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Topological network alignment
20131216
Statistics journal
Result
G

H

G(V, E)

H(U, F)

EC = 0.089
Motivation
large-scale networks such as interactome
Yeast

Human

Are two networks the same or similar?
Theoretical background
Network or Graph
Collection of nodes (vertex) and connections between them (edges).
Biology, social communication, and web pages
Theoretical background
G

H

G(V, E)

H(U, F)
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)
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
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
n-node graphlet and automorphism orbits
n-node graphlet and automorphism orbits

orbit

Topologically
relevant

graphlet

Topologically
relevant

Topologically
relevant
Graphlet Degree Vector
Graphlet Degree Vector
Graphlet Degree Vector
Graphlet Degree Vector
n-node graphlet and automorphism orbits
Signature similarity
Signature similarity
GRAph ALigner algorithm (GRAAL)

density

topology

*
G

H

G(V, E)

H(U, F)
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)
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)
GRAAL
Make spheres and align.

G(V, E)

H(U, F)
GRAAL
Make spheres and align.

G(V, E)

H(U, F)
GRAAL
Make spheres and align.

G(V, E)

H(U, F)
GRAAL
Expand radii of spheres and align.

Aligned node
G(V, E)

H(U, F)
GRAAL
Expand radii of spheres and align.

Aligned node
G(V, E)

H(U, F)
GRAAL
Expand radii of spheres up to 3.

Aligned node
G(V, E)

H(U, F)
GRAAL
Expand radii of spheres up to 3.

Aligned node
G(V, E)

H(U, F)
GRAAL
Expand radii of spheres up to 3.

Aligned node
G(V, E)

H(U, F)
Some nodes are not aligned.
Aligned node
Aligned node
Aligned node

New seed
New seed
Aligned node

New seed
New seed
GRAAL
Nodes in G are aligned to exactly one node in H.

Aligned node
G(V, E)

H(U, F)
Alignment score

G

H

G(V, E)

GRAAL function
The correct node mapping G to H

H(U, F)
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)
Result
G

H

G(V, E)

H(U, F)

EC = 0.089

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