如今,網(wǎng)絡(luò)數(shù)據(jù)在社會(huì)科學(xué)、生物學(xué)、經(jīng)濟(jì)學(xué)和計(jì)算機(jī)科學(xué)等許多領(lǐng)域都很常見。研究網(wǎng)絡(luò)的生成機(jī)制,探索網(wǎng)絡(luò)結(jié)構(gòu)的各種性質(zhì),具有重要的意義。許多網(wǎng)絡(luò)模型被提出來研究網(wǎng)絡(luò)數(shù)據(jù)的特征并對網(wǎng)絡(luò)數(shù)據(jù)進(jìn)行擬合。網(wǎng)絡(luò)數(shù)據(jù)的非標(biāo)準(zhǔn)結(jié)構(gòu)使得統(tǒng)計(jì)推斷變得困難,特別是在漸近理論中。本文主要研究網(wǎng)絡(luò)模型中的三個(gè)問題,如下所述。首先研究了p0模型中所有極大似然估計(jì)（MLEs）線性組合的漸近分布。p0模型是一個(gè)指數(shù)隨機(jī)圖模型,其中雙度序列是唯一的充分統(tǒng)計(jì)量。在p0模型中,極大似然估計(jì)量的一致相合性和具有固定數(shù)目的極大似然估計(jì)量的漸近正態(tài)性被證明。在之前的工作基礎(chǔ)上,本論文進(jìn)一步得到了網(wǎng)絡(luò)邊取二值、連續(xù)值和離散值時(shí),所有維數(shù)為遞增的MLEs線性組合的中心極限定理。模擬研究被用來說明理論結(jié)果。其次,我們研究了對數(shù)線性模型與隱式對數(shù)線性模型的等價(jià)問題。邏輯線性模型實(shí)際上是p0模型。我們使用“邏輯線性”這個(gè)符號,因?yàn)樗幸粋�(gè)邏輯線性表示。隱式對數(shù)線性模型可以看作是期望度模型的有向版本,其中頂點(diǎn)i和j之間的邊形成概率pij為di+b+j/g++,其中di=∑j≠i ai,j為頂點(diǎn)i的出度,bj=∑i≠jai,j為頂點(diǎn)j的入度,∑... 

【文章來源】：華中師范大學(xué)湖北省 211工程院校 教育部直屬院校

【文章頁數(shù)】：118 頁

【學(xué)位級別】：博士

【文章目錄】：
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


【參考文獻(xiàn)】：
期刊論文
[1]一種基于譜聚類的共指消解方法[J]. 謝永康,周雅倩,黃萱菁.  中文信息學(xué)報(bào). 2009(03)



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