Talk given at neo4j conference "Graph Connect" - discussing some graph theory (old and new), and why knowing your stuff can come in handy on a software project.

1. G RA P H T H E O RY I N
P RACT I S E
D A V I D S I M O N S
@ S W A M W I T H T U R T L E S

2. W H O A M I ?
• David Simons
• @SwamWithTurtles
• github.com/
SwamWithTurtles
• Technical Lead at Softwire
and part-time hacker
• Statistician in a past life

3. T O S E E D ATA D O N E R I G H T
M Y PA S S I O N …

4. W H AT I S D ATA
D O N E R I G H T ?
• Choosing the right
database;
• Using the right
mathematical and
statistical techniques to
leverage its power

5. S Q L
• SQL has had 40 years of
academic set theory
applied to it…
• Let’s do the same with
neo4j!

6. T O D AY…
• Concepts in Graph Theory
• Theory;
• Use Cases;
• Implementation Details
• Reward: What shape is
the internet?

7. W H AT I S A G R A P H ?
G R A P H T H E O RY

8. W H AT I S A G R A P H ?
Taken from Jim Webber’s Dr. Who Dataset

9. W H AT I S A G R A P H ?
{ (V, E) : V = [n], E ⊆ V(2) }

10. W H AT I S A G R A P H ?
{ (V, E) : V = [n], E ⊆ V(2) }
Made up of two parts,
“V” and “E”

11. W H AT I S A G R A P H ?
{ (V, E) : V = [n], E ⊆ V(2) }
V is a set of n items

12. W H AT I S A G R A P H ?
Vertex Set

13. W H AT I S A G R A P H ?
{ (V, E) : V = [n], E ⊆ V(2) }
E is made up of pairs
of elements of V
(Ordered and
not necessarily distinct)

14. W H AT I S A G R A P H ?
Edge Set

15. G I V I N G R E A L
W O R L D
M E A N I N G S T O V
A N D E
W H A T I S G R A P H I C A L
M O D E L L I N G ?

16. B R I D G E S AT K Ö N I G S B E R G

17. B R I D G E S AT K Ö N I G S B E R G
V =
bits of land
E =
bridges

18. E L E C T I O N D ATA

19. E L E C T I O N D ATA

20. E L E C T I O N D ATA
E =
(e.g.) member of, held in,
stood in…
V =
elections, constituencies,
years, politicians and parties

21. W H E R E D O E S
N E O 4 J F I T I N ?
• Stores both the vertex set
and the edge set as first
class objects:
• Queryable
• Can store properties
• “Typed”

22. W H Y L E A R N
T H E T H E O RY ?
• Tells us what we can do
• Let’s us utilise many years
of academics
• Gives us a common
language

23. C A S E S T U D Y
T H E B R E A K D O W N …

24. T H E B R I T I S H
I S L E S
A G R A P H O F

25. W H AT I S A G R A P H ?
{ (V, E) : V = [n], E ⊆ V(2) }

26. W H AT I S A G R A P H ?
{ (V, E) :
V = Places of Interest,
E = Places that are connected}

27. T H E B R I T I S H I S L E S
L O N D O N
L A N D ’ S
E N D
O X F O R D
Y O R K
S T.
I V E S

28. T H E B R I T I S H I S L E S
L O N D O N
L A N D ’ S
E N D
O X F O R D
Y O R K
S T.
I V E S

29. P L A N A R I T Y
• A planar graph is one that
can be drawn on paper
with its edges crossing
• There are easy theories
that tell you when a graph
is planar
• Used for planning
construction of roads

30. C O N N E C T I V I T Y
• A graph is connected if
there is a path between
any two points
• A graph is k-connected if
you need to remove at
least k vertices to stop it
being connected
• Used for infrastructure
robustness studies

31. S PA N N I N G
T R E E
• A tree is a graph with no
loops
• A spanning tree is a
graph with tree with every
vertex connected
• Ensure resources flow
through a network

32. C O L O U R I N G
G R A P H T H E O RY

33. W E L I K E T H E
S I M P L E T H I N G S
I N L I F E
M A T H E M A T I C I A N S …

34. C O L O U R I N G
I N …
M A T H E M A T I C I A N S …

35. C O L O U R I N G
I N …
• Take your graph (V, E)
• Vertex Colouring
• Assign every vertex a
colour such that no two
adjacent vertices have
the same colour.

36. T H AT ’ S A L L V E RY
W E L L …

37. O R G A N I S I N G
S P O R T S
T O U R N A M E N T S
W H Y ?

38. O R G A N I S I N G
S P O R T S
T O U R N A M E N T S
• Graph Model
• V = all matches that
must be played
• E = a team is the same
across two matches
• Two vertices the same
colour => they can be
played simultaneously

39. O R G A N I S I N G S P O R T S T O U R N A M E N T S

40. O R G A N I S I N G S P O R T S T O U R N A M E N T S

41. O T H E R
U S E S …
• Mobile Phone Tower
frequency assignment
• V = mobile phone
towers
• E = towers so close
their waves will
interfere
• Colours = frequencies

42. O T H E R
U S E S …
• Solving SuDokus
• V = Squares on a
SuDoku grid
• E = Knowledge that
they must be different
numbers
• Colours = numbers 1
to 9

43. O T H E R U S E S …
http://watch.neo4j.org/video/74870401
Avoiding Deadlocks in Neo4j on Z-Platform

44. N O J AVA F R A M E W O R K …
Y E T !

45. R A N D O M G R A P H S
G R A P H T H E O RY

46. R A N D O M N E S S
S E E M S C A RY…
B U T WA I T…

47. R A N D O M N E S S
S E E M S C A RY…
• It can be!
• Someone should do a
talk about that…
• https://
www.youtube.com/
watch?v=rV9dqR0P0lQ

48. A graph with a fixed number of vertices, whose
edges are generated non-deterministically

49. U S E C A S E S
R A N D O M G R A P H S S T I L L H A V E …

50. S T U B B E D T E S T
D ATA
U S E C A S E S

51. S T U B B E D
T E S T D ATA
• Suppose you have a
method that coloured the
vertices of a graph…
• How could you test that?

52. S T U B B E D
T E S T D ATA
S T U B B E D D ATA S E T
A P P LY M E T H O D
A S S E RT T H AT:
* E V E RY N O D E H A S A
C O L O U R
* N O T W O A D J A C E N T
N O D E S S H A R E A
C O L O U R

53. S T U B B E D
T E S T D ATA
R A N D O M LY
G E N E R AT E D D ATA S E T
A P P LY M E T H O D
A S S E RT T H AT:
* E V E RY N O D E H A S A
C O L O U R
* N O T W O A D J A C E N T
N O D E S S H A R E A
C O L O U R

54. S I M U L AT I O N
A L G O R I T H M S
U S E C A S E S

55. - N A S D A Q . C O M
“solving a problem by performing a large number
of trail runs… and inferring a solution from the
collective results of the trial runs.”

56. W H Y
S I M U L AT I O N ?
• Modelling underlying
randomness
• Underlying question is
impossible (or hard) to
solve
• Trying to model something
of which we cannot have
full knowledge

57. A N D …
• It’s possible to use
randomness and always be
correct
• cf. ‘Probabilistic
Combinatorics’ by Paul
Erdős

58. H O W C A N W E
A C C O M P L I S H I T I N N E O 4 J ?

59. D I Y
I N T H E O RY …

60. D I Y

61. G R A P H A W A R E
I N P R A C T I S E …

62. G R A P H A W A R E
• “#1 Neo4j Consultancy”
• Open-sourced a lot of
projects under GPL3
including:
• TimeTree
• Reco
• Algorithms

63. G R A P H A W A R E

64. G R A P H A W A R E

65. A graph with a fixed number of vertices, whose
edges are generated non-deterministically

66. E R D Ő S - R E N Y I
• Take a graph with n vertices;
• For each pair of vertices, randomly connect them with
probability p

67. E R D Ő S - R E N Y I

68. I WA N T T O M O D E L
D ATA A B O U T
K E V I N B A C O N
B U T …

69. I WA N T T O M O D E L
D ATA A B O U T
S P R E A D O F H I V
B U T …

70. I WA N T T O M O D E L
D ATA A B O U T
S C A L E F R E E
N E T W O R K S
B U T …

71. S C A L E F R E E
N E T W O R K S
• As the system grows, we
have:
• A small number of
highly connected
hubs
• A large number of
sparsely connected
nodes

72. S C A L E F R E E N E T W O R K S
H U B S S PA R S E N O D E S
A C T O R
C O W O R K E R S
Blockbuster stars,
like Kevin Bacon
Drama college graduate
#1828, #1829, #1830…
S P R E A D O F
H I V
Patriarchs
Less privileged society
members
C H E M I C A L
R E A C T I O N S
Catalysts Inert Chemicals

73. S C A L E F R E E N E T W O R K S

74. B A R A B A S I - A L B E R T
• Take a graph with 2 (connected) vertices
• Add vertices one at a time such that it is more likely to
add vertices to a node that is already connected
• Repeat until you have n vertices

75. B A R A B A S I - A L B E R T

76. Y O U R R E WA R D
R E M E M B E R …

77. I WA N T T O M O D E L
D ATA A B O U T
T H E I N T E R N E T
B U T …

78. O V E R V I E W
• Looking at graph theory
can give us a common
language
• Utilising techniques means
we don’t have to solve
problems from scratch each
time (e.g. colouring,
simulation)
• The internet looks like
Kevin Bacon’s career

79. A N Y Q U E ST I O N S ?
@ S W A M W I T H T U R T L E S
S W A M W I T H T U R T L E S . C O M