Bootstrap quantification of estimation uncertainties in network degree distributions

Abstract

We propose a new method of nonparametric bootstrap to quantify estimation uncertainties in functions of network degree distribution in large ultra sparse networks. Both network degree distribution and network order are assumed to be unknown. The key idea is based on adaptation of the “blocking” argument, developed for bootstrapping of time series and re-tiling of spatial data, to random networks. We first sample a set of multiple ego networks of varying orders that form a patch, or a network block analogue, and then resample the data within patches. To select an optimal patch size, we develop a new computationally efficient and data-driven cross-validation algorithm. The proposed fast patchwork bootstrap (FPB) methodology further extends the ideas for a case of network mean degree, to inference on a degree distribution. In addition, the FPB is substantially less computationally expensive, requires less information on a graph, and is free from nuisance parameters. In our simulation study, we show that the new bootstrap method outperforms competing approaches by providing sharper and better-calibrated confidence intervals for functions of a network degree distribution than other available approaches, including the cases of networks in an ultra sparse regime. We illustrate the FPB in application to collaboration networks in statistics and computer science and to Wikipedia networks.

Assessing network structure with practical sampling methods

“…This issue is related to another important research topic of generating representative samples (e.g., bootstrap samples) from a network. For bootstrap the variability is critical, but at the same time there is a need to reproduce distributions of certain network characteristics such as centrality Gel et al (2017). At the same time, bootstrap techniques that preserve centrality might not be the best ones to represent clustering.…”

Snowboot: Bootstrap Methods for Network Inference

“…While the first results on sampling on networks go back to 1960s (see, e.g., Goodman, 1961;Frank, 1968;Granovetter, 1976) and while nowadays there exist numerous graph sampling procedures (see overviews by Scott and Carrington, 2011;Ahmed et al, 2014;Kolaczyk, 2009;Simpson et al, 2015;Zhang et al, 2015, and references therein), still surprisingly little is known on how to reliably and efficiently quantify sampling uncertainties, without imposing typically unverifiable model specification constraints. In this section, we discuss the new method of patchwork sampling and bootstrap (based on algorithms of Thompson et al, 2016 andGel et al, 2017) that enables us to quantify sampling estimation uncertainties for network degree distribution and its functions, while using only a small proportion of network information.…”

“…Detailed steps of the patch formation are given in Figure 3, which is a modified version of respective algorithms by Thompson et al (2016) and Gel et al (2017). The advantage of the algorithm in Figure 3 is that we explore patches with a smaller number of seeds by taking a subset from the seeds we have sampled, rather than by sampling new seeds from a network.…”

“…Vertex bootstrap is widely used in social studies and is implemented in such highly popular software for social network analysis as UCINET (Borgatti et al, 2002). However, vertex bootstrap remains largely unknown in statistics and there still exists no implementation of this procedure in R. The next attempt to draw bootstrap inference on complex networks, with a focus on uncertainty quantification in network degree estimation, has been suggested by Thompson et al (2016) and Gel et al (2017). The idea is to borrow the "blocking" argument, developed for bootstrapping of time series and re-tiling of spatial data, and adapt it to random networks.…”

“…Challenges of parametric model specification and validation inspire a search for more data-driven and flexible nonparametric approaches for inference of complex networks. In this paper we discuss methodology and R implementation of two bootstrap procedures on random networks, that is, patchwork bootstrap of Thompson et al (2016) and Gel et al (2017) and vertex bootstrap of Snijders and Borgatti (1999). To our knowledge, the new R package snowboot is the first implementation of the vertex and patchwork bootstrap inference on networks in R. Our new package is accompanied with a detailed user's manual, and is compatible with the popular R package on network studies igraph.…”