Statistical analysis of networks with community structure and bootstrap methods for big data

This dissertation is divided into two parts, concerning two areas of statistical methodology. The first part of this dissertation concerns statistical analysis of networks with community structure. The second part of this dissertation concerns bootstrap methods for big data. Statistical analysis of networks with community structure: Networks are ubiquitous in today's world --- network data appears from varied fields such as scientific studies, sociology, technology, social media and the Internet, to name a few. An interesting aspect of many real-world networks is the presence of community structure and the problem of detecting this community structure. In the first chapter, we consider heterogeneous networks which seems to have not been considered in the statistical community detection literature. We propose a blockmodel for heterogeneous networks with community structure, and introduce a heterogeneous spectral clustering algorithm for community detection in heterogeneous networks. Theoretical properties of the clustering algorithm under the proposed model are studied, along with simulation study and data analysis. A network feature that is closely associated with community structure is the popularity of nodes in different communities. Neither the classical stochastic blockmodel nor its degree-corrected extension can satisfactorily capture the dynamics of node popularity. In the second chapter, we propose a popularity-adjusted blockmodel for flexible modeling of node popularity. We establish consistency of likelihood modularity for community detection under the proposed model, and illustrate the improved empirical insights that can be gained through this methodology by analyzing the political blogs network and the British MP network, as well as in simulation studies. Bootstrap methods for big data: Resampling methods provide a powerful method of evaluating the precision of a wide variety of statistical inference methods. The complexity and massive size of big data makes it infeasible to apply traditional resampling methods for big data. In the first chapter, we consider the problem of resampling for irregularly spaced dependent data. Traditional block-based resampling or subsampling schemes for stationary data are difficult to implement when the data are irregularly spaced, as it takes careful programming effort to partition the sampling region into complete and incomplete blocks. We develop a resampling method called Dependent Random Weighting (DRW) for irregularly spaced dependent data, where instead of using blocks we use random weights to resample the data. By allowing the random weights to be dependent, the dependency structure of the data can be preserved in the resamples. We study the theoretical properties of this resampling methods as well as its numerical performance in simulations. In the second chapter, we consider the problem of resampling in massive data, where traditional methods like bootstrap (for independent data) or moving block bootstrap (for dependent data) can be computationally infeasible since each resample has effective size of the same order as the sample. We develop a new resampling method called subsampled double bootstrap (SDB) for both independent and stationary data. SDB works by choosing small random subsets of the massive data, and then constructing a single resample from that subset using bootstrap (for independent data) or moving block bootstrap (for stationary data). We study theoretical properties of SDB as well as its numerical performance in simulated data and real data. Extending the underlying ideas of the second chapter, we introduce two new resampling strategies for big data in Chapter 3. The first strategy is called aggregation of little bootstraps or ALB, a generalized resampling technique that includes the SDB as a special case. The second strategy is called subsampled residual bootstrap or SRB, a fast version of residual bootstrap intended for massive regression models. We study both methods through simulations.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Statistics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2016

Author dc:creator

• Sengupta, Srijan

Contributors dc:contributor

• Chen, Yuguo
• Shao, Xiaofeng
• Simpson, Douglas G.
• Marden, John I.

• network data
• resampling
• community structure
• big data

Statement dc:rights

• Copyright 2016 Srijan Sengupta

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/92763

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/92763