Borderline Personality Disorder: A Life-Changing Guide to Successfully Manage BPD, Protect Your Mental Health, and Cultivate Healthy Relationships

The study of networks, including computer networks, social networks, and biological networks, has attracted enormous interest in recent years. The rise of the Internet and the wide availability of inexpensive computers have made it possible to gather and analyse network data on an unprecendented scale, and the development of new theoretical tools has allowed us to extract knowledge from networks of many different kinds. The study of networks is broadly interdisciplinary and developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social science. This book brings together the most important breakthroughts in each of these fields and presents them in a unified fashion, highlighting the strong interconnections between work in different areas. Topics covered include the measurement of networks; methods for analysing network data, including methods developed in physics, statistics, and sociology; fundamentals of graph theory; computer algorithms, including spectral algorithms and community detection; mathematical models of networks such as random graph models and generative models; and models of processes taking place on networks. Cover......Page 1 Networks......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 10 Examples of networks......Page 14 What can we learn from networks?......Page 20 Properties of networks......Page 21 Outline of this book......Page 24 Part I. The Empirical study of networks......Page 26 Chapter 2. Technological networks......Page 27 2.1 The Internet......Page 28 2.1.1 Measuring Internet structure using traceroute......Page 30 2.1.2 Measuring Internet structure using routing tables......Page 34 2.2 The telephone network......Page 38 2.3 Power grids......Page 40 2.4 Transportation networks......Page 41 2.5 Delivery and distribution networks......Page 42 3.1 The World Wide Web......Page 45 3.2 Citation networks......Page 50 3.2.1 Patent and legal citations......Page 52 3.3 Other information networks......Page 54 3.3.1 Peer-to-peer networks......Page 55 3.3.2 Recommender networks......Page 57 3.3.3 Keyword indexes......Page 58 4.1 The empirical study of social networks......Page 60 4.2 Interviews and questionnaires......Page 64 4.2.1 Ego-centered networks......Page 68 4.3 Direct observation......Page 70 4.4 Data from archival or third-party records......Page 71 4.5 Affiliation networks......Page 73 4.6 The small-world experiment......Page 75 4.7 Snowball sampling, contact tracing, and random walks......Page 78 5.1.1 Metabolic networks......Page 83 5.1.2 Protein–protein interaction networks......Page 89 5.1.3 Genetic regulatory networks......Page 93 5.1.4 Other biochemical networks......Page 99 5.2.1 Networks of neurons......Page 101 5.2.2 Networks of functional connectivity in the brain......Page 107 5.3 Ecological networks......Page 108 5.3.1 Food webs......Page 109 5.3.2 Other ecological networks......Page 113 Part II. Fundamentals of network theory......Page 116 6.1 Networks and their representation......Page 118 6.2 The adjacency matrix......Page 119 6.3 Weighted networks......Page 121 6.4 Directed networks......Page 123 6.4.1 Acyclic networks......Page 124 6.5 Hypergraphs......Page 127 6.6 Bipartite networks......Page 128 6.6.1 The incidence matrix and network projections......Page 129 6.7 Multilayer and dynamic networks......Page 131 6.8 Trees......Page 134 6.9 Planar networks......Page 136 6.10 Degree......Page 139 6.10.1 Density and sparsity......Page 141 6.10.2 Directed networks......Page 143 6.11 Walks and paths......Page 144 6.11.1 Shortest paths......Page 145 6.12 Components......Page 146 6.12.1 Components in directed networks......Page 147 6.13 Independent paths, connectivity, and cut sets......Page 150 6.13.1 Maximum flows and cut sets on weighted networks......Page 154 6.14 The graph Laplacian......Page 155 6.14.1 Graph partitioning......Page 156 6.14.2 Network visualization......Page 158 6.14.3 Random walks......Page 160 6.14.4 Resistor networks......Page 161 6.14.5 Properties of the graph Laplacian......Page 163 Exercises......Page 166 Chapter 7. Measures and metrics......Page 171 7.1.2 Eigenvector centrality......Page 172 7.1.3 Katz centrality......Page 176 7.1.4 PageRank......Page 178 7.1.5 Hubs and authorities......Page 181 7.1.6 Closeness centrality......Page 183 7.1.7 Betweenness centrality......Page 186 7.2 Groups of nodes......Page 190 7.2.2 Cores......Page 191 7.2.3 Components and k-components......Page 193 7.3 Transitivity and the clustering coefficient......Page 196 7.3.1 Local clustering and redundancy......Page 199 7.4 Reciprocity......Page 202 7.5 Signed edges and structural balance......Page 203 7.6 Similarity......Page 207 7.6.1 Measures of structural equivalence......Page 208 7.6.2 Measures of regular equivalence......Page 211 7.7 Homophily and assortative mixing......Page 214 7.7.1 Assortative mixing by unordered characteristics......Page 216 7.7.2 Assortative mixing by ordered characteristics......Page 219 7.7.3 Assortative mixing by degree......Page 222 Exercises......Page 224 Chapter 8. Computer algorithms......Page 231 8.1 Software for network analysis and visualization......Page 232 8.2 Running time and computational complexity......Page 234 8.3 Storing network data......Page 238 8.3.1 The adjacency matrix......Page 239 8.3.2 The adjacency list......Page 242 8.3.3 Other network representations......Page 246 8.4 Algorithms for basic network quantities......Page 250 8.4.1 Degrees......Page 251 8.4.2 Clustering coefficients......Page 252 8.5.1 Description of the breadth-first search algorithm......Page 254 8.5.2 A naive implementation......Page 256 8.5.3 A better implementation......Page 257 8.5.4 Variants of breadth-first search......Page 260 8.5.5 Finding shortest paths......Page 262 8.5.6 Betweenness centrality......Page 265 8.6 Shortest paths in networks with varying edge lengths......Page 270 8.7 Maximum flows and minimum cuts......Page 275 8.7.1 The augmenting path algorithm......Page 276 8.7.2 Implementation and running time......Page 278 8.7.3 Why the algorithm gives correct answers......Page 279 8.7.4 Finding independent paths and minimum cut sets......Page 281 8.7.5 Node-independent paths......Page 282 Exercises......Page 285 Chapter 9. Network statistics and measurement error......Page 288 9.1 Types of error......Page 289 9.2 Sources of error......Page 291 9.3.1 Conventional statistics of measurement error......Page 294 9.3.2 The method of maximum likelihood......Page 295 9.3.3 Errors in network data......Page 298 9.3.4 The EM algorithm......Page 299 9.3.5 Independent edge errors......Page 303 9.3.6 Example......Page 306 9.3.7 Estimation of other quantities......Page 308 9.3.8 Other error models......Page 309 9.4 Correcting errors......Page 310 9.4.1 Link prediction......Page 311 9.4.2 Node disambiguation......Page 313 Exercises......Page 314 10.1 Components......Page 317 10.1.1 Components in directed networks......Page 321 10.2 Shortest paths and the small-world effect......Page 323 10.3 Degree distributions......Page 326 10.4 Power laws and scale-free networks......Page 330 10.4.1 Detecting and visualizing power laws......Page 332 10.4.2 Properties of power-law distributions......Page 338 10.5 Distributions of other centrality measures......Page 343 10.6 Clustering coefficients......Page 345 10.6.1 Local clustering coefficient......Page 347 10.7 Assortative mixing......Page 348 Exercises......Page 350 Part III. Network models......Page 354 Chapter 11. Random graphs......Page 355 11.1 Random graphs......Page 356 11.2 Mean number of edges and mean degree......Page 358 11.3 Degree distribution......Page 359 11.5 Giant component......Page 360 11.5.1 Can there be more than one giant component?......Page 367 11.6 Small components......Page 368 11.7 Path lengths......Page 373 11.8 Problems with the random graph......Page 377 Exercises......Page 379 Chapter 12. The configuration model......Page 382 12.1 The configuration model......Page 383 12.1.1 Edge probability in the configuration model......Page 386 12.1.2 Random graphs with given expected degree......Page 388 12.2 Excess degree distribution......Page 390 12.3 Clustering coefficient......Page 394 12.4 Locally tree-like networks......Page 395 12.5 Number of second neighbors of a node......Page 396 12.6 Giant component......Page 397 12.6.1 Example......Page 400 12.6.2 General solution for the size of the giant component......Page 402 12.7 Small components......Page 404 12.7.1 Degrees of nodes in the small components......Page 405 12.7.2 Average number of nodes reached along an edge......Page 406 12.8 Networks with power-law degree distributions......Page 408 12.9 Diameter......Page 412 12.10.1 Generating functions......Page 414 12.10.2 Examples......Page 415 12.10.3 Power-law distributions......Page 416 12.10.4 Normalization and moments......Page 417 12.10.5 Products of generating functions......Page 419 12.10.6 Generating functions for degree distributions......Page 420 12.10.7 Number of second neighbors of a node......Page 421 12.10.8 Generating functions for the small components......Page 424 12.10.9 Complete distribution of small component sizes......Page 427 12.11.1 Directed networks......Page 429 12.11.3 Acyclic networks......Page 432 12.11.4 Degree correlations......Page 433 12.11.6 Assortative mixing and community structure......Page 434 12.11.7 Dynamic networks......Page 437 12.11.8 The small-world model......Page 438 Exercises......Page 441 Chapter 13. Models of network formation......Page 447 13.1 Preferential attachment......Page 448 13.1.1 Degree distribution of Price’s model......Page 451 13.1.2 Computer simulation of Price’s model......Page 456 13.2 The model of Barabasi and Albert......Page 461 13.3 Time evolution of the network and the first mover effect......Page 464 13.4 Extensions of preferential attachment models......Page 471 13.4.1 Addition of extra edges......Page 472 13.4.2 Removal of edges......Page 474 13.4.3 Non-linear preferential attachment......Page 479 13.5 Node copying models......Page 485 13.6 Network optimization models......Page 492 13.6.1 Trade-offs between travel time and cost......Page 493 Exercises......Page 500 Part IV. Applications......Page 506 Chapter 14. Community structure......Page 507 14.1 Dividing networks into groups......Page 508 14.2 Modularity maximization......Page 511 14.2.1 The form of the modularity function......Page 513 14.2.2 A simple modularity maximization algorithm......Page 515 14.2.3 Spectral modularity maximization......Page 518 14.2.4 Division into more than two groups......Page 522 14.2.5 The Louvain algorithm......Page 524 14.2.6 Resolution limit for modularity maximization......Page 525 14.3 Methods based on information theory......Page 528 14.4 Methods based on statistical inference......Page 533 14.4.1 Community detection using statistical inference......Page 535 14.5 Other algorithms for community detection......Page 542 14.5.1 Betweenness-based methods......Page 543 14.5.2 Hierarchical clustering......Page 547 14.6 Measuring algorithm performance......Page 551 14.6.1 Tests on real-world networks......Page 552 14.6.2 Artificial test networks......Page 555 14.6.3 Quantifying performance......Page 557 14.6.4 Comparison of community detection algorithms......Page 563 14.7.1 Overlapping communities......Page 564 14.7.2 Hierarchical communities......Page 566 14.7.3 Core–periphery structure......Page 568 14.7.4 Latent spaces, stratified networks, and rank structure......Page 571 Exercises......Page 579 15.1 Percolation......Page 582 15.2 Uniform random removal of nodes......Page 584 15.2.1 Uniform removal in the configuration model......Page 587 15.3 Non-uniform removal of nodes......Page 599 15.4 Percolation in real-world networks......Page 606 15.5 Computer algorithms for percolation......Page 607 15.5.1 Results for real-world networks......Page 612 Exercises......Page 615 Chapter 16. Epidemics on networks......Page 620 16.1 Models of the spread of infection......Page 621 16.1.1 The SI model......Page 622 16.1.2 The SIR model......Page 625 16.1.3 Solution of the SIR model......Page 627 16.1.4 Basic reproduction number......Page 629 16.1.5 The SIS model......Page 630 16.1.6 The SIRS model......Page 632 16.1.8 Combinations of diseases......Page 633 16.1.9 Complex contagion and the spread of information......Page 635 16.2 Epidemic models on networks......Page 637 16.3 Outbreak sizes and percolation......Page 638 16.3.1 Outbreak sizes in the SIR model......Page 639 16.3.2 SIR model and the configuration model......Page 642 16.3.3 Coexisting diseases......Page 646 16.3.4 Coinfection......Page 650 16.3.5 Complex contagion......Page 652 16.4 Time-dependent properties of epidemics on networks......Page 658 16.5 Time-dependent properties of the SI model......Page 659 16.5.1 Pair approximation......Page 663 16.5.2 Degree-based approximation for the SI model......Page 667 16.6 Time-dependent properties of the SIR model......Page 673 16.6.1 Degree-based approximation for the SIR model......Page 675 16.7 Time-dependent properties of the SIS model......Page 680 16.7.1 Degree-based approximation for the SIS model......Page 682 Exercises......Page 684 Chapter 17. Dynamical systems on networks......Page 688 17.1 Dynamical systems......Page 689 17.1.1 Fixed points and linearization......Page 690 17.2 Dynamics on networks......Page 698 17.2.1 Linear stability analysis......Page 699 17.2.2 Special cases......Page 701 17.2.3 An example......Page 706 17.3 Dynamics with more than one variable per node......Page 707 17.3.1 Special cases......Page 709 17.4 Spectra of networks......Page 711 17.5 Synchronization......Page 714 Exercises......Page 719 18.1 Web search......Page 723 18.2 Searching distributed databases......Page 726 18.3.1 Kleinberg’s model......Page 731 18.3.2 A hierarchical model for messages......Page 736 Exercises......Page 743 References......Page 745 Index......Page 764 "The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet and the wide availability of inexpensive computers have made it possible to gather and analyze network data on an unprecedented scale, and the development of new theoretical tools has allowed us to extract new knowledge from networks of many different kinds. The study of networks is broadly interdisciplinary and `developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. This book brings together the most important breakthroughs in each of these fields and presents them in a unified fashion, highlighting the strong interconnections between work in different areas. Topics covered include the measurement of networks; methods for analyzing network data, including methods developed in physics, statistics, and sociology; fundamentals of graph theory; computer algorithms, including spectral algorithms and community detection; mathematical models of networks such as random graph models; and models of processes taking place on networks"--Back cover The study of networks, including computer networks, social networks, and biological networks, has attracted enormous interest in the last few years. The rise of the Internet and the wide availability of inexpensive computers have made it possible to gather and analyze network data on an unprecedented scale, and the development of new theoretical tools has allowed us to extract knowledge from networks of many different kinds. The study of networks is broadly interdisciplinary and central developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. This book brings together the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Topics covered include the measurement of networks; methods for analyzing network data, including methods developed in physics, statistics, and sociology; fundamentals of graph theory; computer algorithms; mathematical models of networks, including random graph models and generative models; and theories of dynamical processes taking place on networks.