Barabási–Albert model (Preferential Attachment)

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Table of contents

1 Introduction 
1.1 Graphs: definitions and notations
1.2 Some statistical uses of random graphs
1.3 Random graph models
1.3.1 Erdős-Rényi graph model
1.3.2 Configuration model
1.3.3 Barabási–Albert model (Preferential Attachment)
1.3.4 Exponential Random Graph Models (ERGM)
1.3.5 Stochastic Block Model (SBM)
1.3.6 Latent Block Model (LBM)
1.3.7 Latent Position Model (LPM)
1.3.8 Graphon and W-graph model
1.4 Node clustering techniques
1.4.1 Spectral clustering
1.4.2 Modularity
1.4.3 Other community detection methods
1.4.4 Model-based clustering
1.4.5 Choice of the number of classes
1.4.6 Theoretical results
1.5 Time-evolving networks
1.5.1 Discrete-time dynamic networks
1.5.2 Continuous-time dynamic networks
1.6 Dynamic SBM with Markov membership evolution
1.6.1 Label switching and identifiability in the dynamic SBM
1.6.2 Estimation
1.6.3 Contributions in the dynamic SBM
1.7 Markov Random Fields (MRF)
1.7.1 Definition
1.7.2 Autologistic model and Potts model
1.7.3 Strength of interaction and phase transition
1.7.4 Simulation with a Gibbs sampler
1.7.5 Likelihood estimation and approximations
1.7.6 Hidden Markov random field
1.7.7 EM with mean field or mean field like approximation
1.7.8 Other methods
1.7.9 Choice of the number of classes for hidden MRF
1.8 Space-evolving networks
1.8.1 Contributions in the spatial SBM
2 Consistency of the maximum likelihood and variational estimators in a dynamic stochastic block model 
2.1 Introduction
2.2 Model and notation
2.2.1 Dynamic stochastic block model
2.2.2 Assumptions
2.2.3 Finite time case
2.2.4 Likelihood
2.3 Consistency of the maximum likelihood estimator
2.3.1 Connectivity parameter
2.3.2 Latent transition matrix
2.4 Variational estimators
2.4.1 Connectivity parameter
2.4.2 Latent transition matrix
2.5 Proofs of main results
2.5.1 Proof of Theorem 2.3.1
2.5.2 Proof of Corollary 2.3.1
2.5.3 Proof of Theorem 2.3.2
2.5.4 Proof of Theorem 2.3.3
2.5.5 Proof of Corollary 2.3.3
2.5.6 Proof of Theorem 2.4.1
2.5.7 Proof of Corollary 2.4.1
2.5.8 Proof of Theorem 2.4.2
3 Estimation of parameters in a space-evolving graph model based on Markov random fields 
3.1 Introduction
3.2 Model and notation
3.2.1 Definition of the model
3.2.2 Assumptions
3.3 Identifiability
3.4 Estimation
3.4.1 Likelihood
3.4.2 Maximum likelihood approach
3.4.3 EM algorithm
3.4.4 Mean-field like approximation
3.4.5 Simulated EM Algorithm
3.4.6 Step 1: Simulation of a configuration for the mean-field like approximation
3.4.7 Step 2: EM iteration
3.4.8 Initialisation and stopping criterion of the algorithm
3.5 Illustration of the method on synthetic datasets
3.6 Proofs
Conclusions and perspectives
References