
$r $p++
(Min. Age, $c@age,MIN)
(Avg. Rtg., $r@stars,AVG)
authors$a
rates$e
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 11
Measures
A measure is a derived fact
� combining several facts
� computed by a speci�ed function(e.g., scalar, aggregation).
Ô Annotate the fact patternÔ Introduce representative vertex
Example� Average product rating byproduct category
� Minimum age of customersby nationality
$c
$r $p++
(Min. Age, $c@age,MIN)
(Avg. Rtg., $r@stars,AVG)
authors$a
rates$e
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 11
Measures
A measure is a derived fact
� combining several facts
� computed by a speci�ed function(e.g., scalar, aggregation).
Ô Annotate the fact patternÔ Introduce representative vertex
Example� Average product rating byproduct category
� Minimum age of customersby nationality
$c
$r $p++
(Min. Age, $c@age,MIN)
(Avg. Rtg., $r@stars,AVG)
authors$a
rates$e
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 11
Operations: Roll-up, Drill-down, Slice & Dice
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 12
Roll-up/Drill-down
Granularity of the Cube� Represents the �grouping�: the current levels of interest
� Initially: the lowest level of each dimension
Roll-up� Reduces the granularity
� For dimension d, move up one level from li to li+1
Drill-down� Increases the granularity
� For dimension d, move down one level from li to li−1
Ô Introduce representative vertex for each groupÔ Expose computed values for measures as attributes
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 13
Roll-up/Drill-down
Granularity of the Cube� Represents the �grouping�: the current levels of interest
� Initially: the lowest level of each dimension
Roll-up� Reduces the granularity
� For dimension d, move up one level from li to li+1
Drill-down� Increases the granularity
� For dimension d, move down one level from li to li−1
Ô Introduce representative vertex for each groupÔ Expose computed values for measures as attributes
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 13
Roll-up/Drill-down
Granularity of the Cube� Represents the �grouping�: the current levels of interest
� Initially: the lowest level of each dimension
Roll-up� Reduces the granularity
� For dimension d, move up one level from li to li+1
Drill-down� Increases the granularity
� For dimension d, move down one level from li to li−1
Ô Introduce representative vertex for each groupÔ Expose computed values for measures as attributes
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 13
Slice & Dice
Function �lter transforms fact base of cube
� evaluates level-predicate pairs
� removes facts not matching the predicates
For a single predicate applied to one dimension Ô slice
Example
Slice product reviews by German customers from the cube c:filter(c, {(Nationality, λ = �DE�)}).
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 14
Slice & Dice
Function �lter transforms fact base of cube
� evaluates level-predicate pairs
� removes facts not matching the predicates
For a single predicate applied to one dimension Ô slice
Example
Slice product reviews by German customers from the cube c:filter(c, {(Nationality, λ = �DE�)}).
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 14
Challenges: Unbalanced Hierarchies & OLAP Anomalies
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 15
Unbalanced Hierarchies
Facts with di�erent granularities
Relative dimension speci�cation:
Product category:
$p-[@type='in']->
Product group:
$p-[@type='in']->-[@type='part-of']->
Product area:
$p-[@type='in']->-[@type='part-of']->(2)
Ô Absolute instead of relativedimension speci�cation required
Example
Products in categories andgroups
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5 �Phones�
6
�Computers &Accessories�
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part of
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© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 16
Unbalanced Hierarchies
Facts with di�erent granularities
Relative dimension speci�cation:
Product category:
$p-[@type='in']->
Product group:
$p-[@type='in']->-[@type='part-of']->
Product area:
$p-[@type='in']->-[@type='part-of']->(2)
Ô Absolute instead of relativedimension speci�cation required
Example
Products in categories andgroups
15red 16 GB
�Google Nexus 5�16black
�SamsungE1200�
4�Cell Phones& Accessories�
5 �Phones�
6
�Computers &Accessories�
7�Tablets�
12�Smartphones�in in
part of
part ofpart of
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 16
Unbalanced Hierarchies
Facts with di�erent granularities
Relative dimension speci�cation:
Product category:
$p-[@type='in']->
Product group:
$p-[@type='in']->-[@type='part-of']->
Product area:
$p-[@type='in']->-[@type='part-of']->(2)
Ô Absolute instead of relativedimension speci�cation required
Example
Products in categories andgroups
15red 16 GB
�Google Nexus 5�16black
�SamsungE1200�
4�Cell Phones& Accessories�
5 �Phones�
6
�Computers &Accessories�
7�Tablets�
12�Smartphones�in in
part of
part ofpart of
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 16
Unbalanced Hierarchies
Facts with di�erent granularities
Relative dimension speci�cation:
Product category:
$p-[@type='in']->
Product group:
$p-[@type='in']->-[@type='part-of']->
Product area:
$p-[@type='in']->-[@type='part-of']->(2)
Ô Absolute instead of relativedimension speci�cation required
Example
Products in categories andgroups
15red 16 GB
�Google Nexus 5�16black
�SamsungE1200�
4�Cell Phones& Accessories�
5 �Phones�
6
�Computers &Accessories�
7�Tablets�
12�Smartphones�in in
part of
part ofpart of
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 16
Unbalanced Hierarchies
Facts with di�erent granularities
Relative dimension speci�cation:
Product category:
$p-[@type='in']->
Product group:
$p-[@type='in']->-[@type='part-of']->
Product area:
$p-[@type='in']->-[@type='part-of']->(2)
Ô Absolute instead of relativedimension speci�cation required
Example
Products in categories andgroups
15red 16 GB
�Google Nexus 5�16black
�SamsungE1200�
4�Cell Phones& Accessories�
5 �Phones�
6
�Computers &Accessories�
7�Tablets�
12�Smartphones�in in
part of
part ofpart of
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 16
Unbalanced Hierarchies
Facts with di�erent granularities
Relative dimension speci�cation:
Product category:
$p-[@type='in']->
Product group:
$p-[@type='in']->-[@type='part-of']->
Product area:
$p-[@type='in']->-[@type='part-of']->(2)
Ô Absolute instead of relativedimension speci�cation required
Example
Products in categories andgroups
15red 16 GB
�Google Nexus 5�16black
�SamsungE1200�
4�Cell Phones& Accessories�
5 �Phones�
6
�Computers &Accessories�
7�Tablets�
12�Smartphones�in in
part of
part ofpart of
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 16
Unbalanced Hierarchies
Solution: Pre-process the graph
Data Cleansing� Balance hierarchies
� Add missing root nodes
Tagging� Add attributes for absolutereferencing
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�Google Nexus 5�16black
�SamsungE1200�
4�Cell Phones& Accessories�
5 �Phones�
6
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7�Tablets�
12�Smartphones�in in
part of
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© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 17
Unbalanced Hierarchies
Solution: Pre-process the graph
Data Cleansing� Balance hierarchies
� Add missing root nodes
Tagging� Add attributes for absolutereferencing
15red 16 GB
�Google Nexus 5�16black
�SamsungE1200�
4�Cell Phones& Accessories�
5 �Phones�
6
�Computers &Accessories�
7�Tablets�
12�Smartphones�in
part of
part ofpart of
13 �Dumbphones�
14�ConsumerElectronics�
in
part of
part ofpart of
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 17
Unbalanced Hierarchies
Solution: Pre-process the graph
Data Cleansing� Balance hierarchies
� Add missing root nodes
Tagging� Add attributes for absolutereferencing
15red 16 GB
�Google Nexus 5�16black
�SamsungE1200�
43 �Cell Phones& Accessories�
5
2
�Phones�
62
�Computers &Accessories�
7 1�Tablets�
12 1�Smartphones�in in
part of
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© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 17
OLAP Anomalies
It depends on the model:
� double counting can occur, if acardinality assumption isviolated (1:1 vs. 1:nrelationship)
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15128GB
�Apple iPad Air�
16 black
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inin in
� incompleteness can occur, if aconnectivity assumption isviolated
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15128GB
�Apple iPad Air�
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inin
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 18
OLAP Anomalies
It depends on the model:
� double counting can occur, if acardinality assumption isviolated (1:1 vs. 1:nrelationship)
7�Tablets�5 �Phones�
15128GB
�Apple iPad Air�
16 black
�SamsungE1200�
inin in
� incompleteness can occur, if aconnectivity assumption isviolated
7�Tablets�5 �Phones�
15128GB
�Apple iPad Air�
16 black
�SamsungE1200�
inin
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 18
Conclusion
Powerful Mapping of Multidimensional Analytics� Expose well-known concepts and operations
� Emphasize challenges posed by graph data
Ô Open up the graph world to Business Intelligence
Flexible Work�ow for the Big Graph Data Era� No up-front schema design
� Adapt to changing data and requirements
Ô What is a fact today can be a dimension tomorrow
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 19
1 Additional Material & References
References I
Chen Chen, Xifeng Yan, Feida Zhu, Jiawei Han, and Philip S. Yu.Graph OLAP: Towards Online Analytical Processing on Graphs.In Proceedings of the Eighth International Conference on Data Mining, pages 103�112, Pisa,Italy, December 2008. IEEE.
Microsoft Research.The Fourth Paradigm: Data-Intensive Scienti�c Discovery.Microsoft Press, 2009.
Marko A. Rodriguez and Peter Neubauer.Constructions from Dots and Lines.Bulletin of the American Society for Information Science and Technology, 36(6):35�41, 2010.
Yuanyuan Tian and Jignesh M. Patel.TALE: A Tool for Approximate Large Graph Matching.In 2008 IEEE 24th International Conference on Data Engineering, pages 963�972. IEEE, April2008.
Peixiang Zhao, Xiaolei Li, Dong Xin, and Jiawei Han.Graph Cube: On Warehousing and OLAP Multidimensional Networks.In Proceedings of the ACM SIGMOD International Conference on Management of Data, pages853�864, Athens, Greece, 2011. ACM.
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 2
References II
Ning Zhang, Yuanyuan Tian, and Jignesh M. Patel.Discovery-Driven Graph Summarization.In Proceedings of the 26th International Conference on Data Engineering, pages 880�891,Long Beach, CA, USA, 2010. IEEE.
© Michael Rudolf | SynopSys: Foundations for Multidimensional Graph Analytics | 3