20131216 stat journal
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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
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