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Probabilistic Foundations of Statistical Network Analysis

Probabilistic Foundations of Statistical Network Analysis

Authors
Publisher Taylor & Francis Ltd
Year 2018
Pages 236
Version paperback
Readership level General/trade
Language English
ISBN 9781138630154
Categories Probability & statistics
$56.92 (with VAT)
253.05 PLN / €54.25 / £47.10
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Book description

Probabilistic Foundations of Statistical Network Analysis presents a fresh and insightful perspective on the fundamental tenets and major challenges of modern network analysis. Its lucid exposition provides necessary background for understanding the essential ideas behind exchangeable and dynamic network models, network sampling, and network statistics such as sparsity and power law, all of which play a central role in contemporary data science and machine learning applications. The book rewards readers with a clear and intuitive understanding of the subtle interplay between basic principles of statistical inference, empirical properties of network data, and technical concepts from probability theory. Its mathematically rigorous, yet non-technical, exposition makes the book accessible to professional data scientists, statisticians, and computer scientists as well as practitioners and researchers in substantive fields. Newcomers and non-quantitative researchers will find its conceptual approach invaluable for developing intuition about technical ideas from statistics and probability, while experts and graduate students will find the book a handy reference for a wide range of new topics, including edge exchangeability, relative exchangeability, graphon and graphex models, and graph-valued Levy process and rewiring models for dynamic networks.





The author's incisive commentary supplements these core concepts, challenging the reader to push beyond the current limitations of this emerging discipline. With an approachable exposition and more than 50 open research problems and exercises with solutions, this book is ideal for advanced undergraduate and graduate students interested in modern network analysis, data science, machine learning, and statistics.





Harry Crane is Associate Professor and Co-Director of the Graduate Program in Statistics and Biostatistics and an Associate Member of the Graduate Faculty in Philosophy at Rutgers University. Professor Crane's research interests cover a range of mathematical and applied topics in network science, probability theory, statistical inference, and mathematical logic. In addition to his technical work on edge and relational exchangeability, relative exchangeability, and graph-valued Markov processes, Prof. Crane's methods have been applied to domain-specific cybersecurity and counterterrorism problems at the Foreign Policy Research Institute and RAND's Project AIR FORCE. "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
reader."
~Silvano Romano, ISCB Newsletter

Probabilistic Foundations of Statistical Network Analysis

Table of contents

Dedication











Preface











Acknowledgements

























Orientation













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

























Binary relational data













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































Network sampling













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



















Generative models













Specification of generative models





Preferential Attachment model





Random walk models





Erd os-Renyi-Gilbert model





General sequential construction





Further reading



















Statistical modeling paradigm













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



















Vertex exchangeable models













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



















Getting beyond graphons













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



















Relatively exchangeable models













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



















Edge exchangeable models













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



















Relationally exchangeable models













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



















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

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