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Abstract Latent space models are often used to model network data by embedding a network’s nodes into a low-dimensional latent space; however, choosing the dimension of this space remains a challenge. To this end, we begin by formalizing a class of latent space models we call generalized linear network eigenmodels that can model various edge types (binary, ordinal, nonnegative continuous) found in scientific applications. This model class subsumes the traditional eigenmodel by embedding it in a generalized linear model with an exponential dispersion family random component and fixes identifiability issues that hindered interpretability. We propose a Bayesian approach to dimension selection for generalized linear network eigenmodels based on an ordered spike-and-slab prior that provides improved dimension estimation and satisfies several appealing theoretical properties. We show that the model’s posterior is consistent and concentrates on low-dimensional models near the truth. We demonstrate our approach’s consistent dimension selection on simulated networks, and we use generalized linear network eigenmodels to study the effect of covariates on the formation of networks from biology, ecology, and economics and the existence of residual latent structure. 

Reciprocity, or the stochastic tendency for actors to form mutual relationships, is an essential characteristic of directed network data. Existing latent space approaches to modelling directed networks are severely limited by the assumption that reciprocity is homogeneous across the network. In this work, we introduce a new latent space model for directed networks that can model heterogeneous reciprocity patterns that arise from the actors' latent distances. Furthermore, existing conditionally edge‐independent latent space models are nested within the proposed model class, which allows for meaningful model comparisons. We introduce a Bayesian inference procedure to infer the model parameters using Hamiltonian Monte Carlo. Lastly, we use the proposed method to infer different reciprocity patterns in an advice network among lawyers, an information‐sharing network between employees at a manufacturing company and a friendship network between high school students. 

One of the first steps in applications of statistical network analysis is frequently to produce summary charts of important features of the network. Many of these features take the form of sequences of graph statistics counting the number of realized events in the network, examples of which are degree distributions, edgewise shared partner distributions, and more. We provide conditions under which the empirical distributions of sequences of graph statistics are consistent in the L-infinity-norm in settings where edges in the network are dependent. We accomplish this task by deriving concentration inequalities that bound probabilities of deviations of graph statistics from the expected value under weak dependence conditions. We apply our concentration inequalities to empirical distributions of sequences of graph statistics and derive non-asymptotic bounds on the L-infinity-error which hold with high probability. Our non-asymptotic results are then extended to demonstrate uniform convergence almost surely in selected examples. We illustrate theoretical results through examples, simulation studies, and an application. 

In this work, we explore the extent to which the spectrum of the graph Laplacian can characterize the probability distribution of random graphs for the purpose of model evaluation and model selection for network data applications. Network data, often represented as a graph, consist of a set of pairwise observations between elements of a population of interests. The statistical network analysis literature has developed many different classes of network data models, with notable model classes including stochastic block models, latent node position models, and exponential families of random graph models. We develop a novel methodology which exploits the information contained in the spectrum of the graph Laplacian to predict the data-generating model from a set of candidate models. Through simulation studies, we explore the extent to which network data models can be differentiated by the spectrum of the graph Laplacian. We demonstrate the potential of our method through two applications to well-studied network data sets and validate our findings against existing analyses in the statistical network analysis literature.