The document discusses the application of graph algebra in use cases such as real-time recommendation and fraud detection, emphasizing efficient algorithms and implementations like RedisGraph. It illustrates concepts like graph traversal methods and matrix operations in the context of graph theory, including age filtering in social networks to find connections. The document also presents performance benchmarks comparing various graph databases for node insertion and query processing times.

1. Graph AlgebraGraph Algebra

2. !2

3. Use casesUse cases

4. Realtime
Recommendation

5. Fraud detection

6. Navigation & Traffic
management
Navigation & Traffic
management

7. RedisGraph

8. ImplementationImplementation

9. Adjacency list

10. Adjacency matrix
0 0 1
1 0 1
0 0 0

11. Graph Algorithms in the
language of Linear Algebra

12. !12
readth-first search example
A(i, j) = 1 for edge (j, i)
A is binary; dot (.) is zero for clarity
. . . 1 . . .
1 . . . . . .
. . . 1 . 1 1
1 . . . . . 1
. 1 . . . . 1
. . 1 . 1 . .
. 1 . . . . .
A(i, j) = 1 for edge (
A is binary; dot (.) is
. . . 1 . . .
1 . . . . . .
. . . 1 . 1 1
1 . . . . . 1
. 1 . . . . 1
. . 1 . 1 . .
. 1 . . . . .

13. !13
h-first search: initializations
v = zeros (n,1) ; // result
q = false (n,1) ; // current level
q (source) = true ;
v: q:
. .
. .
. .
. 1
. .
. .
. .
readth-first search example
A(i, j) = 1 for edge (j, i)
A is binary; dot (.) is zero for clarity
. . . 1 . . .
1 . . . . . .
. . . 1 . 1 1
1 . . . . . 1
. 1 . . . . 1
. . 1 . 1 . .
. 1 . . . . .

14. !14
GrB assign (v, q, NULL, level, GrB ALL, n, NULL)
v <q> = level ; // assig
v: q:
. .
. .
. .
1 1
. .
. .
. .
gn (v, q, NULL, level, GrB ALL, n, NULL)
v <q> = level ; // assign level
v: q:
. .
. .
. .
1 1
. .
. .
. .

15. !15
GrB mxv (q, v, NULL, GxB LOR LAND BOOL, A, q, desc)
first part of q<!>=A*q:
t = A*q ;
xv (q, v, NULL, GxB LOR LAND BOOL, A, q, desc)
first part of q<!>=A*q:
t = A*q ;

16. !16
GrB mxv (q, v, NULL, GxB LOR LAND BOOL, A, q, desc)
second part of q<!>=A*q:
q = false (n,1) ;
q <!v> = t ;
v: t=A*q: q<!v>=t
. 1 1
. . .
. 1 1
1 . .
. . .
. . .
. . .
v (q, v, NULL, GxB LOR LAND BOOL, A, q, desc)
second part of q<!>=A*q:
q = false (n,1) ;
q <!v> = t ;
v: t=A*q: q<!v>=t
. 1 1
. . .
. 1 1
1 . .
. . .
. . .
. . .

17. gn (v, q, NULL, level, GrB ALL, n, NULL)
v <q> = level ; // assign level
v: q:
2 1
. .
2 1
1 .
. .
. .
. .
GrB assign (v, q, NULL, level, GrB ALL, n, NULL)
v <q> = level ; // a
v: q:
2 1
. .
2 1
1 .
. .
. .
. .

18. GrB mxv (q, v, NULL, GxB LOR LAND BOOL, A, q, desc
first part of q<!>=A*q:
t = A*q ;
q, v, NULL, GxB LOR LAND BOOL, A, q, desc)
first part of q<!>=A*q:
t = A*q ;

19. Binary matrix
1 bit per matrix cell
1,000,000 X 1,000,000
One 10^12 (trillion) bits = 125GB
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
……………………………………………………….
!19

20. Real world graphs
Most real world graphs are sparse
Facebook
2 billion users
338 friends for user on average
2,000,000,000 * 338 / 2,000,000,000^2
0.000000169% utilisation
!20

21. Sparse matrixSparse matrix

22. Graph BLAS
Tim Davis John GilbertJeremy Kepner

23. GrB_Info GrB_mxm
(
GrB_Matrix C,
const GrB_Matrix Mask,
const GrB_BinaryOp accum,
const GrB_Semiring semiring,
const GrB_Matrix A,
const GrB_Matrix B,
const GrB_Descriptor desc
);

24. RedisGraph
+
GraphBLAS

25. Friend of a friend
MATCH (src)-[:friend]->(f)-[:friend]->(foaf)
WHERE src.age > 30
RETURN foaf
src f foaf
friend friend
!25

26. Friend of a friend
MATCH (src)-[:friend]->(f)-[:friend]->(foaf)
WHERE src.age > 30
RETURN foaf
src f foaf
friend friend
!26

27. Friend of a friend
MATCH (src)-[:friend]->(f)-[:friend]->(foaf)
WHERE src.age > 30
RETURN foaf
src f foaf
friend friend
!27

28. Friend of a friend
MATCH (src)-[:friend]->(f)-[:friend]->(foaf)
WHERE src.age > 30
RETURN foaf
src f foaf
friend friend
!28

29. Execution plan
Index scan
Expand
Expand
Project
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf
!29

30. Execution plan
Index scan
Expand
Expand
Project
!30
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf

31. Execution plan
Index scan
Expand
Expand
Project
!31
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf

32. Execution plan
Index scan
Expand
Expand
Project
!32
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf

33. Execution plan
Index scan
Expand
Expand
Project
!33
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf

34. Execution plan
Index scan
Expand
Expand
Project
!34
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf

35. Execution plan
Index scan
Expand
Expand
Project
!35
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf

36. Execution plan
Index scan
Expand
Expand
Project
!36
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf

37. Execution plan
Index scan
Expand
Expand
Project
!37
src.age > 30
(src)-[:follows]->(f)
(f)-[:follows]->(foaf)
RETURN foaf

38. Cypher to
Algebraic expressions

39. MATCH
(src)-[:friend]->(f)-[:friend]->(foaf)
WHERE src.age > 30
RETURN foaf
=
Age_Filter * Friendship * Friendship
!39

40. 0 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 1
Age Filter
0 1 0 0 1 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
1 0 1 0 0 0
0 1 0 1 0 0
Friendships
0 1 0 0 1 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
1 0 1 0 0 0
0 1 0 1 0 0
Friendships
* *
!40

41. Matrix multiplication
is associative
(A*B)*C = A*(B*C)
!41

42. 0 1 0 0 1 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
1 0 1 0 0 0
0 1 0 1 0 0
Friendships
0 1 0 0 1 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
1 0 1 0 0 0
0 1 0 1 0 0
Friendships
*
1 0 1 0 0 0
1 0 0 0 1 1
1 1 1 1 1 0
1 0 1 0 0 0
1 1 0 0 1 1
0 0 1 0 1 0
Friendships ^2
=
number of none zero values = 18
!42

43. Age Filter
0 1 0 0 1 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
1 0 1 0 0 0
0 1 0 1 0 0
Friendships
0 0 0 0 0 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
0 0 0 0 0 0
0 1 0 1 0 0
Filtered friendships
src > 30
* =
number of none zero values = 7
0 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 1
!43

44. 0 1 0 0 1 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
1 0 1 0 0 0
0 1 0 1 0 0
Friendships
0 0 0 0 0 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
0 0 0 0 0 0
0 1 0 1 0 0
Filtered friendships
src > 30
* =
0 0 0 0 0 0
1 0 0 0 1 1
1 1 1 1 1 0
1 0 1 0 0 0
0 0 0 0 0 0
0 0 1 0 1 0
FOF
!44

45. 0 0 0 0 0 0
1 0 0 0 1 1
1 1 1 1 1 0
1 0 1 0 0 0
0 0 0 0 0 0
0 0 1 0 1 0
1
5
4
2
63
!45

46. 0 0 0 0 0 0
1 0 0 0 1 1
1 1 1 1 1 0
1 0 1 0 0 0
0 0 0 0 0 0
0 0 1 0 1 0
1
5
4
2
63
!46

47. Variable length
MATCH (src)-[:friend*2..4]->(foaf)
WHERE src.age > 30
RETURN foaf
src F2 foaf
friend
F3 F4
!47

48. MATCH (src)-[:friend*2..4]->(foaf)
WHERE src.age > 30
RETURN foaf
=
Age_Filter * (Friendship^2 + Friendship^3 + Friendship^4)
M = AF;
R = 0;
For i=0; i < 3; i++
M = M*F
R = R+M
!48

49. bidirectional edge
MATCH (A)-[:follows]-(B)
RETURN A, B
A B
Knows
!49

50. Follows + Follows^T
!50

51. 0 1 0 0 1 0
0 0 1 0 0 0
1 0 0 0 1 1
0 0 0 0 1 0
1 0 1 0 0 0
0 1 0 1 0 0
Follows
0 0 1 0 1 0
1 0 0 0 0 1
0 1 0 0 1 0
0 0 0 0 0 1
1 0 1 1 0 0
0 0 1 0 0 0
Follows^T
0 1 1 0 1 0
1 0 1 0 0 1
1 1 0 0 1 1
0 0 0 0 1 1
1 0 1 1 0 0
0 1 1 1 0 0
Symmetric Follows
+ *
!51

52. 1
5
4
2
63
0 1 1 0 1 0
1 0 1 0 0 1
1 1 0 0 1 1
0 0 0 0 1 1
1 0 1 1 0 0
0 1 1 1 0 0
!52

53. Graph distribution
Block multiplication
A*B=C
A B C
A1
A3 A4
A2 B1 B2
B4B3
C1 C2
C3 C4
!53

54. Graph distribution
Block multiplication
A*B=C
A B C
A1
A3 A4
A2 B1 B2
B4B3
A1*B1+
A2*B3
A1*B2+
A2*B4
A3*B1+
A4*B3
A3*B2+
A4*B4
!54

55. BenchmarksBenchmarks

56. Massive Insertion Workload
!56
~1M nodes, ~3M edges
RedisGraph
Titan
OrientDB
Neo4j
Insert time in seconds
0 75 150 225
24.69
252.15
104.27
0.53

57. Find Neighbours
!57
~1M nodes, ~3M edges
RedisGraph
Titan
OrientDB
Neo4j
Query processing time in seconds
0 7.5 15 22.5
4.51
9.34
20.71
0.05

58. Find Adjacent Nodes
!58
~1M nodes, ~3M edges
RedisGraph
Titan
OrientDB
Neo4j
Query processing time in seconds
0 12.5 25 37.5
1.46
6.15
42.82
0.05

59. Shortest Path
!59
~1M nodes, ~3M edges
RedisGraph
Titan
OrientDB
Neo4j
Query processing time in seconds
0 7.5 15 22.5
0.08
23.47
24.87
0.001

60. Thanks
@roilipman