1 Networks and Fundamental Concepts ............................ 1
1.1 Network Adjacency Matrix ................................ 1
1.1.1 Connectivity and Related Concepts ................ 2
1.1.2 Social Network Analogy: Affection Network ........ 2
1.2 Analysis Tasks Amenable to Network Methods .............. 3
1.3 Fundamental Network Concepts ............................ 4
1.3.1 Matrix and Vector Notation ....................... 5
1.3.2 Scaled Connectivity .............................. 5
1.3.3 Scale-Free Topology Fitting Index ................ 6
1.3.4 Network Heterogeneity ............................ 8
1.3.5 Maximum Adjacency Ratio .......................... 8
1.3.6 Network Density .................................. 9
1.3.7 Quantiles of the Adjacency Matrix ............... 10
1.3.8 Network Centralization .......................... 10
1.3.9 Clustering Coefficient .......................... 11
1.3.10 Hub Node Significance ........................... 11
1.3.11 Network Significance Measure .................... 12
1.3.12 Centroid Significance and Centroid Conformity ... 12
1.3.13 Topological Overlap Measure ..................... 13
1.3.14 Generalized Topological Overlap for Unweighted
Networks ........................................ 14
1.3.15 Multinode Topological Overlap Measure ........... 16
1.4 Neighborhood Analysis in PPI Networks .................. 18
1.4.1 GTOM Analysis of Fly Protein-Protein
Interaction Data ................................ 18
1.4.2 MTOM Analysis of Yeast Protein-Protein
Interaction Data ................................ 20
1.5 Adjacency Function Based on Topological Overlap ........ 21
1.6 R Functions for the Topological Overlap Matrix ......... 21
1.7 Network Modules ........................................ 22
1.8 Intramodular Network Concepts .......................... 24
1.9 Networks Whose Nodes Are Modules ....................... 25
1.10 Intermodular Network Concepts .......................... 26
1.11 Network Concepts for Comparing Two Networks ............ 27
1.12 R Code for Computing Network Concepts .................. 29
1.13 Exercises .............................................. 30
References .................................................. 32
2 Approximately Factorizable Networks ......................... 35
2.1 Exactly Factorizable Networks .......................... 35
2.2 Conformity for a Non-Factorizable Network .............. 36
2.2.1 Algorithm for Computing the Node Conformity ..... 37
2.3 Module-Based and Conformity-Based Approximation
of a Network ........................................... 39
2.4 Exercises .............................................. 42
References ............................................. 43
3 Different Types of Network Concepts ......................... 45
3.1 Network Concept Functions .............................. 46
3.2 CF-Based Network Concepts .............................. 48
3.3 Approximate CF-Based Network Concepts .................. 49
3.4 Fundamental Network Concepts Versus CF-Based Analogs ... 50
3.5 CF-Based Concepts Versus Approximate CF-Based Analog ... 51
3.6 Higher Order Approximations of Fundamental Concepts .... 52
3.7 Fundamental Concepts Versus Approx. CF-Based Analogs ... 53
3.8 Relationships Among Fundamental Network Concepts ....... 54
3.8.1 Relationships for the Topological Overlap
Matrix .......................................... 55
3.9 Alternative Expression of the Factorizability F(A) ..... 56
3.10 Approximately Factorizable PPI Modules ................. 56
3.11 Studying Block Diagonal Adjacency Matrices ............. 61
3.12 Approximate CF-Based Intermodular Network Concepts ..... 63
3.13 CF-Based Network Concepts for Comparing Two Networks ... 64
3.14 Discussion ............................................. 65
3.15 RCode .................................................. 67
3.16 Exercises .............................................. 69
References .................................................. 74
4 Adjacency Functions and Their Topological Effects ........... 77
4.1 Definition of Important Adjacency Functions ............ 77
4.2 Topological Effects of the Power Transformation
AFpower ................................................ 79
4.2.1 Studying the Power AF Using Approx. CF-Based
Concepts ........................................ 80
4.2.2 MAR Is a Nonincreasing Function of β ............ 80
4.3 Topological Criteria for Choosing AF Parameters ........ 82
4.4 Differential Network Concepts for Choosing AF
Parameters ............................................. 83
4.5 Power AF for Calibrating Weighted Networks ............. 84
4.6 Definition of Threshold-Preserving Adjacency
Functions .............................................. 84
4.7 Equivalence of Network Construction Methods ............ 86
4.8 Exercises .............................................. 87
References .................................................. 89
5 Correlation and Gene Co-Expression Networks ................. 91
5.1 Relating Two Numeric Vectors ........................... 91
5.1.1 Pearson Correlation ............................. 93
5.1.2 Robust Alternatives to the Pearson Correlation .. 94
5.1.3 Biweight Midcorrelation ......................... 95
5.1.4 C-Index ......................................... 96
5.2 Weighted and Unweighted Correlation Networks ........... 97
5.2.1 Social Network Analogy: Affection Network ....... 98
5.3 General Correlation Networks ........................... 99
5.4 Gene Co-Expression Networks ........................... 101
5.5 Mouse Tissue Gene Expression Data from of an F2
Intercross ............................................ 103
5.6 Overview of Weighted Gene Co-Expression Network
Analysis .............................................. 108
5.7 Brain Cancer Network Application ...................... 110
5.8 R Code for Studying the Effect of Thresholding ........ 112
5.9 Gene Network (Re-)Construction Methods ................ 114
5.10 RCode ................................................. 115
5.11 Exercises ............................................. 117
References ................................................. 118
6 Geometric Interpretation of Correlation Networks
Using the Singular Value Decomposition ..................... 123
6.1 Singular Value Decomposition of a Matrix datX ......... 123
6.1.1 Signal Balancing Based on Right Singular
Vectors ........................................ 124
6.1.2 Eigenvectors, Eigengenes, and Left Singular
Vectors ........................................ 125
6.2 Characterizing Approx. Factorizable Correlation
Networks .............................................. 126
6.3 Eigenvector-Based Network Concepts .................... 129
6.3.1 Relationships Among Density Concepts in
Correlation Networks ........................... 131
6.4 Eigenvector-Based Approximations of Intermodular
Concepts .............................................. 132
6.5 Networks Whose Nodes are Correlation Modules .......... 134
6.6 Dictionary for Fundamental-Based and Eigenvector-
Based Concepts ........................................ 135
6.7 Geometric Interpretation .............................. 136
6.7.1 Interpretation of Eigenvector-Based Concepts ... 136
6.7.2 Interpretation of a Correlation Network ........ 137
6.7.3 Interpretation of the Factorizability .......... 138
6.8 Network Implications of the Geometric Interpretation .. 139
6.8.1 Statistical Significance of Network Concepts ... 140
6.8.2 Intramodular Hubs Cannot be Intermediate
Nodes .......................................... 140
6.8.3 Characterizing Networks Where Hub Nodes
Are Significant ................................ 140
6.9 Data Analysis Implications of the Geometric
Interpretation ........................................ 141
6.10 Brain Cancer Network Application ...................... 143
6.11 Module and Hub Significance in Men, Mice, and Yeast ... 147
6.12 Summary ............................................... 150
6.13 R Code for Simulating Gene Expression Data ............ 153
6.14 Exercises ............................................. 157
References ................................................. 159
7 Constructing Networks from Matrices ........................ 161
7.1 Turning a Similarity Matrix into a Network ............ 161
7.2 Turning a Symmetric Matrix into a Network ............. 162
7.3 Turning a General Square Matrix into a Network ........ 163
7.4 Turning a Dissimilarity or Distance into a Network .... 164
7.5 Networks Based on Distances Between Vectors ........... 165
7.6 Correlation Networks as Distance-Based Networks ....... 166
7.7 Sample Networks for Outlier Detection ................. 167
7.8 KL Dissimilarity Between Positive Definite Matrices ... 169
7.9 KL Pre-Dissimilarity for Parameter Estimation ......... 170
7.10 Adjacency Function Based on Distance Properties ....... 171
7.11 Constructing Networks from Multiple Similarity
Matrices .............................................. 172
7.11.1 Consensus and Preservation Networks ............ 173
7.12 Exercises ............................................. 175
References ................................................. 178
8 Clustering Procedures and Module Detection ................. 179
8.1 Cluster Object Scatters Versus Network Densities ...... 179
8.2 Partitioning-Around-Medoids Clustering ................ 181
8.3 Ј-Means Clustering .................................... 182
8.4 Hierarchical Clustering ............................... 184
8.5 Cophenetic Distance Based on a Hierarchical Cluster
Tree .................................................. 186
8.6 Defining Clusters from a Hierarchical Cluster Tree:
The Dynamictreecut Library for R ...................... 188
8.7 Cluster Quality Statistics Based on Network Concepts .. 192
8.8 Cross-Tabulation-Based Cluster (Module) Preservation
Statistics ............................................ 193
8.9 Rand Index and Similarity Measures Between Two
Clusterings ........................................... 195
8.9.1 Co-Clustering Formulation of the Rand Index .... 196
8.9.2 R Code for Cross-Tabulation and
Co-Clustering .................................. 197
8.10 Discussion of Clustering Methods ...................... 198
8.11 Exercises ............................................. 200
References ................................................. 205
9 Evaluating Whether a Module is Preserved in Another
Network .................................................... 207
9.1 Introduction .......................................... 207
9.2 Module Preservation Statistics ........................ 209
9.2.1 Summarizing Preservation Statistics and
Threshold Values ............................... 212
9.2.2 Module Preservation Statistics for General
Networks ....................................... 213
9.2.3 Module Preservation Statistics for
Correlation Networks ........................... 214
9.2.4 Assessing Significance of Observed Module
Preservation Statistics by Permutation Tests ... 218
9.2.5 Composite Preservation Statistic Zsummary ...... 218
9.2.6 Composite Preservation Statistic medianRank .... 220
9.3 Cholesterol Biosynthesis Module Between Mouse
Tissues ............................................... 221
9.4 Human Brain Module Preservation in Chimpanzees ........ 224
9.5 KEGG Pathways Between Human and Chimpanzee Brains ..... 231
9.6 Simulation Studies of Module Preservation ............. 233
9.7 Relationships Among Module Preservation Statistics .... 239
9.8 Discussion of Module Preservation Statistics .......... 242
9.9 R Code for Studying the Preservation of Modules ....... 244
9.10 Exercises ............................................. 245
References ................................................. 245
10 Association Measures and Statistical Significance
Measures ................................................... 249
10.1 Different Types of Random Variables ................... 249
10.2 Permutation Tests for Calculating p Values ............ 250
10.3 Computing p Values for Correlations ................... 252
10.4 R Code for Calculating Correlation Test p Values ...... 254
10.5 Multiple Comparison Correction Procedures for
p Values .............................................. 255
10.6 False Discovery Rates and q-values .................... 258
10.7 R Code for Calculating g-values ....................... 260
10.8 Multiple Comparison Correction as p Value
Transformation ........................................ 262
10.9 Alternative Approaches for Dealing with Many
p Values .............................................. 265
10.10 R Code for Standard Screening ........................ 266
10.11 When Are Two Variable Screening Methods
Equivalent? .......................................... 267
10.12 Threshold-Equivalence of Linear Significance
Measures ............................................. 269
10.13 Network Screening .................................... 271
10.14 General Definition of an Association Network ......... 272
10.15 Rank-Equivalence and Threshold-Equivalence ........... 272
10.16 Threshold-Equivalence of Linear Association
Networks ............................................. 273
10.17 Statistical Criteria for Choosing the Threshold ...... 274
10.18 Exercises ............................................ 274
References ................................................. 277
11 Structural Equation Models and Directed Networks ........... 279
11.1 Testing Causal Models Using Likelihood Ratio Tests ... 279
11.1.1 Depicting Causal Relationships in a Path
Diagram ....................................... 280
11.1.2 Path Diagram as Set of Structural Equations ... 282
11.1.3 Deriving Model-Based Predictions of
Covariances ................................... 283
11.1.4 Maximum Likelihood Estimates of Model
Parameters .................................... 285
11.1.5 Model Fitting p Value and Likelihood Ratio
Tests ......................................... 287
11.1.6 Model Fitting Chi-Square Statistics and LRT ... 287
11.2 R Code for Evaluating an SEM Model .................... 289
11.3 Using Causal Anchors for Edge Orienting ............... 294
11.3.1 Single Anchor Local Edge Orienting Score ...... 295
11.3.2 Multi-Anchor LEO Score ........................ 297
11.3.3 Thresholds for Local Edge Orienting Scores .... 299
11.4 Weighted Directed Networks Based on LEO Scores ........ 299
11.5 Systems Genetic Applications .......................... 300
11.6 The Network Edge Orienting Method ..................... 301
11.6.1 Step 1: Combine Quantitative Traits and
SNPs .......................................... 301
11.6.2 Step 2: Genetic Marker Selection and
Assignment to Traits .......................... 303
11.6.3 Step 3: Compute Local Edge Orienting Scores
for Aggregating the Genetic Evidence
in Favor of a Causal Orientation .............. 305
11.6.4 Step 4: For Each Edge, Evaluate the
Fit of the Underlying Local SEM Models ........ 305
11.6.5 Step 5: Robustness Analysis with Respect
to SNP Selection Parameters ................... 305
11.6.6 Step 6: Repeat the Analysis for the Next
A-B Trait-Trait Edge and Apply Edge Score
Thresholds to Orient the Network .............. 307
11.6.7 NEO Software and Output ....................... 307
11.6.8 Screening for Genes that Are Reactive
to Insigl ..................................... 308
11.6.9 Discussion of NEO ............................. 308
11.7 Correlation Tests of Causal Models .................... 310
11.8 R Code for LEO Scores ................................. 311
11.8.1 R Code for the LEO.SingleAnchor Score ......... 311
11.8.2 R Code for the LEO.CPA ........................ 313
11.8.3 R Code for the LEO.OCA Score .................. 315
11.9 Exercises ............................................. 317
References ................................................. 318
12 Integrated Weighted Correlation Network Analysis
of Mouse Liver Gene Expression Data ........................ 321
12.1 Constructing a Sample Network for Outlier Detection ... 321
12.2 Co-Expression Modules in Female Mouse Livers .......... 324
12.2.1 Choosing the Soft Threshold j3 Via
Scale-Free Topology ........................... 324
12.2.2 Automatic Module Detection Via Dynamic
Tree Cutting .................................. 326
12.2.3 Blockwise Module Detection for Large
Networks ...................................... 327
12.2.4 Manual, Stepwise Module Detection ............. 328
12.2.5 Relating Modules to Physiological Traits ...... 330
12.2.6 Output File for Gene Ontology Analysis ........ 333
12.3 Systems Genetic Analysis with NEO ..................... 334
12.4 Visualizing the Network ............................... 337
12.4.1 Connectivity, TOM, and MDS Plots .............. 337
12.4.2 VisANT Plot and Software ...................... 339
12.4.3 Cytoscape and Pajek Software .................. 339
12.5 Module Preservation Between Female and Male Mice ...... 340
12.6 Consensus modules Between Female and Male Liver
Tissues ............................................... 344
12.6.1 Relating Consensus Modules to the Traits ....... 345
12.6.2 Manual Consensus Module Analysis ............... 348
12.7 Exercises ............................................. 350
References ................................................. 351
13 Networks Based on Regression Models and Prediction
Methods .................................................... 353
13.1 Least Squares Regression and MLE ...................... 353
13.2 R Commands for Simple Linear Regression ............... 355
13.3 Likelihood Ratio Test for Linear Model Fit ............ 356
13.4 Polynomial and Spline Regression Models ............... 358
13.5 R Commands for Polynomial Regression and Spline
Regression ............................................ 360
13.6 Conditioning on Additional Covariates ................. 363
13.7 Generalized Linear Models ............................. 364
13.8 Model Fitting Indices and Accuracy Measures ........... 365
13.9 Networks Based on Predictors and Linear Models ........ 365
13.10 Partial Correlations and Related Networks ............ 366
13.11 R Code for Partial Correlations ...................... 368
13.12 Exercises ............................................ 368
References ................................................. 372
14 Networks Between Categorical or Discretized Numeric
Variables .................................................. 373
14.1 Categorical Variables and Statistical Independence .... 373
14.2 Entropy ............................................... 375
14.2.1 Estimating the Density of a Random Variable .... 376
14.2.2 Entropy of a Discretized Continuous Variable ... 378
14.3 Association Measures Between Categorical Vectors ...... 379
14.3.1 Association Measures Expressed in Terms of
Counts ........................................ 381
14.3.2 R Code for Relating Categorical Variables ..... 381
14.3.3 Chi-Square Statistic Versus Cor in Case of
Binary Variables .............................. 382
14.3.4 Conditional Mutual Information ................ 383
14.4 Relationships Between Networks of Categorical
Vectors ............................................... 384
14.5 Networks Based on Mutual Information 385
14.6 Relationship Between Mutual Information and
Correlation ........................................... 387
14.6.1 Applications for Relating MI with Cor ......... 390
14.7 ARACNE Algorithm ...................................... 391
14.7.1 Generalizing the ARACNE Algorithm .............. 393
14.7.2 Discussion of Mutual Information Networks ...... 394
14.7.3 R Packages for Computing Mutual Information .... 395
14.8 Exercises ............................................. 396
References ................................................. 399
15 Network Based on the Joint Probability Distribution
of Random Variables ........................................ 401
15.1 Association Measures Based on Probability Densities ... 401
15.1.1 Entropy(X) Versus Entropy(Discretize(X)) ...... 403
15.1.2 Kullback-Leibler Divergence for Assessing
Model Fit ..................................... 405
15.1.3 KL Divergence of Multivariate Normal
Distributions ................................. 406
15.1.4 KL Divergence for Estimating Network
Parameters .................................... 407
15.2 Partitioning Function for the Joint Probability ....... 408
15.3 Discussion ............................................ 409
References ................................................. 410
Index ......................................................... 413
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