Estimating the Number of Communities in a Bipartite Network
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1 Estimating the Number of Communities in a Bipartite Network Tzu-Chi Visitor at Santa Fe Institute, and Institute of Theoretical Physics, Chinese Academy of Sciences Joint work with: Daniel Larremore (Santa Fe Institute) SINM NetSci2017 Indianapolis, IN, USA June 20, 2017
2 Model Selection in Community Detection For SBM, point estimate of K via the Minimum Description Length Principle (MDL) [Peixoto (PRL, 2013), Peixoto (PRX, 2014)] or the Bayesian Information Criterion (BIC) [Yan et al. (J. Stat. Mech., 2014), Yan (ASONAM, 2016)] is established. MDL is equivalent to BIC. [Peixoto (PRX, 2014), Yan (ASONAM, 2016)]. Integrated data likelihood approach, which directly samples the marginal distribution. Order selection goes beyond point estimates [Newman et al. (PRL, 2016)]. model: Stochastic Block Model SBM: #communities: K Given the graph, find the labeling! Risk: We do not know K! Goal: P (G ) k K! P (,K G) 1/8 bisbm: #communities: Ka, Kb Risk: We do not know (Ka, Kb)!
3 2/8 Outline (1) Can we use the MDL criterion to decide the number of communities (Ka, Kb)? And, can we use any tricks to make inference more efficient? (2) Can we make use of partial knowledge of the number of communities say if we know the number of type-a communities and still estimate Kb? (3) Is it even necessary to build separate model selection techniques for bipartite SBMs?
4 3/8 Minimum Description Length (MDL) Criterion Description Length: = S + L Entropy: How well one fits the data given the parameters? Complexity: How complex is the model defined by the parameters? Trick: b = (data) - (random graph) b = Eh K ak b E + n a ln K a + n b ln K b EÎ Profile likelihood: I = rs m rs ln m rs m r m s Î = I ln (2) m r = s m rs I 2 [ln (2), ln (2 min{k a,k b })] (bipartite network) MDL criterion: (K a,k b ) = argmin ( b ) AND b < 0
5 MDL Criterion + Efficient Heuristic in 2D (K A*, K B* ) = (4, 6) (a) (b) Iterations Introducing, where usually = 0.1, s.t., g! g blue traces: approx. row-and-column merging (cheaper) S t+1 S t /S t < trade-off between efficiency and precision g! g Stars: [green crosses] true, at lower K we compute the full inference (expensive) Independent of inference algorithms used (Kernighan-Lin, BP, MCMC, etc.) 4/8 Please visit our poster: P1114, Starting Wednesday (6/20)
6 5/8 Can we go beyond the MDL point estimate, and ask how likely each (Ka,Kb) is to appear? YES. We can use Gibbs sampling.
7 5/8 MCMC Sampling for (Uni-partite) Posterior Probability P(K P (K A) Karate Club Network Number of Communities MEJN and GR, PRL 117, (2016)
8 6/8 Ergodic MCMC Sampling for P(g, P K K A) All parameters are treated equally, Thus the Bayes theorem is used. At each sweep, either changes g or K / RR!, P (A,!,g,,K) d!d integrated data likelihood SBM Newman-Reinert, PRL(2016) bisbm Yen-Larremore, In preparation P (change g) n n+1 n n+2 How? heat-bath: P (g i = r K, A) = P (K,g i=r A) s P (K,g i =s A) P (change K) 1 n+1 1 n+2 (for K a or K b ) How? P (K! K 1) = 1 (always accepted, if possible) P (K! K + 1) = P (K+1,g A) P (K,g A) = k n+k
9 Gibbs Sampling: Newman-Reinert v.s. Ours Posterior Probability P (K A) hki = 6 and Ir = 1.5 Newman-Reinert, PRL (2016) Planted as a (4, 6)-bipartite network Under-fits! Number of Communities Number of Sweeps 7/8 2* Algorithm improves! (4, 6) (4, 5) (5, 5) Ka Relative Likelihood (4, 6) (4, 5) (5, 5) Kb Entropy (2, 2) Yen-Larremore, In preparation K = Ka + Kb = 6
10 8/8 Main Message YES. (1) Can we use the MDL criterion to decide the number of communities (Ka, Kb)? And, can we use any tricks to make inference more efficient? YES. (2) Can we make use of partial knowledge of the number of communities say if we know the number of type-a communities and still estimate Kb? YES, but work in progress. (3) Is it even necessary to build separate model selection techniques for bipartite SBMs? More proofs, statements, and experimental results: Please visit our poster: P1114, Starting Wednesday (6/20) Pre-released code: Please visit this GitHub repo: github.com/junipertcy/det_k_bisbm
11 Acknowledgement Tiago Peixoto (Univ. Bath, invention of graph-tool and research inspirations) Jean-Gabriel Young (Univ. Laval, his useful code released on GitHub) Jie Tang (Tsinghua Univ., financial support; Apr Dec. 2016) Pan Zhang (Chinese Academy of Sciences, financial support and comments on block model detectability; Jan Jun. 2017) Daniel Larremore (SFI, provided invaluable comments and constant support throughout the work) Computational Resources at SFI.
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