Probabilistic foundations of statistical network analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton ; London ; New York
CRC Press, Taylor & Francis Group
[2018]
|
| Series: | Monographs on statistics and applied probability
157 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xx, 236 Seiten Diagramme |
| ISBN: | 9781138630154 9781138585997 |
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Contents
Preface xiii
Acknowledgments xix
1 Orientation 1
1.1 Analogy: Bernoulli trials 2
1.2 What it is: Graphs vs. Networks 5
1.3 How to look at it: Labeling and representation 6
1.4 Where it comes from: Context 7
1.5 Making sense of it all: Coherence 8
1.6 What we're talking about: Examples of network data 8
1.6.1 Internet 9
1.6.2 Social networks 9
1.6.3 Karate club 9
1.6.4 Enron email corpus 10
1.6.5 Collaboration networks 10
1.6.6 Blockchain and cryptocurrency networks 10
1.6.7 Other networks 11
1.6.8 Some common scenarios 11
1.7 Major open questions 12
1.7.1 Sparsity 12
1.7.2 Modeling network complexity 13
1.7.3 Sampling issues 13
1.7.4 Modeling network dynamics 14
1.8 Toward a Probabilistic Foundation for Statistical Network Analysis 14
2 Binary relational data 15
2.1 Scenario: Patterns in international trade 17
2.1.1 Summarizing network structure 18
2.2 Dyad independence model 18
2.3 Exponential random graph models (ERGMs) 20
2.4 Scenario: Friendships in a high school 21
2.5 Network inference under sampling 21
2.6 Further reading 23
vii
CONTENTS
viii
3 Network sampling 25
3.1 Opening example 25
3.2 Consistency under selection 27
3.2.1 Consistency of the p\ model 29
3.3 Significance of sampling consistency 31
3.3.1 Toward a coherent framework for network modeling 32
3.4 Selection from sparse networks 33
3.5 Scenario: Ego networks in high school friendships 35
3.6 Network sampling schemes 36
3.6.1 Relational sampling 37
3.6.1.1 Edge sampling 37
3.6.1.2 Hyperedge sampling 39
3.6.1.3 Path sampling 40
3.6.2 Snowball sampling 42
3.7 Units of observation 43
3.8 What is the sample size? 44
3.9 Consistency under subsampling 46
3.10 Further reading 48
3.11 Solutions to exercises 48
3.11.1 Exercise 3.1 48
3.11.2 Exercise 3.2 49
3.11.3 Exercise 3.3 49
3.11.4 Exercise 3.4 50
4 Generative models 51
4.1 Specification of generative models 51
4.2 Generative model 1: Preferential attachment model 52
4.3 Generative model 2: Random walk models 56
4.4 Generative model 3: Erdos-Renyi-Gilbert model 57
4.5 Generative model 4: General sequential construction 57
4.6 Further reading 58
5 Statistical modeling paradigm 59
5.1 The quest for coherence 60
5.2 An incoherent model 62
5.3 What is a statistical model? 63
5.3.1 Population model 64
5.3.2 Finite sample models 64
5.4 Coherence 66
5.4.1 Coherence in sampling models 67
5.4.2 Coherence in generative models 68
5.5 Statistical implications of coherence 69
5.6 Examples 71
5.6.1 Example 1: Erdos-Renyi-Gilbert model under selection
sampling 71
CONTENTS
IX
5.6.2 Example 2: ERGM under selection sampling 72
5.6.3 Example 3: Erdos—Rényi—Gilbert model under edge sam-
pling 72
5.7 Invariance principles 73
5.8 Further reading 74
5.9 Solutions to exercises 75
5.9.1 Exercise 5.1 75
6 Vertex exchangeable models 77
6.1 Preliminaries: Formal definition of exchangeability 77
6.2 Implications of exchangeability 78
6.3 Finite exchangeable random graphs 82
6.3.1 Exchangeable ERGMs 84
6.4 Countable exchangeable models 86
6.4.1 Graphon models 86
6.4.1.1 Generative model 86
6.4.2 Aldous-Hoover theorem 89
6.4.3 Graphons and vertex exchangeability 90
6.4.4 Subsampling description 91
6.5 Viability of graphon models 94
6.5.1 Implication 1: Dense structure 95
6.5.2 Implication 2: Representative sampling 96
6.5.3 The emergence of graphons 97
6.6 Potential benefits of graphon models 99
6.6.1 Connection to de Finetti’s theorem 99
6.6.2 Graphon estimation 102
6.7 Further reading 104
6.8 Solutions to exercises 104
6.8.1 Exercise 6.1 104
6.8.2 Exercise 6.2 105
6.8.3 Exercise 6.3 105
6.8.4 Exercise 6.4 106
6.8.5 Exercise 6.5 107
6.8.6 Exercise 6.6 107
6.8.7 Exercise 6.7 108
6.8.8 Exercise 6.8 108 7
7 Getting beyond graphons 111
7.1 Something must go 112
7.2 Sparse graphon models 114
7.3 Completely random measures and graphex models 116
7.3.1 Scenario: Formation of Facebook friendships 117
7.3.2 Network representation 118
7.3.3 Interpretation of vertex labels 119
7.3.4 Exchangeable point process models 120
X
CONTENTS
7.3.5 Oxymoron: ‘Sparse exchangeable graphs’ 121
7.3.6 Graphex representation 122
7.3.7 Sampling context 123
7.3.8 Further discussion 126
7.4 Variants of invariance 127
7.4.1 Relatively exchangeable models (Chapter 8) 127
7.4.2 Edge exchangeable models (Chapter 9) 127
7.4.3 Relationally exchangeable models (Chapter 10) 128
7.5 Solutions to exercises 128
7.5.1 Exercise 7.1 128
7.5.2 Exercise 7.2 128
7.5.3 Exercise 7.3 129
7.5.4 Exercise 7.4 129
8 Relatively exchangeable models 131
8.1 Scenario: Heterogeneity in social networks 132
8.2 Stochastic blockmodels 132
8.2.1 Generalized blockmodels 134
8.2.2 Community detection and Bayesian versions of SBM 136
8.2.3 Beyond SBMs and community detection 138
8.3 Exchangeability relative to another network 139
8.3.1 Scenario: High school social network revisited 139
8.3.2 Exchangeability relative to a social network 139
8.3.3 Lack of interference 140
8.3.4 Label equivariance 142
8.4 Latent space models 143
8.5 Relatively exchangeable random graphs 144
8.5.1 Relatively exchangeable ^-processes 145
8.6 Relative exchangeability under arbitrary sampling 147
8.7 Relatively invariant graphex models 149
8.8 Final remarks and further reading 150
8.9 Solutions to exercises 151
8.9.1 Exercise 8.1 151
8.9.2 Exercise 8.2 152
8.9.3 Exercise 8.3 153
8.9.4 Exercise 8.4 154
9 Edge exchangeable models 155
9.1 Scenario: Monitoring phone calls 155
9.2 Edge-centric view 156
9.3 Edge exchangeability 159
9.4 Interaction propensity processes 161
9.5 Characterizing edge exchangeable random graphs 164
9.6 Vertex components models 168
9.6.1 Stick-breaking constructions for vertex components 169
CONTENTS xi
9.7 Hollywood model 170
9.7.1 The Hollywood process 172
9.7.2 Role of parameters in the Hollywood model 173
9.7.3 Statistical properties of the Hollywood model 174
9.7.4 Prediction from the Hollywood model 175
9.8 Contexts for edge sampling 175
9.9 Relative edge exchangeability 176
9.10 Thresholding 177
9.11 Comparison: Edge exchangeability v. graphex 179
9.12 Further reading 181
9.13 Solutions to exercises 182
9.13.1 Exercise 9.1 182
9.13.2 Exercise 9.2 182
9.13.3 Exercise 9.3 182
9.13.4 Exercise 9.4 182
9.13.5 Exercise 9.5 183
9.13.6 Exercise 9.6 183
9.13.7 Exercise 9.7 183
9.13.8 Exercise 9.8 183
10 Relationally exchangeable models 185
10.1 Sampling multiway interactions (hyperedges) 185
10.1.1 Collaboration networks 185
10.1.2 Coauthorship networks 187
10.2 Representing multiway interaction networks 187
10.3 Hyperedge exchangeability 188
10.3.1 Interaction propensity process 189
10.3.2 Characterization of hyperedge exchangeable network mod-
els 191
10.4 Scenario: Traceroute sampling of Internet topology 192
10.4.1 Representing the data 194
10.4.2 Path exchangeability 195
10.4.3 Relational exchangeability 197
10.5 General Hollywood model 198
10.6 Markovian vertex components models 200
10.7 Contexts for relational sampling 201
10.8 Concluding remarks and further reading 201
11 Dynamic network models 203
11.1 Scenario: Dynamics in social media activity 205
11.2 Modeling considerations 205
11.2.1 Network dynamics: Markov property 206
11.2.1.1 Modeling the initial state 207
11.2.1.2 Is the Markov property a good assumption? 208
Xll
CONTENTS
11.2.1.3 Temporal Exponential Random Graph Model
(TERGM) 208
11.2.2 Projectivity and sampling 209
11.2.2.1 Example: A TERGM for triangle counts 210
11.2.2.2 Projective Markov property 212
11.3 Rewiring chains and Markovian graphons 213
11.3.1 Exchangeable rewiring processes (Markovian graphons) 215
11.4 Graph-valued Lévy processes 216
11.4.1 Inference from graph-valued Lévy processes 218
11.5 Continuous time processes 219
11.5.1 Poissonian construction 220
11.6 Further reading 221
11.7 Solutions to exercises 222
11.7.1 Exercise 11.1 222
References 223
Index
233 |
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| spelling | Crane, Harry Verfasser (DE-588)1161811508 aut Probabilistic foundations of statistical network analysis Harry Crane (Rutgers University, New Jersey, USA) Boca Raton ; London ; New York CRC Press, Taylor & Francis Group [2018] © 2018 xx, 236 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Monographs on statistics and applied probability 157 System analysis Mathematical models Network analysis (Planning) Statistische Analyse (DE-588)4116599-8 gnd rswk-swf Netzwerkanalyse (DE-588)4075298-7 gnd rswk-swf Netzwerkanalyse (DE-588)4075298-7 s Statistische Analyse (DE-588)4116599-8 s DE-604 Monographs on statistics and applied probability 157 (DE-604)BV002494005 157 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030561802&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Crane, Harry Probabilistic foundations of statistical network analysis Monographs on statistics and applied probability System analysis Mathematical models Network analysis (Planning) Statistische Analyse (DE-588)4116599-8 gnd Netzwerkanalyse (DE-588)4075298-7 gnd |
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| title | Probabilistic foundations of statistical network analysis |
| title_auth | Probabilistic foundations of statistical network analysis |
| title_exact_search | Probabilistic foundations of statistical network analysis |
| title_full | Probabilistic foundations of statistical network analysis Harry Crane (Rutgers University, New Jersey, USA) |
| title_fullStr | Probabilistic foundations of statistical network analysis Harry Crane (Rutgers University, New Jersey, USA) |
| title_full_unstemmed | Probabilistic foundations of statistical network analysis Harry Crane (Rutgers University, New Jersey, USA) |
| title_short | Probabilistic foundations of statistical network analysis |
| title_sort | probabilistic foundations of statistical network analysis |
| topic | System analysis Mathematical models Network analysis (Planning) Statistische Analyse (DE-588)4116599-8 gnd Netzwerkanalyse (DE-588)4075298-7 gnd |
| topic_facet | System analysis Mathematical models Network analysis (Planning) Statistische Analyse Netzwerkanalyse |
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