009_20150201_Structural Inference for Uncertain Networks
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This paper develops methods for analyzing networks where the connections between nodes are only known probabilistically rather than exactly. It presents a maximum likelihood method for inferring community structure in such uncertain networks by fitting a generative model using EM and belief propagation algorithms. Evaluations on synthetic and real-world networks demonstrate the ability to accurately detect communities and recover the underlying network structure from uncertain edge probability data.
009_20150201_Structural Inference for Uncertain Networks
- 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