This document describes a new algorithm called GRAph ALigner (GRAAL) for topological network alignment. GRAAL aims to provide a unique global alignment of nodes between two networks based solely on topological similarity, without using other a priori information. It works by first finding a dense core between the networks to seed the alignment, then expanding outwards in spheres of increasing radius to align additional nodes. The alignment score is based on the percentage of edges from the first network that are correctly aligned to edges in the second network. The algorithm is tested on sample networks and achieves an edge correctness of 0.089.

1. Topological network alignment
20131216
Statistics journal

3. Result
G
H
G(V, E)
H(U, F)
EC = 0.089

4. Motivation
large-scale networks such as interactome
Yeast
Human
Are two networks the same or similar?

5. Theoretical background
Network or Graph
Collection of nodes (vertex) and connections between them (edges).
Biology, social communication, and web pages

6. Theoretical background
G
H
G(V, E)
H(U, F)

7. Theoretical background
Graph comparison
Subgraph isomorphism
Is G an exact subgraph of H?
NP-complete
Efficient algorithms are not known.
G
H
G(V, E)
Graph alignment
Fitting G into H
Edge correctness (EC): the % of E aligned to F
NP-hard
H(U, F)

8. Previous approaches
Local alignment : ambiguous, different pairing
Mapping are chosen independently for local regions of similarity.
PathBLAST : homology information
NetworkBLAST : conserved protein clusters with likelihood method
MaWISh : evolution (sequence alignment)
GRAEMLIN : dense conserved subgraph with phylogeny
Global alignment
Provide unique alignment from each node in smaller graph to
exactly one node in larger graph
ISORANK : maximize overall match
GRAEMLIN : training from known graph alignments and phylogeny

9. New approaches
Never use a priori information
Sequence, Homology, Clusters, Phylogeny ,and Known alignments
Topological similarity
Orbit, graphlet, and signature similarity
Of course, a priori information can be used.
そう、GRAAL ならね

10. n-node graphlet and automorphism orbits

11. n-node graphlet and automorphism orbits
orbit
Topologically
relevant
graphlet
Topologically
relevant
Topologically
relevant

12. Graphlet Degree Vector

13. Graphlet Degree Vector

14. Graphlet Degree Vector

15. Graphlet Degree Vector

16. n-node graphlet and automorphism orbits

17. Signature similarity

18. Signature similarity

19. GRAph ALigner algorithm (GRAAL)
density
topology
*
G
H
G(V, E)
H(U, F)

20. GRAAL
Search the densest part and align.
Search the minimal cost nodes pair (seed).
If multi-minimal cost pairs, chosen randomly.
G(V, E)
H(U, F)

21. GRAAL
Search the densest part and align.
Search the minimal cost nodes pair (seed).
If multi-minimal cost pairs, chosen randomly.
G(V, E)
H(U, F)

22. GRAAL
Make spheres and align.
G(V, E)
H(U, F)

23. GRAAL
Make spheres and align.
G(V, E)
H(U, F)

24. GRAAL
Make spheres and align.
G(V, E)
H(U, F)

25. GRAAL
Expand radii of spheres and align.
Aligned node
G(V, E)
H(U, F)

26. GRAAL
Expand radii of spheres and align.
Aligned node
G(V, E)
H(U, F)

27. GRAAL
Expand radii of spheres up to 3.
Aligned node
G(V, E)
H(U, F)

28. GRAAL
Expand radii of spheres up to 3.
Aligned node
G(V, E)
H(U, F)

29. GRAAL
Expand radii of spheres up to 3.
Aligned node
G(V, E)
H(U, F)
Some nodes are not aligned.

30. Aligned node

31. Aligned node

32. Aligned node
New seed
New seed

33. Aligned node
New seed
New seed

34. GRAAL
Nodes in G are aligned to exactly one node in H.
Aligned node
G(V, E)
H(U, F)

35. Alignment score
G
H
G(V, E)
GRAAL function
The correct node mapping G to H
H(U, F)

36. Statistical significance
The number of node pairs in H.
Edge correctness
The number of edges from G that are aligned to edges in H.
G
H
G(V, E)
H(U, F)

37. Result
G
H
G(V, E)
H(U, F)
EC = 0.089