Search

We propose a semiparametric model for dyadic link formations in directed networks. The model contains a set of degree parameters that measure different effects of popularity or outgoingness across nodes, a regression parameter vector that reflects the homophily effect resulting from the nodal attributes or pairwise covariates associated with edges, and a set of latent random noises with unknown distributions. Our interest lies in inferring the unknown degree parameters and homophily parameters. The dimension of the degree parameters increases with the number of nodes. Under the high-dimensional regime, we develop a kernel-based least squares approach to estimate the unknown parameters. The major advantage of our estimator is that it does not encounter the incidental parameter problem for the homophily parameters. We prove consistency of all the resulting estimators of the degree parameters and homophily parameters. We establish high-dimensional central limit theorems for the proposed estimators and provide several applications of our general theory, including testing the existence of degree heterogeneity, testing sparse signals and recovering the support. Simulation studies and a real data application are conducted to illustrate the finite sample performance of the proposed methods., Comment: 28 pages, 3 figures

Latent space models (LSMs) are frequently 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 LSMs we call generalized linear network eigenmodels (GLNEMs) that can model various edge types (binary, ordinal, non-negative 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. Next, we propose a Bayesian approach to dimension selection for GLNEMs based on an ordered spike-and-slab prior that provides improved dimension estimation and satisfies several appealing theoretical properties. In particular, we show that the model's posterior concentrates on low-dimensional models near the truth. We demonstrate our approach's consistent dimension selection on simulated networks. Lastly, we use GLNEMs to study the effect of covariates on the formation of networks from biology, ecology, and economics and the existence of residual latent structure., Comment: 58 pages, 13 figures

Continuous time network data have been successfully modeled by multivariate counting processes, in which the intensity function is characterized by covariate information. However, degree heterogeneity has not been incorporated into the model which may lead to large biases for the estimation of homophily effects. In this paper, we propose a degree-corrected Cox network model to simultaneously analyze the dynamic degree heterogeneity and homophily effects for continuous time directed network data. Since each node has individual-specific in- and out-degree effects in the model, the dimension of the time-varying parameter vector grows with the number of nodes, which makes the estimation problem non-standard. We develop a local estimating equations approach to estimate unknown time-varying parameters, and establish consistency and asymptotic normality of the proposed estimators by using the powerful martingale process theories. We further propose test statistics to test for trend and degree heterogeneity in dynamic networks. Simulation studies are provided to assess the finite sample performance of the proposed method and a real data analysis is used to illustrate its practical utility., Comment: 60 pages, 10 figures

We investigate the effects of chain flexibility on the self-assembly behavior of symmetric diblock copolymers (BCPs) when they are confined as a thin film between two surfaces. Employing worm-like chain (WLC) self-consistent field theory, we study the relative stability of parallel (L$_{\parallel}$) and perpendicular (L$_{\perp}$) orientations of BCP lamellar phases, ranging in chain flexibility from flexible Gaussian chains to semi-flexible and rigid chains. For flat and neutral bounding surfaces (no surface preference for one of the two BCP components), the stability of the L$_{\perp}$ lamellae increases with chain rigidity. When the top surface is flat and the bottom substrate is corrugated, increasing the surface roughness enhances the stability of the L$_{\perp}$ lamellae for flexible Gaussian chains. However, an opposite behavior is observed for rigid chains, where the L$_{\perp}$ stability decreases as the substrate roughness increases. We further show that as the substrate roughness increases, the critical value of the substrate preference, $u^{*}$, corresponding to an L$_{\perp}$-to-L$_{\parallel}$ transition, decreases for rigid chains, while it increases for flexible Gaussian chains. Our results highlight the physical mechanism of tailoring the orientation of lamellar phases in thin-film setups. This is of importance, in particular, for short (semi-flexible or rigid) chains that are in high demand in emerging nanolithography and other industrial applications.

Latent space models are popular for analyzing dynamic network data. We propose a variational approach to estimate the model parameters as well as the latent positions of the nodes in the network. The variational approach is much faster than Markov chain Monte Carlo algorithms, and is able to handle large networks. Theoretical properties of the variational Bayes risk of the proposed procedure are provided. We apply the variational method and latent space model to simulated data as well as real data to demonstrate its performance., Comment: 50 pages

Dynamic multilayer networks frequently represent the structure of multiple co-evolving relations; however, statistical models are not well-developed for this prevalent network type. Here, we propose a new latent space model for dynamic multilayer networks. The key feature of our model is its ability to identify common time-varying structures shared by all layers while also accounting for layer-wise variation and degree heterogeneity. We establish the identifiability of the model's parameters and develop a structured mean-field variational inference approach to estimate the model's posterior, which scales to networks previously intractable to dynamic latent space models. We demonstrate the estimation procedure's accuracy and scalability on simulated networks. We apply the model to two real-world problems: discerning regional conflicts in a data set of international relations and quantifying infectious disease spread throughout a school based on the student's daily contact patterns., Comment: 55 pages, 18 figures

Dynamic networks are used in a variety of fields to represent the structure and evolution of the relationships between entities. We present a model which embeds longitudinal network data as trajectories in a latent Euclidean space. A Markov chain Monte Carlo algorithm is proposed to estimate the model parameters and latent positions of the actors in the network. The model yields meaningful visualization of dynamic networks, giving the researcher insight into the evolution and the structure, both local and global, of the network. The model handles directed or undirected edges, easily handles missing edges, and lends itself well to predicting future edges. Further, a novel approach is given to detect and visualize an attracting influence between actors using only the edge information. We use the case-control likelihood approximation to speed up the estimation algorithm, modifying it slightly to account for missing data. We apply the latent space model to data collected from a Dutch classroom, and a cosponsorship network collected on members of the U.S. House of Representatives, illustrating the usefulness of the model by making insights into the networks.

It is often of interest to perform clustering on longitudinal data, yet it is difficult to formulate an intuitive model for which estimation is computationally feasible. We propose a model-based clustering method for clustering objects that are observed over time. The proposed model can be viewed as an extension of the normal mixture model for clustering to longitudinal data. While existing models only account for clustering effects, we propose modeling the distribution of the observed values of each object as a blending of a cluster effect and an individual effect, hence also giving an estimate of how much the behavior of an object is determined by the cluster to which it belongs. Further, it is important to detect how explanatory variables affect the clustering. An advantage of our method is that it can handle multiple explanatory variables of any type through a linear modeling of the cluster transition probabilities. We implement the generalized EM algorithm using several recursive relationships to greatly decrease the computational cost. The accuracy of our estimation method is illustrated in a simulation study, and U.S. Congressional data is analyzed.

Embedding dyadic data into a latent space has long been a popular approach to modeling networks of all kinds. While clustering has been done using this approach for static networks, this paper gives two methods of community detection within dynamic network data, building upon the distance and projection models previously proposed in the literature. Our proposed approaches capture the time-varying aspect of the data, can model directed or undirected edges, inherently incorporate transitivity and account for each actor's individual propensity to form edges. We provide Bayesian estimation algorithms, and apply these methods to a ranked dynamic friendship network and world export/import data.

The formation of social networks and the evolution of their structures have been of interest to researchers for many decades. We wish to answer questions about network stability, group formation and popularity effects. We propose a latent space model for ranked dynamic networks that can be used to intuitively frame and answer these questions. The well known data collected by Newcomb in the 1950's is very well suited to analyze the formation of a social network. We applied our model to this data in order to investigate the network stability, what groupings emerge and when they emerge, and how individual popularity is associated with individual stability.

Longitudinal binary relational data can be better understood by implementing a latent space model for dynamic networks. This approach can be broadly extended to many types of weighted edges by using a link function to model the mean of the dyads, or by employing a similar strategy via data augmentation. To demonstrate this, we propose models for count dyads and for non-negative real dyads, analyzing simulated data and also both mobile phone data and world export/import data. The model parameters and latent actors' trajectories, estimated by Markov chain Monte Carlo algorithms, provide insight into the network dynamics.

The evolution of communities in dynamic (time-varying) network data is a prominent topic of interest. A popular approach to understanding these dynamic networks is to embed the dyadic relations into a latent metric space. While methods for clustering with this approach exist for dynamic networks, they all assume a static community structure. This paper presents a Bayesian nonparametric model for dynamic networks that can model networks with evolving community structures. Our model extends existing latent space approaches by explicitly modeling the additions, deletions, splits, and mergers of groups with a hierarchical Dirichlet process hidden Markov model. Our proposed approach, the hierarchical Dirichlet process latent position clustering model (HDP-LPCM), incorporates transitivity, models both individual and group level aspects of the data, and avoids the computationally expensive selection of the number of groups required by most popular methods. We provide a Markov chain Monte Carlo estimation algorithm and apply our method to synthetic and real-world networks to demonstrate its performance., Comment: 52 pages, 24 figures