A talk presented at WWW 2015 on how to label blank nodes in an RDF graph in a deterministic way. Applications include Skolemisation (mapping blank nodes to IRIs); detecting RDF graph isomorphism; as well as RDF graph hashing, signing, canonicalisation etc. (The slides make heavy use of animation and when flattened, they will not make much sense. Hence I recommend to open them as a slideshow in .ppt format.)

1. Skolemising Blank Nodes while
Preserving Isomorphism
Aidan Hogan – DCC, Universidad de Chile

2. WHY? BLANK NODES ARE GREAT!

3. When life gives you blank nodes …

4. Blank Nodes are glue!

5. Blank Nodes names aren’t important …
(Isomorphic)

6. Blank nodes are common in real-world data …
Aidan Hogan, Marcelo Arenas, Alejandro Mallea and Axel Polleres
"Everything You Always Wanted to Know About Blank Nodes".
Journal of Web Semantics 27: pp. 42–69, 2014

7. BLANK NODES ENABLE SYNTAX SHORTCUTS
They represent implicit nodes in the graph
They help specify order, higher-arity relations, reification, etc., succinctly
They are common in real-world data

8. BLANK NODES:
WHAT’S THE PROBLEM?

9. Are two RDF graphs isomorphic?

10. Are two RDF graphs isomorphic?

11. RDF ISOMORPHISM IS GI-COMPLETE
A general algorithm to see if two RDF graphs are the “same” will
(probably) not be tractable

12. BLANK NODES ADD COMPLEXITY?
WHAT TO DO?

13. RDF 1.1 proposes Skolemisation

14. But fresh IRIs every time is not ideal

15. But fresh IRIs every time is not ideal

16. Would prefer a “consistent” labelling

17. Would prefer a “consistent” labelling

18. Compute isomorphically-unique graph hash

19. Finding duplicate documents from a crawler

20. CANONICAL LABELLING USEFUL FOR:
1. Mapping blank nodes to IRIs
2. Computing unique hashes for RDF graphs

21. OLD BUT RECURRING QUESTION

22. An old question that won’t go away …
Jeremy J. Carroll. “Signing RDF Graphs.” ISWC 2003.
Edzard Höfig, Ina Schieferdecker. “Hashing of RDF Graphs
and a Solution to the Blank Node Problem.” URSW 2014.

23. NO EXISTING APPROACH IS GENERAL
• Hard cases seem unlikely in practice
• Let’s build a general (and thus worst-case exponential) algorithm
that’s efficient for practical cases

24. NAÏVE CANONICAL LABELLING SCHEME

25. (Naïve) Canonical labels for blank nodes

26. But wait … what happens if ... ?

27. Or another case …

28. Or another case …

29. Or another case …

30. Fixpoint does not distinguish all blank nodes!

31. NAÏVE: COLOUR BLANK NODES RECURSIVELY
UNTIL FIXPOINT
• Efficient
• Incomplete

32. CANONICAL LABELLING SCHEME:
ALWAYS DISTINGUISH ALL BLANK NODES
Brendan D. McKay. "Practical graph isomorphism". Congressus Numerantium 30: pp. 45–87, 1981.

33. Start with a (non-distinguished) colouring …

34. Let’s distinguish a node …

35. Let’s distinguish a node …

36. Colouring is no longer a fixpoint!

37. Rerun colouring to fixpoint

38. Rerun colouring to fixpoint

39. Rerun colouring to fixpoint

40. Rerun colouring to fixpoint

41. Fixpoint reached: still not finished!

42. So again let’s distinguish another …

43. … and rerun colouring to fixpoint

44. … and rerun colouring to fixpoint

45. … and rerun colouring to fixpoint

46. … and rerun colouring to fixpoint

47. … and rerun colouring to fixpoint

48. … and rerun colouring to fixpoint

49. Now all blank nodes are distinguished!

50. Blank node labels computed from colour

51. Let’s go back: first, why pick _:a and _:c?

52. Okay so: why _:a …

53. Adapt ideas from the Nauty algorithm
(for standard graph isomorphism)

54. Adapt ideas from the Nauty algorithm
(for standard graph isomorphism)

55. Check all leafs for minimum graph

56. What happened?

57. What happened?

58. What happened?

59. Automorphisms cause repetitions

60. CORE ALGORITHM: FIND MINIMAL GRAPH
FOLLOWING FIXED COLOURING RULES
• Complete
• Efficient for many cases?

61. OKAY … SO WHAT HASHING TO USE?

62. What about hash collisions?
128 bit: MD5, Murmur3_128
160 bit: SHA1

63. HASHING MAY LEAD TO COLLISIONS
• Don’t care what hashing you want to use
• 128-bit hash shortest hash with acceptable collision probability
• For cryptographic use-cases, SHA-256 or better might be needed

64. EVALUATION

65. Evaluation: Real-world Graphs

66. Evaluation: Nasty Synthetic Graphs

67. CONCLUSIONS

68. In loving memory of
Linked Data
2007–2012
Survived by its research
community
_:b
1999–2015

69. Conclusions

70. Aside: Why GI-Hard?

71. Aside: Why GI-Hard?
(Can Encode Graph Isomorphism as RDF Isomorphism)
if and only if

72. Aside: Why GI-Complete?
(Can we encode RDF isomorphism as graph isomorphism?)
if and only if
?
?

73. Aside: Why GI-Complete?
(Yes: We can encode RDF isomorphism as graph isomorphism)

74. Aside: Why GI-Complete?
(Yes: We can encode RDF isomorphism as graph isomorphism)
if and only if

75. COMPLETE CANONICAL LABELLING SCHEME

76. A complete canonical labelling?

77. Find a canonical labelling for H

78. Choose the lowest possible graph

79. COMPLETE: FIND MINIMUM POSSIBLE
GRAPH USING FIXED BLANK NODE LABELS
• Complete
• Inefficient

80. The need for a graph-level hash

81. OPTIMISATION: PRUNE THE TREE USING
AUTOMORPHISMS

82. Trim the search tree
using “found” automorphisms
Found Automorphisms …

83. PRUNING PER AUTOMORPHISMS AVOIDS
SYMMETRIC REPETITIONS
• Automorphisms are found naturally
• Makes very “regular” structures (like cliques) a lot easier
• Need to be careful how to manage the automorphism group