Sparse and structured networks: Statistical theory and algorithms

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The proposal focuses on Markov random fields and directed graphicalmodels, classes of statistical models that are based on a marriagebetween graph theory and probability theory, and allow for flexiblemodeling of network-structured data. The core of the proposalconsists of multiple research thrusts, all centered around the goal ofdeveloping practical algorithms and theory for statistical estimationwith network-structured data. One research thrust concerns variousissues associated with model selection in undirected graphical models,also known as Gibbs distributions or Markov random fields. Problemsinclude determining the information-theoretic limitations of graphicalmodel selection in high dimensions (where the number of vertices maybe larger than the sample size), not only for i.i.d. data but alsodependent data; developing methods for tracking sequences of networksthat evolve over time; and developing methods for data with hiddenvariables. Another research thrust concerns various statisticalproblems associated with directed acyclic graphical structures (DAGs),including estimating equivalence classes of DAGs in thehigh-dimensional setting; estimating causal relationships via designedinterventions; and efficient computational methods for DAG selectionusing the Lasso and related methods. Overall, the proposed researchis inter-disciplinary in nature, drawing on techniques frommathematical statistics, convex optimization, information theory,concentration of measure, and graph theory.Science and engineering abounds with different types of networks.Examples include social networks such as FaceBook and Twitter,networks of genes and proteins in molecular biology, network modelsfor economic and market dynamics, neural networks in brain imaging,networks of disease transmission in epidemiology, and informationnetworks in law enforcement. In the real-world, the structure of theunderlying network is not known, but instead one observes samples ofthe network behavior (e.g., packet counts in a computer network;instances of infection at given time instances of an epidemic; emailsor text messages sent among a group of people), and the goal is toinfer the network structure. Methods for solving this networkinference problem have a broad range of applications. Examplesinclude inferring brain connectivity and disease etiology inneuroimaging studies, detecting terrorist cells in social networks,monitoring intrusions in computer networks, and understanding thebasis of gene-protein interactions in systems biology.