关于网络模型中的若干问题的研究

发布时间：2021-07-06 18:33

　　如今,网络数据在社会科学、生物学、经济学和计算机科学等许多领域都很常见。研究网络的生成机制,探索网络结构的各种性质,具有重要的意义。许多网络模型被提出来研究网络数据的特征并对网络数据进行拟合。网络数据的非标准结构使得统计推断变得困难,特别是在渐近理论中。本文主要研究网络模型中的三个问题,如下所述。首先研究了p0模型中所有极大似然估计（MLEs）线性组合的渐近分布。p0模型是一个指数随机图模型,其中双度序列是唯一的充分统计量。在p0模型中,极大似然估计量的一致相合性和具有固定数目的极大似然估计量的渐近正态性被证明。在之前的工作基础上,本论文进一步得到了网络边取二值、连续值和离散值时,所有维数为递增的MLEs线性组合的中心极限定理。模拟研究被用来说明理论结果。其次,我们研究了对数线性模型与隐式对数线性模型的等价问题。逻辑线性模型实际上是p0模型。我们使用“逻辑线性”这个符号,因为它有一个逻辑线性表示。隐式对数线性模型可以看作是期望度模型的有向版本,其中顶点i和j之间的边形成概率pij为di+b+j/g++,其中di=∑j≠i ai,j为顶点i的出度,bj=∑i≠jai,j为顶点j的入度,∑... 

【文章来源】：华中师范大学湖北省 211工程院校 教育部直属院校

【文章页数】：118 页

【学位级别】：博士

【文章目录】：
Abstract
Chinese Abstract
Chapter 1 Introduction to network models and the content of the thesis
    1.1 What are networks
    1.2 Graphs
        1.2.1 Some characteristics of vertex and edge
        1.2.2 Density, Clustering,Transitivity
    1.3 Models of random graphs
        1.3.1 Erdos-Renyi model
        1.3.2 Giant component of Erdos-Renyi
        1.3.3 Example of application of the Erdos-Renyi model
        1.3.4 Small-world model
        1.3.5 The Watts-Strogatz
        1.3.6 Scale-free network model
        1.3.7 Exponential Random Graph Models (ERGMs)
        1.3.8 Distribution of the degrees
        1.3.9 Between-group connectivity
        1.3.10 Examples of ERGMs
        1.3.11 The p_1 model
        1.3.12 Markov graph model
    1.4 Stochastic bloc models (SBMs)
        1.4.1 Degree-Corrected Stochastic Block Model(DCSBM)
    1.5 Random geometric graph
    1.6 Latent Position Cluster Model (LPCM)
        1.6.1 Item response theory models
    1.7 The content of this thesis
Chapter 2 Asymptotic distributions in directed ERGMs with bi-degree sequences
    2.1 Introduction
    2.2 Asymptotic distributions
        2.2.1 Binary weights
        2.2.2 Continuous weights
        2.2.3 Discrete weights
    2.3 Simulations
    2.4 Proofs of theorems
        2.4.1 Preliminaries
        2.4.2 Proof of Theorem 2.1
    2.5 Proof of Theorem 2.2
    2.6 Proof of Theorem 2.3
Chapter 3 Approximate estimation in a class of directed network models
    3.1 Introduction
    3.2 Null models for directed network data
        3.2.1 Model
        3.2.2 Approximate Estimation
    3.3 Approximation
        3.3.1 Approximate results
    3.4 Numerical studies
        3.4.1 Simulation studies
        3.4.2 The data example
    3.5 Proofs
        3.5.1 Preliminaries
        3.5.2 Proof of Theorem 3.1
Chapter 4 Community detection
    4.1 Spectral clustering
        4.1.1 Laplacian matrix
        4.1.2 Unnormalized spectral clustering
        4.1.3 The Ng, Jordan and Weiss (NJW)algorithm
        4.1.4 Normalized spectral clustering
    4.2 Estimating the number of communities
        4.2.1 Non-Backtracking matrix
        4.2.2 The Bethe Hessian matrix
        4.2.3 Spectrum of Bethe Hessian
        4.2.4 Relation of the Bethe Hessian and the non-backtracking matrix
        4.2.5 Estimate the number of communities k using the non-backtracking matrix
        4.2.6 Estimate the number of communities k using the Bethe Hessian matrix
        4.2.7 Cross-validation
        4.2.8 A modified Bethe Hessian matrix to estimate the number of communities K
        4.2.9 Numerical simulations
        4.2.10 Real world network
Chapter 5 Summary and discussion
References
Acknowledgements
List of publications


【参考文献】：
期刊论文
[1]一种基于谱聚类的共指消解方法[J]. 谢永康,周雅倩,黄萱菁.  中文信息学报. 2009(03)



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