Topological network alignment

20131216Statistics journal

Result

G H

G(V, E) H(U, F)

EC = 0.089

Motivation

HumanYeast

Are two networks the same or similar?

large-scale networks such as interactome

Theoretical background

Network or GraphCollection of nodes (vertex) and connections between them (edges).Biology, social communication, and web pages

Theoretical background

Graph G and HNode set V and U (V U)Edge set E V*V and F U*UPossible graphs: for G

G H

G(V, E) H(U, F)

Theoretical background

Graph comparisonSubgraph isomorphismIs G an exact subgraph of H?NP-completeEfficient algorithms are not known.

Graph alignmentFitting G into HEdge correctness (EC): the % of E aligned to FNP-hard

G H

G(V, E) H(U, F)

Previous approaches

Local alignment : ambiguous, different pairingMapping are chosen independently for local regions of similarity.PathBLAST : homology informationNetworkBLAST : conserved protein clusters with likelihood methodMaWISh : evolution (sequence alignment)GRAEMLIN : dense conserved subgraph with phylogeny

Global alignmentProvide unique alignment from each node in smaller graph to exactly one node in larger graphISORANK : maximize overall matchGRAEMLIN : training from known graph alignments and phylogeny

New approaches

Never use a priori informationSequence, Homology, Clusters, Phylogeny ,and Known alignments

Topological similarityOrbit, 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

graphlet

orbit

Topologically relevant

Topologically relevant

Topologically relevant

Graphlet Degree Vector

Graphlet Degree Vector

Graphlet Degree Vector

Graphlet Degree Vector

n-node graphlet and automorphism orbits

Orbit 15 in touches orbit 0, 1, 4, and 15 once.

Signature similarityWeight vector

[0, 1] 1 means is not affected by any other orbit.

𝑜15=4 𝑜44=5

Signature similarity

Node , denotes the i-th coordinates of its signature vector. The distance is the i-th orbits of nodes and is

The total distance between and is

The signature similarity is

S = 1 is that and are identical (D = 0).

GRAph ALigner algorithm (GRAAL)

Compute costs of aligning each node with each node .

This matrix is row V and col U (all pairs of nodes).Align the densest parts (the minimal cost nodes, seed).Greedily alignment in the sphere.Repeat * while all nodes in the smaller graph will be aligned.

GRAAL uses only topological information.Biological information can be used by the equation

G H

G(V, E) H(U, F)

density topology

: degree of node

*

GRAALSearch 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)

GRAALSearch the densest part and align.

Search the minimal cost nodes pair (seed).If multi-minimal cost pairs, chosen randomly.

Seed nodes pair

G(V, E) H(U, F)

GRAALMake spheres and align.

Make sphere .Greedily align and with the minimal cost.

𝑢𝑣

G(V, E) H(U, F)

: length of the shortest path

GRAALMake spheres and align.

Make sphere .Greedily align and with the minimal cost.

𝑢𝑣

G(V, E) H(U, F)

𝑆𝑮 (𝑣 ,𝑟 )

𝑆𝑯 (𝑢 ,𝑟 )

: length of the shortest path

GRAALMake spheres and align.

Make sphere .Greedily align and with the minimal cost.

𝑢𝑣

G(V, E) H(U, F)

𝑆𝑮 (𝑣 ,𝑟 )

𝑆𝑯 (𝑢 ,𝑟 )

Align

: length of the shortest path

GRAALExpand radii of spheres and align.

𝑢𝑣

: length of the shortest path

G(V, E) H(U, F)

𝑆𝑮 (𝑣 ,𝑟 )

𝑆𝑯 (𝑢 ,𝑟 )

Make sphere .Greedily align and with the minimal cost.

Aligned node

GRAALExpand radii of spheres and align.

𝑢𝑣

: length of the shortest path

G(V, E) H(U, F)

𝑆𝑮 (𝑣 ,𝑟 )

𝑆𝑯 (𝑢 ,𝑟 )

Make sphere .Greedily align and with the minimal cost.

Aligned node

radii :

GRAALExpand radii of spheres up to 3.

𝑢𝑣

: length of the shortest path

G(V, E) H(U, F)

𝑆𝑮 (𝑣 ,𝑟 )

𝑆𝑯 (𝑢 ,𝑟 )

Make sphere .Greedily align and with the minimal cost.

Aligned node

GRAALExpand radii of spheres up to 3.

𝑢𝑣

: length of the shortest path

G(V, E) H(U, F)

𝑆𝑮 (𝑣 ,𝑟 )

𝑆𝑯 (𝑢 ,𝑟 )

Make sphere .Greedily align and with the minimal cost.

Aligned node

radii :

GRAALExpand radii of spheres up to 3.

𝑢𝑣

: length of the shortest path

G(V, E) H(U, F)

𝑆𝑮 (𝑣 ,𝑟 )

𝑆𝑯 (𝑢 ,𝑟 )

Some nodes are not aligned.

Make sphere .Greedily align and with the minimal cost.

Aligned node

radii :

GRAALRepeat with new edge networks .

𝑮𝑝 (𝑽 ,𝑬𝑝 )

The distance between and , Aligned node

𝑝 ≤2𝑯 𝑝 (𝑼 ,𝑭𝑝 )

𝑆𝑯 𝑝 (𝑢 ,𝑟 )

: length of the shortest path

𝑆𝑮𝑝 (𝑣 ,𝑟 )

𝑝 ≤2

𝑟=1

𝑟=1

GRAALRepeat with new edge networks .

𝑮𝑝 (𝑽 ,𝑬𝑝 )

The distance between and , Aligned node

𝑝 ≤2

𝑆𝑮𝑝 (𝑣 ,𝑟 )𝑟=1

: length of the shortest path

edge()

edge

Path: 6 12 25 can be replaced by , which is analogous for insertion or deletion.

GRAALRepeat with new edge networks .

𝑮𝑝 (𝑽 ,𝑬𝑝 )

The distance between and , Aligned node

𝑝 ≤2

New seed

𝑯 𝑝 (𝑼 ,𝑭𝑝 )

𝑆𝑯 𝑝 (𝑢 ,𝑟 )

: length of the shortest path

New seed

𝑆𝑮𝑝 (𝑣 ,𝑟 )

𝑝 ≤2

𝑟=1

𝑟=1

GRAALRepeat with new edge networks .

𝑮𝑝 (𝑽 ,𝑬𝑝 )

The distance between and , Aligned node

𝑝 ≤2

New seed

𝑯 𝑝 (𝑼 ,𝑭𝑝 )

𝑆𝑯 𝑝 (𝑢 ,𝑟 )

: length of the shortest path

New seed

𝑆𝑮𝑝 (𝑣 ,𝑟 )

𝑝 ≤2

𝑟=1

𝑟=1

GRAALNodes in G are aligned to exactly one node in H.

The distance between and , Aligned node

: length of the shortest path

G(V, E) H(U, F)

Alignment scoreEdge correctness: the % of edges in G are aligned to edges in H.

Node correctness: the % of nodes in G are aligned to nodes in H.Correct mapping is needed.

Interaction correctness: the % of interactions that aligned correctly.Correct interaction is needed.

G H

G(V, E) H(U, F)

GRAAL function

The correct node mapping G to H𝑔 :𝑽→𝑼𝑓 :𝑽→𝑼

Statistical significance

: a random mapping between nodes in G(V, E) and H(U, F).The probability P of successfully aligning k or more edges by chance is the tail of the hypergeometric distribution:

G H

G(V, E) H(U, F)

𝑛1=|𝑉|𝑃=∑𝑖=𝑘

𝑚 2 (𝑚2

𝑖 )(𝑝−𝑚2

𝑚1−𝑖 )( 𝑝𝑚1

)

The number of edges from G that are aligned to edges in H.

The number of node pairs in H.

Edge correctness

Result

G H

G(V, E) H(U, F)

EC = 0.089