Consistency of maximum-likelihood and variational estimators in the Stochastic Block Model

By Alain Celisse, Jean-Jacques Daudin, Laurent Pierre. 2011

211 citations recorded by the source corpus.

The stochastic block model (SBM) is a probabilistic model de- signed to describe heterogeneous directed and undirected graphs. In this paper, we address the asymptotic inference on SBM by use of maximum- likelihood and variational approaches. The identi ability of SBM is proved, while asymptotic properties of maximum-likelihood and variational esti- mators are provided. In particular, the consistency of these estimators is settled, which is, to the best of our knowledge, the rst result of this type for variational estimators with random graphs.

Subfields: Artificial Intelligence, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Anima Anandkumar
2. Michael I. Jordan
3. Sham M. Kakade
4. Rong Ge
5. David M. Blei
6. Daniel J. Hsu
7. Alp Kucukelbir
8. Jon D. McAuliffe

In Statistics and Probability

1. Jing Lei
2. Alessandro Rinaldo
3. Tommaso Cai
4. Arash Ali Amini
5. David M. Blei
6. Martin J. Wainwright
7. Alp Kucukelbir
8. Daniel J. Hsu

Citations in the corpus

• Variational Inference: A Review for Statisticians (2016, background): “Celisse et al. (2012) and Bickel et al. (2013) analyzed network data using stochastic blockmodels.”
• Consistency of spectral clustering in stochastic block models (2013, methodology): “Other computationally efficient procedures with strong theoretical guarantees, which include in particular the ones proposed and analyzed in McSherry (2001); Chen et al. (2012); Channarond et al. (2012); Sarkar & Bickel (2013), require instead the degrees to be of larger order than log n .”
• Statistical physics of inference: thresholds and algorithms (2015, methodology): “[45] Alain Celisse, Jean-Jacques Daudin, Laurent Pierre, et al.”
• Asymptotic Normality of Maximum Likelihood and its Variational Approximation for Stochastic Blockmodels (2012, background): “To some extent, we are following a pioneering paper of Celisse et al. [Celisse, Daudin and Pierre (2011)], in which the dense model was considered, and consistency was established, but only for a subset of the parameters.”
• Pseudo-likelihood methods for community detection in large sparse networks (2012, methodology): “Asymptotic behavior of variational methods is studied in [10] and [8], and in [14] this belief propagation method is analyzed for both the sparse [ λ n = O (1)] and the dense ( λ n → ∞ ) regimes, by nonrigorous cavity methods from physics, and a phase transition threshold, below which the labels…”
• Learning Latent Block Structure in Weighted Networks (2014, background): “This structure has led to theoretical guarantees, including consistency of the SBM estimators [7] and the identifiability and consistency of latent block models [3, 4].”
• A tensor approach to learning mixed membership community models (2013, result): “A recent work [Celisse et al., 2012] establishes consistency of maximum likelihood and variational estimators for stochastic block models, which are special cases of the MMSB model.”