1. 2004-11-04 1 F. Olken, SC32WG2 Graph Theoretic Characterization of Metadata Structures Frank Olken, Bruce Bargmeyer, Kevin D. Keck Lawrence Berkeley National Laboratory Computational Research Division presented to ISO JTC1 SC32 WG2 Mtg. Washington, DC Nov. 8, 2004

2. 2004-11-04 2 F. Olken, SC32WG2 Our Goal ● Create a taxonomy of metadata structures: – Terminologies, Taxonomies, partonomies, ontologies,... ● Taxonomy will be framed in graph theoretic terms: – What kind of graph would be used to represent a given metadata structure

3. 2004-11-04 3 F. Olken, SC32WG2 Why use a graph theoretic characterization? ● Readily comprehensible characterization of metadata structures ● Graph structure has implications for: – Integrity Constraint Enforcement – Data structures – Query languages – Combining metadata sets – Algorithms for query processing

4. 2004-11-04 4 F. Olken, SC32WG2 Definition of a graph ● Graph = vertex (node) set + edge set ● Nodes, edges may be labeled ● Edge set = binary relation over nodes – cf. NIAM ● Labeled edge set – RDF triples (subject, predicate, object) – predicate = edge label ● Typically edges are directed

5. 2004-11-04 5 F. Olken, SC32WG2 Example of a graph infectious disease influenza measles is-a

6. 2004-11-04 6 F. Olken, SC32WG2 Types of Metadata Graph Structures ● Trees ● Partially Ordered Trees ● Ordered Trees ● Faceted Classifications ● Directed Acyclic Graphs ● Partially Ordered Graphs ● Lattices ● Bipartite Graphs ● Directed Graphs ● Cliques ● Compound Graphs

7. 2004-11-04 7 F. Olken, SC32WG2 Graph Taxonomy Directed Graph Directed Acyclic Graph Graph Undirected Graph Bipartite Graph Partial Order Graph Faceted Classification Clique Partial Order Tree Tree Lattice Ordered Tree Note: not all bipartite graphs are undirected.

8. 2004-11-04 8 F. Olken, SC32WG2 Trees ● In metadata settings trees are almost invariably directed – edges indicate direction ● In metadata settings trees are usually partial orders – Transtivity is implied (see next slide) – Not true for some trees with mixed edge types. – Not always true for all partonomies

9. 2004-11-04 9 F. Olken, SC32WG2 Example: Tree Oakland Berkeley Alameda County California part-of Santa Clara San Jose Santa Clara County part-of

10. 2004-11-04 10 F. Olken, SC32WG2 Trees - cont. ● Uniform vs. non-uniform height subtrees ● Uniform height subtrees – fixed number of levels – common in dimensions of multi-dimensional data models ● Non-uniform height subtrees – common terminologies

11. 2004-11-04 11 F. Olken, SC32WG2 Partially Ordered Trees ● A conventional directed tree ● Plus, assumption of transitivity ● Usually only show immediate ancestors (transitive reduction) ● Edges of transitive closure are implied ● Classic Example: – Simple Taxonomy, “is-a” relationship

12. 2004-11-04 12 F. Olken, SC32WG2 Example: Partial Order Tree polio smallpox Infectious Disease Disease is-a diabetes heart disease chronic disease is-a Signifies inferred is-a relationship

13. 2004-11-04 13 F. Olken, SC32WG2 Ordered Trees ● Order here refers to order among sibling nodes (not related to partial order discussed elsewhere) ● XML documents are ordered trees – Ordering of “sub-elements” is to support classic linear encoding of documents

14. 2004-11-04 14 F. Olken, SC32WG2 Example: Ordered Tree part-of Paper Title page Section Bibliography Note: implicit ordering relation among parts of paper.

15. 2004-11-04 15 F. Olken, SC32WG2 Faceted Classification ● Classification scheme has mulitple facets ● Each facet = partial order tree ● Categories = conjunction of facet values (often written as [facet1, facet2, facet3]) ● Faceted classification = a simplified partial order graph ● Introduced by Ranganathan in 19 th century, as Colon Classification scheme ● Faceted classification can be descirbed with Description Logc, e.g., OWL-DL

16. 2004-11-04 16 F. Olken, SC32WG2 Example: Faceted Classification Wheeled Vehicle Facet 2 wheeled 4 wheeled 3 wheeled is-a Internal Combustion Human Powered Vehicle Propulsion Facet Bicycle TricycleMotorcycle Auto is-a is- is-a

17. 2004-11-04 17 F. Olken, SC32WG2 Faceted Classifications and Multi- dimensional Data Model ● MDM – a.k.a. OLAP data model – Online Analytical Processing data model – Star / Snowflake schemas ● Fact Tables – fact = function over Cartesian product of dimensions – dimensions = facets ● geographic region, product category, year,...

18. 2004-11-04 18 F. Olken, SC32WG2 Directed Acylic Graphs ● Graph: – Directed edges – No cycles – No assumptions about transitivity (e.g., mixed edge types, some partonomies) – Nodes may have multiple parents ● Examples: – Partonomies (“part-of”) - transitivity is not always true

19. 2004-11-04 19 F. Olken, SC32WG2 Example: Directed Acyclic Graph Propelled Vehicle Wheeled Vehicle 2 wheeled vehicle 4 wheeled vehicle 3 wheeled vehicle is-a Internal Combustion Vehicle Human Powered Vehicle Bicycle Tricycle Motorcycle Auto is-a is- is-a Vehicle is-a Observe: multiple inheritance.

20. 2004-11-04 20 F. Olken, SC32WG2 Partial Order Graphs ● Directed acyclic graphs + inferred transitivity ● Nodes may have multiple parents ● Most taxonomies drawn as transitive reduction, transitive closure edges are implied. ● Examples: – all taxonomies – most partonomies – multiple inheritance ● POGs can be described in Description Logic, e.g., OWL-DL

21. 2004-11-04 21 F. Olken, SC32WG2 Example: Partial Order Graph Propelled Vehicle Wheeled Vehicle 2 wheeled vehicle 4 wheeled vehicle 3 wheeled vehicle is-a Internal Combustion Vehicle Human Powered Vehicle Bicycle TricycleMotorcycle Auto is-a is- is-a Vehicle is-a Dashed line = inferred is-a (transitive closure)

22. 2004-11-04 22 F. Olken, SC32WG2 Directed Graph ● Generalization of DAG (directed acyclic graph) ● Cycles are allowed ● Arises when many edge types allowed ● Example: UMLS

23. 2004-11-04 23 F. Olken, SC32WG2 Lattices ● A partial order ● For every pair of elements A and B – There exists a least upper bound – There exists a greatest lower bound ● Example: – The power set (all possible subsets) of a finite set – LUB(A,B) = union of two sets A, B – GLB(A,B) = intersect of two sets A,B

24. 2004-11-04 24 F. Olken, SC32WG2 Example Lattice: Powerset of 3 element set empty set = {}empty set = {} {a,b,c} {c} {a} {b} {b,c}{a,b} {a,c} Denotes subset

25. 2004-11-04 25 F. Olken, SC32WG2 Lattices - Applications ● Formal Concept Analysis – synthesizing taxonomies ● Machine Learning – concept learning

26. 2004-11-04 26 F. Olken, SC32WG2 Bipartite Graphs ● Vertices = two disjoint sets, V and W ● All edges connect one vertex from V and one vertex from W ● Examples: – mappings among value representations – mappings among schemas – (entity/attribute, relationship) nodes in Conceptual Graphs

27. 2004-11-04 27 F. Olken, SC32WG2 Example Bipartite Graph California Massachusetts Oregon CA MA OR States Two-letter state codes

28. 2004-11-04 28 F. Olken, SC32WG2 Clique ● Clique = complete graph (or subgraph) – all possible edges are present ● Used to represent equivalence classes ● Typically, on undirected graphs

29. 2004-11-04 29 F. Olken, SC32WG2 Example of Clique California CA Calif. CAL Here edges denote synonymy.

30. 2004-11-04 30 F. Olken, SC32WG2 Compound Graphs ● Edges can point to/from subgraphs, not just simple nodes ● Used in conceptual graphs ● CG is isomorphic to First Order Logic ● Could be used to specify contexts for subgraphs

31. 2004-11-04 31 F. Olken, SC32WG2 Example Compound Graph Colin Powell claimed Iraq had WMDs

32. 2004-11-04 32 F. Olken, SC32WG2 Conclusions ● We can characterize metadata structure in terms of graph structures ● Partial Order Graphs are the most common structure: – used for taxonomies, partonomies – support multiple inheritance, faceted classification – implicit inclusion of inferred transitive closure edges

33. 2004-11-04 33 F. Olken, SC32WG2 Additional Resources ● Faceted classification – http://www.kmconnection.com/DOC100100.htm http://www.kmconnection.com/DOC100100.htm ● Lattices – http://www.rci.rutgers.edu/~cfs/472_html/Learn/SimpleLattice_472.html http://www.rci.rutgers.edu/~cfs/472_html/Learn/SimpleLattice_472.html

34. 2004-11-04 34 F. Olken, SC32WG2 Contact Information ● Frank Olken – olken@lbl.gov, 510-486-5891 olken@lbl.gov – http://www.lbl.gov/~olken http://www.lbl.gov/~olken ● XMDR Project – http://hpcrd.lbl.gov/SDM/XMDR/ http://hpcrd.lbl.gov/SDM/XMDR/ ● Bruce Bargmeyer – bebargmeyer@lbl.gov bebargmeyer@lbl.gov ● Kevin D. Keck – mailto:kdkeck@lbl.gov mailto:kdkeck@lbl.gov