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9ヶ月前 (2016/02/01)にアップロードin学び

1 Feb Paper Alert, Hasegawa Lab., Tokyo

- Structural Inference for

Uncertain Networks

Travis martin, Brian Ball, and M. E. J. Newman

Phys. Rev. E 93, 012306 – Published 15 January 2016

Tran Quoc Hoan

The University of Tokyo

haduonght.wordpress.com/

@k09ht

1 February 2016, Paper Alert, Hasegawa lab., Tokyo - Abstract

“… Rather than knowing the structure of a network exactly,

we know the connections between nodes only with a

certain probability. In this paper we develop methods for

the analysis of such uncertain data, focusing particularly

on the problem of community detection…”

“…We give a principled maximum-likelihood method for

inferring community structure and demonstrate how the

results can be used to make improved estimates of the true

structure of the network.…”

Structural Inference for Uncertain Networks

2 - Outline

• Motivation

- Analyze the networks represented by uncertain

measurements of their edges

• Proposal

- Fitting a generative network model to the data using a

combination of an EM algorithm and belief propagation

• Applications

- Reconstruct underlaying structure of network

(community detection, edge recovery, …)

Structural Inference for Uncertain Networks

3 - Focus problem

Noisy representation of

true network

• Uncertain structure network

prob of exist edge = Qij

i

j

• Community detection

- Classify the nodes into non-overlapping communities

- Communities = groups of nodes with dense connection within

groups and sparse connections between groups

Trivial approach = threshold

Generative model for uncertain

Fit model to

Community

community-structured networks

observed data

structure

Structural Inference for Uncertain Networks

4 - Model

• Stochastic block model

- n nodes are distributed at random among k groups

- γr : probability to assign to group r

k

X

r = 1

r=1

- wrs : probability to place undirected edges (depends only to group r, s)

- If wrr >> wrs (r ≠ s) then the network has traditional assortative

community structure

- Probability to generate a network (given γr wrs) in which node i is

assigned to group gi, and with the adjacency matrix A

(1)

Aij = 1 if there is

an edge

Structural Inference for Uncertain Networks

5 - Model

• Generative model

- Each pair of nodes i, j a probability Qij of being connected by an edge,

drawn from diﬀerent distributions for edges Aij = 1 and non-edges Aij = 0

- Probability that a true network represented by A = {Aij} become to a

matrix of observed edge probabilities Q = {Qij}

Structural Inference for Uncertain Networks

6 - Model

• Generative model

m: total number of edges in

Number of edges with observed

underlying true network

probability between Q and Q + dQ

Number of non-edges with observed

A value of Qij (assumed

probability between Q and Q + dQ

independent)

then

X

where

and m can be approximated by

Qij

i<j

Structural Inference for Uncertain Networks

7 - Model

Constant

• Generative model

From (2) and (4)

From (1)

and (6)

Likelihood

Structural Inference for Uncertain Networks

8 - Methods

• Fitting to empirical data

Maximize margin

likelihood

Jensen’s

inequality

Structural Inference for Uncertain Networks

9 - Methods

• Fitting to empirical data

Structural Inference for Uncertain Networks

10 - Methods

• Equality condition of (11)

Could be use to detect communities

• EM algorithm, repeat:

- E-step: Fix γ, w and find q(g) by (14)

- M-step: Find γ, w by maximize the right hand side of (11)

Structural Inference for Uncertain Networks

11 - Methods

• M-step: Maximum the right-hand side of (11)

Apply EM algorithm again to find optimal w

Structural Inference for Uncertain Networks

12 - Methods

• M-step: Update equations of parameters

Structural Inference for Uncertain Networks

13 - Methods

• Physical interpretation of t

The posterior probability that

there is an edge between notes

i and j, given that they are in

groups r and s.

Structural Inference for Uncertain Networks

14 - Methods

• E-step: Compute q(g)

- It’s unpractical to compute

denominator of eq. (14)

Approximate q(g) by importance sampling or MCMC

However, in this paper, they use “Belief Propagation” method

Message = the probability that node i below to

⌘i!j

r

community r if node j is removed from network

current best estimate

Structural Inference for Uncertain Networks

15 - Belief propagation equation

Solve by iterate to converge

Our target q(g)

Two-node

marginal prob

Structural Inference for Uncertain Networks

16 - Degree corrected stochastic block model

- The stochastic block model gives poor performance for community

detection in real-world problem (because the assumed model is Poisson

degree distribution).

• Degree corrected

stochastic block model

- Probability to place

undirected edges

between nodes i, j that

fall into groups r, s is

didjwrs

Structural Inference for Uncertain Networks

17 - Result - synthetic network

The delta function makes the matrix Q of

edge probabilities realistically sparse, in

keeping with the structure of real-world

data sets, with a fraction 1 − c of non-

edges having exactly zero probability in

the observed data, on average.

To satisfy e.q. (4)

Structural Inference for Uncertain Networks

18 - Result - synthetic network

Structural Inference for Uncertain Networks

19 - Result - protein interaction network

Structural Inference for Uncertain Networks

20 - Edge Recovery

• Given the matrix Q of edge probabilities, can we make an

informed guess about the adjacency matrix A?

- Simple approach: predict the edges with the highest probability

- Better approach: if we know that network has community structure,

given two pairs of nodes with similar values of Qij, the pair that are

in the same community should be more likely to be connected by

an edge than the pair that are not

Compute in EM step

Structural Inference for Uncertain Networks

21 - Edge Recovery

Structural Inference for Uncertain Networks

22 - Conclusion

• Motivation

- Analyze the networks represented by uncertain

measurements of their edges

• Proposal

- Fitting a generative network model to the data using a

combination of an EM algorithm and belief propagation

• Applications

- Reconstruct underlaying structure of network

(community detection, edge recovery, …)

Structural Inference for Uncertain Networks

23