Probabilistic Foundations of Statistical Network Analysis


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Table of Contents



  1. Orientation
  2. Analogy: Bernoulli trials What it is: Graphs vs Networks Moving beyond graphs How to look at it: Labeling and representation Where it comes from: Context Making sense of it all: Coherence What we're talking about: Common examples of network data Internet Social networks Karate club Enron email corpus Collaboration networks Other networks Some common scenarios Major Open Questions Sparsity Modeling network complexity Sampling issues Modeling temporal variation Chapter synopses and reading guide Binary relational data Network sampling Generative models Statistical modeling paradigm Vertex exchangeable models Getting beyond graphons Relatively exchangeable models Edge exchangeable models Relationally exchangeable models DEDICATION Dynamic network models
  3. Binary relational data
  4. Scenario: Patterns in international trade Summarizing network structure Dyad independence model Exponential random graph models (ERGMs) Scenario: Friendships in a high school Network inference under sampling Further reading
  5. Network sampling
  6. Opening example Consistency under selection Consistency in the p model Significance of sampling consistency Toward a coherent framework of network modeling Selection from sparse networks Scenario: Ego networks in high school friendships Network sampling schemes Relational sampling Edge sampling Hyperedge sampling Path sampling Snowball sampling Units of observation What is the sample size? Consistency under subsampling Further reading
  7. Generative models
  8. Specification of generative models Preferential Attachment model Random walk models Erd os-Renyi-Gilbert model General sequential construction Further reading
  9. Statistical modeling paradigm
  10. The quest for coherence An incoherent model What is a statistical model? Population model Finite sample models Coherence Coherence in sampling models Coherence in generative models Statistical implications of coherence Examples Erd os-Renyi-Gilbert model under selection sampling ERGM with selection sampling Erd os-Renyi-Gilbert model under edge sampling Invariance principles Further reading
  11. Vertex exchangeable models
  12. Preliminaries: Formal definition of exchangeability Implications of exchangeability Finite exchangeable random graphs Exchangeable ERGMs Countable exchangeable models Graphon models Generative model Exchangeability of graphon models Aldous-Hoover theorem Graphons and vertex exchangeability Subsampling description Viability of graphon models Implication: Dense structure Implication: Representative sampling The emergence of graphons Potential benefits of graphon models Connection to de Finetti's theorem Graphon estimation Further reading
  13. Getting beyond graphons
  14. Something must go Sparse graphon models Completely random measures and graphex models Scenario: Formation of Facebook friendships Network representation Interpretation of vertex labels Exchangeable point process models Graphex representation Sampling context Further discussion Variants of invariance Relatively exchangeable models DEDICATION Edge exchangeable models Relationally exchangeable models
  15. Relatively exchangeable models
  16. Scenario: heterogeneity in social networks Stochastic blockmodels Generalized blockmodels Community detection and Bayesian versions of SBM Beyond SBMs and community detection Relative exchangeability with respect to another network Scenario: high school social network revisited Exchangeability relative to a social network Lack of interference Label equivariance Latent space models Relatively exchangeable random graphs Relatively exchangeable f-processes Relative exchangeability under arbitrary sampling Final remarks and further reading
  17. Edge exchangeable models
  18. Scenario: Monitoring phone calls Edge-centric view Edge exchangeability Interaction propensity process Characterizing edge exchangeable random graphs Vertex components models Stick-breaking constructions for vertex components Hollywood model The Hollywood process Role of parameters in the Hollywood model Statistical properties of the Hollywood model Prediction from the Hollywood model Thresholding Contexts for edge sampling Concluding remarks Connection to graphex models Further reading
  19. Relationally exchangeable models
  20. Sampling multiway interactions (hyperedges) Collaboration networks Coauthorship networks Representing multiway interaction networks Hyperedge exchangeability Interaction propensity process Characterization for hyperedge exchangeable networks Scenario: Traceroute sampling of Internet topology Representing the data Path exchangeability Relational exchangeability General Hollywood model Markovian vertex components models Concluding remarks and further reading
  21. Dynamic network models
Scenario: Dynamics in social media activity Modeling considerations Network dynamics: Markov property Modeling the initial state Is the Markov property a good assumption? Temporal Exponential Random Graph Model (TERGM) Projectivity and sampling Example: a TERGM for triangle counts Projective Markov property Rewiring chains and Markovian graphons Exchangeable rewiring processes (Markovian graphons) Graph-valued Levy processes Inference from graph-valued Levy processes Continuous time processes Poissonian construction Further reading Bibliography Index


"I believe this book can serve both as a reference and textbook, but primarily should be seen as a textbook for a course built around foundational aspects of statistical modeling for network data. Most prior texts I am aware of focus on statistical methods within existing network models. I really like that this book helps the reader understand the statistical implications of choice of model, both in terms of "coherence" and sampling. Most prior work presents the field of statistical network analysis as a basket of models from which one chooses their preferred method. Crane takes a more foundational approach - showing how choice of model leads to implicit statistical assumptions that too often go unspoken."
~Walter Dempsey, Harvard University

"A set of useful exercises are given in almost all chapters that assists in understanding the topics and - what is very useful and much appreciated - the author also gives their solutions. These are not only a great tool because they allow solutions to be checked, but because somehow they are a complement of the text. Moreover, they provide the opportunity to dive thoroughly into the topics. Finally, the author not only proposes these exercises in each chapter, he also proposes problems that are open research questions. These are very nice inputs for researchers who are working in the field. And in this way, the author opens a door to further research and establishes a dialog between him and the
~Silvano Romano, ISCB Newsletter

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