1 Feb Paper Alert, Hasegawa Lab., Tokyo

1. Structural Inference for
Uncertain Networks
Tran Quoc Hoan
@k09hthaduonght.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

2. 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.…”

3. Outline
3
- 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

4. Focus problem
4Structural Inference for Uncertain Networks
• Uncertain structure network
• Community detection
i 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
Community
structure
Trivial approach = threshold

5. 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)

6. 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}

7. Model
7Structural Inference for Uncertain Networks
• Generative model
Number 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
where
X
i<j
Qijand m can be approximated by

8. Model
8Structural Inference for Uncertain Networks
• Generative model
From (2) and (4)
Constant
Likelihood
From (1)
and (6)

9. Methods
9Structural Inference for Uncertain Networks
• Fitting to empirical data
Maximize margin
likelihood
Jensen’s
inequality

10. Methods
10Structural Inference for Uncertain Networks
• Fitting to empirical data

11. Methods
11Structural Inference for Uncertain Networks
• 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

12. Methods
12Structural Inference for Uncertain Networks
• M-step: Maximum the right-hand side of (11)
Apply EM algorithm again to find optimal w

13. Methods
13Structural Inference for Uncertain Networks
• M-step: Update equations of parameters

14. 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.

15. Methods
15Structural Inference for Uncertain Networks
• 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!j
r
Message = the probability that node i below to
community r if node j is removed from network
current best estimate

16. Belief propagation equation
16Structural Inference for Uncertain Networks
Our target q(g)
Two-node
marginal prob
Solve by iterate to converge

17. Degree corrected stochastic block model
17Structural Inference for Uncertain Networks
- 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

18. 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.

19. Result - synthetic network
19Structural Inference for Uncertain Networks

20. Result - protein interaction network
20Structural Inference for Uncertain Networks

21. Edge Recovery
21Structural Inference for Uncertain Networks
• 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

22. Edge Recovery
22Structural Inference for Uncertain Networks

23. 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