Local Dependence in Random Graph Models: Characterization, Properties and Statistical Inference

Abstract

Summary Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well-defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M-dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with ‘ground truth’.

How Many Communities Are There?

“…Indeed, both the latent space cluster model of Handcock et al (2007) and the local dependence model of Schweinberger and Handcock (2015), as well as any other (composite) likelihood-based approach which requires to select a value of K can employ our proposed CL-BIC methodology for selecting the number of communities. We leave the details of this further investigation for future research.…”

A Popularity Scaled Latent Space Model for Large-Scale Directed Social Network

“…Note that large-scale networks are typically extremely sparse (Watts and Strogatz, 1998;Schweinberger and Handcock, 2015;Huang et al, 2015). Empirically, the observed in-degree for each node is bounded on average.…”

hergm: Hierarchical Exponential-Family Random Graph Models

“…The R (R Core Team 2017) package hergm (Schweinberger, Handcock, and Luna 2018) implements a wide range of random graph models, including stochastic block models (Nowicki and Snijders 2001), exponential-family random graph models (ERGMs; Frank and Strauss 1986;Wasserman and Pattison 1996;Snijders, Pattison, Robins, and Handcock 2006;Hunter and Handcock 2006), and hierarchical exponential-family random graph models (HERGMs; Schweinberger and Handcock 2015), with an emphasis on HERGMs with local dependence. HERGMs with local dependence are motivated by the observation that networks are local local dependence in Section 3.2.…”