Untangling graphs: denoising protein-protein interaction networks

1
Untangling graphs denoising protein-protein
interaction networks

• Quaid Morris
• (joint work with Brendan Frey)

2
Motivation

• High-throughput graph data is noisy, e.g.,
• protein-protein interaction networks
• synthetic lethal interaction networks
• Real world graphs are highly structured

Idea Use prior knowledge about structure to
denoise graphs
3
Protein-Protein interaction network
Jeong et al, Nature 2001
4
Overview

• Illustrative example
• Model and inference algorithm
• Protein-protein interaction network denoising

5
Example spy rings
Suspects
Phone Records
Call

• Spies call exactly two other spies
• Suspects may call other suspects
• Phone records may be lost

6
Example spy rings cont.
Phone Records
Possible Rings
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Denoising example
Noise assumptions

• No lost calls
• Rare, independent social calls

Possible Rings
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Denoising example
Noise assumptions

• No lost calls
• Rare, independent social calls

Possible Rings
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Denoising example
Noise assumptions

• No lost calls
• Rare, independent social calls

Possible Rings
10
Untangling example
Telemarketing Example
Noise assumptions

• Lost calls and rare social calls
• Telemarketing

Possible Decompositions
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Summary

• With structured noise, observed graph is composed
  of different graphs, each with their own
  properties.

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Graph generative model
E
E
1
2

• Sample hidden graphs
• E from P(E )

i
h
h
j
2) Sample x from P(x e , e )
1
2
i,j
i,j
i,j
i,j
X
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Model and inference
Joint
1
2
H
h
1
2
H
P(X, E , E , , E ) P P(E ) P P(x e ,
e , , e )
i,j
i,j
i,j
i,j
h
igtj
Posterior marginal
h
P(e , X)
i,j
h
P(e X)
i,j
Generally Intractable Sums
P(X)
Probability of evidence

1
2
H
P(X) S S S
P(X, E , E , , E )
H
2
1
E
E
E
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Three tricks for tractability

• Degree-based graph priors
• Sum-product approximate inference
• Dynamic programming trick

15
Degree-based graph priors
h
h
h
P(E ) P f (d ) / Z
i
i
Degree of vertex i
Degree potential for graph h
h
i
h
d S e
i,j
i
j

• Real-world network structure captured
• Nice sum-product (loopy belief prop) algorithm
• Introduces dummy degree variable, d

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Two types of random graphs
Exponential
Scale-free
Jeong et al, Nature 2000
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Random graph degree distributions
Scale-free
Exponential
-p
f(k) Ck , p gt 1
Poisson(ltkgt)
Jeong et al, Nature 2000
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Other real-world structure

• Small-worldness
• Degree correlations
• Clustering

Google Mark Newman Michigan for more info
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Factor graph for denoising
e
x
1,2
1,2
e
x
d
1,3
1,3
1
d
e
x
2
1,4
1,4
d
x
e
3
2,3
2,3
e
d
x
2,4
4
2,4
e
x
3,4
3,4
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Factor graph for denoising
e
x
1,2
1,2
e
x
d
1,3
1,3
1
d
e
x
2
1,4
1,4
d
x
e
3
2,3
2,3
e
d
x
2,4
4
2,4
e
x
f(d)
degree potentials
3,4
3,4
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Factor graph for denoising
I(d , Se )
Indicator functions
e
x
j
i,j
1,2
1,2
i
e
x
d
1,3
1,3
1
d
e
x
2
1,4
1,4
d
x
e
3
2,3
2,3
e
d
x
2,4
4
2,4
e
x
f(d)
3,4
3,4
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Factor graph for denoising
I(d , Se )
e
x
j
i,j
1,2
1,2
i
e
x
d
1,3
1,3
1
d
e
x
2
1,4
1,4
d
x
e
3
2,3
2,3
e
d
x
2,4
4
2,4
e
x
f(d)
3,4
3,4
P(xe)
Likelihood functions
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Factor graph for denoising
I(d , Se )
e
x
j
i,j
1,2
1,2
i
e
x
d
1,3
1,3
1
d
e
x
2
1,4
1,4
d
x
e
3
2,3
2,3
e
d
x
2,4
4
2,4
e
x
f(d)
3,4
3,4
P(xe)
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Sum-product approximate inference

• Two types of binary messages
• edge variables a constraint nodes
• constraint nodes a edge variables

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Calculating edge a constraint messages
e 0 or 1
edge e -gt degree I messages
i,j
i
m (e) m (e) m (e)
e -gt I
I -gt e
x -gt e
j
i
i,j
i,j
i,j
i,j
likelihood message
constraint -gt edge message
P(x e e)
i.e.
i,j
i,j
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Calculating constraint a edge messages
degree I -gt edge e messages
i,j
j
m (e) S S f(d) P m (ek)
e -gt I
I -gt e
k j
i
i,k
d
j
i,j
e1, e2, , eN s.t. ej e, and S ek d
degree prior
intractable sum? No, use dynamic programming
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Dynamic programming solution
d
j
I

s
s
s
j
1
2
N


e
e
e
d
1,j
2,j
N,j
j

e
e
e
1,j
2,j
N,j
N
2
O(2 ) time
O(N ) time
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Dynamic programming solution
Constraint
d
j
s
s e
i1
i
i1,j
I

s
s
s
j
1
2
N


e
e
e
d
1,j
2,j
N,j
j

e
e
e
1,j
2,j
N,j
N
2
O(2 ) time
O(N ) time
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Inference for untangling

• Message passing same as denoising, except
  likelihood message needs to be recalculated.
• Likelihood message incorporates edge
  information from other hidden graphs

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Factor graph for untangling
1
2
f (d)
P(xe ,e )
f (d)
1
2
e
1
1
d
e
d
2
2
1,2
1
1,2
1
x
1,2
e
1
1
d
e
d
2
2
1,3
2
1,3
2
x
1,3
e
d
e
d
1
1
2
2
2,3
3
2,3
3
x
2,3
I(d, ee)
I(d, ee)
Graph 1
Graph 2
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Protein-protein interaction network denoising
Von Mering et al (2002) dataset

• Eight PPI networks consisting of
• Low quality direct evidence (high-throughput)
• Indirect evidence
• Gold standard
• A small set of confirmed interactions

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Empirical degree distributions
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Methods

• Split 6k ORFs into training and test set
• On training set
• Fit degree priors to both true graph and
  false graph.
• Construct likelihood function out of all
  observations using Naïve Bayes
• Every observed interaction must be placed in
  exactly one of the two hidden graphs.

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Results
Untangling
Baseline
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Summary

• Generative model for observed graphs, composed of
  many hidden graphs
• Sum-product approximate inference algorithm for
  degree-based priors
• Application to protein-protein interaction
  network noise removal