009_20150201_structural inference for uncertain networks
TRANSCRIPT
Structural Inference for Uncertain Networks
Tran Quoc Hoan
@k09ht haduonght.wordpress.com/
1 February 2016, Paper Alert, Hasegawa lab., Tokyo
The University of Tokyo
Travis martin, Brian Ball, and M. E. J. Newman Phys. Rev. E 93, 012306 – Published 15 January 2016
Abstract
Structural Inference for Uncertain Networks 2
“… 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.…”
Outline
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- Analyze the networks represented by uncertain measurements of their edges
• Motivation
- Fitting a generative network model to the data using a combination of an EM algorithm and belief propagation
• Proposal
- Reconstruct underlaying structure of network (community detection, edge recovery, …)
• Applications
Structural Inference for Uncertain Networks
Focus problem
4Structural Inference for Uncertain Networks
• Uncertain structure network
• Community detectioni j
prob of exist edge = Qij
- Classify the nodes into non-overlapping communities
- Communities = groups of nodes with dense connection within groups and sparse connections between groups
Noisy representation of true network
Generative model for uncertain community-structured networks
Fit model to observed data
Communitystructure
Trivial approach = threshold
Model
5Structural Inference for Uncertain Networks
• Stochastic block model- n nodes are distributed at random among k groups
- γr : probability to assign to group r kX
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
Aij = 1 if there is an edge
(1)
Model
6Structural Inference for Uncertain Networks
• 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}
Model
7Structural Inference for Uncertain Networks
• Generative modelNumber of edges with observed probability between Q and Q + dQ
Number of non-edges with observed probability between Q and Q + dQ
A value of Qij (assumed independent)
m: total number of edges in underlying true network
then
whereX
i<j
Qijand m can be approximated by
Model
8Structural Inference for Uncertain Networks
• Generative model
From (2) and (4)
Constant
Likelihood
From (1) and (6)
Methods
9Structural Inference for Uncertain Networks
• Fitting to empirical data
Maximize margin likelihood
Jensen’s inequality
Methods
10Structural Inference for Uncertain Networks
• Fitting to empirical data
Methods
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• Equality condition of (11)
• 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)
Could be use to detect communities
Methods
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• M-step: Maximum the right-hand side of (11)
Apply EM algorithm again to find optimal w
Methods
13Structural Inference for Uncertain Networks
• M-step: Update equations of parameters
Methods
14Structural Inference for Uncertain Networks
• 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.
Methods
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• 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
⌘i!jr
Message = the probability that node i below to community r if node j is removed from network
current best estimate
Belief propagation equation
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Our target q(g)
Two-node marginal prob
Solve by iterate to converge
Degree corrected stochastic block model
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- 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
Result - synthetic network
18Structural Inference for Uncertain Networks
To satisfy e.q. (4)
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.
Result - synthetic network
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Result - protein interaction network
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Edge Recovery
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• 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
Edge Recovery
22Structural Inference for Uncertain Networks
Conclusion
23
- Analyze the networks represented by uncertain measurements of their edges
• Motivation
- Fitting a generative network model to the data using a combination of an EM algorithm and belief propagation
• Proposal
- Reconstruct underlaying structure of network (community detection, edge recovery, …)
• Applications
Structural Inference for Uncertain Networks