The grade of membership model is a flexible latent variable model for analyzing multivariate categorical data through individual-level mixed membership scores. In many modern applications, auxiliary covariates are collected alongside responses and encode information about the same latent structure. Traditional approaches to incorporating such covariates typically rely on fully specified joint likelihoods, which are computationally intensive and sensitive to misspecification. We introduce a covariate-assisted grade of membership model that integrates response and covariate information by exploiting their shared low-rank simplex geometry, rather than modeling their joint distribution. We propose a likelihood-free spectral estimation procedure that combines heterogeneous data sources through a balance parameter controlling their relative contribution. To accommodate high-dimensional and heteroskedastic noise, we employ heteroskedastic principal component analysis before performing simplex-based geometric recovery. Our theoretical analysis establishes weaker identifiability conditions than those required in the covariate-free model, and further derives finite-sample, entrywise error bounds for both mixed membership scores and item parameters. These results demonstrate that auxiliary covariates can provably improve latent structure recovery, yielding faster convergence rates in high-dimensional regimes. Simulation studies and an application to educational assessment data illustrate the computational efficiency, statistical accuracy, and interpretability gains of the proposed method. The code for reproducing these results is open-source and available at \texttt{https://github.com/Toby-X/Covariate-Assisted-GoM}

Network data is prevalent in many contemporary big data applications in which a common interest is to unveil important latent links between different pairs of nodes. Yet a simple fundamental question of how to precisely quantify the statistical uncertainty associated with the identification of latent links still remains largely unexplored. In this paper, we propose the method of statistical inference on membership profiles in large networks (SIMPLE) in the setting of degree-corrected mixed membership model, where the null hypothesis assumes that the pair of nodes share the same profile of community memberships. In the simpler case of no degree heterogeneity, the model reduces to the mixed membership model for which an alternative more robust test is also proposed. Both tests are of the Hotelling-type statistics based on the rows of empirical eigenvectors or their ratios, whose asymptotic covariance matrices are very challenging to derive and estimate. Nevertheless, their analytical expressions are unveiled and the unknown covariance matrices are consistently estimated. Under some mild regularity conditions, we establish the exact limiting distributions of the two forms of SIMPLE test statistics under the null hypothesis and contiguous alternative hypothesis. They are the chi-square distributions and the noncentral chi-square distributions, respectively, with degrees of freedom depending on whether the degrees are corrected or not. We also address the important issue of estimating the unknown number of communities and establish the asymptotic properties of the associated test statistics. The advantages and practical utility of our new procedures in terms of both size and power are demonstrated through several simulation examples and real network applications.

Recurrent neural networks (RNNs) have become one of the standard models in sequential data analysis [Rumelhart et al., 1986, Elman, 1990]. At each time step of the RNN, an observation is modeled via a neural network using the observations and hidden states from previous time points. Models such as the RNN, and also the hidden Markov model among others, often implicitly assume a sequence as having a fixed time interval between observations. They also often do not account for group-level effects when multiple sequences are observed and each sequence belongs to one of multiple groups. For example, consider data in the form of a sequence of discrete counts by a set of groups-- e.g., a sequence of purchases (market baskets) for a set of customers, with one sequence per customer. A vanilla RNN implementation would model these sequences using a network with the same parameters, which removes the customer-level information, and according to an enumerated indexing, which removes the time interval information between orders. However, this information is important: customer-specific effects can improve predictive performance for each customer, while an interval of one day versus one month between orders significantly impacts the items likely to be purchased next.

We present a probabilistic model for learning from dynamic relational data, wherein the observed interactions among networked nodes are modeled via the Bernoulli Poisson link function, and the underlying network structure are characterized by nonnegative latent node-group memberships, which are assumed to be gamma distributed. The latent memberships evolve according to Markov processes.The optimal number of latent groups can be determined by data itself. The computational complexity of our method scales with the number of non-zero links, which makes it scalable to large sparse dynamic relational data. We present batch and online Gibbs sampling algorithms to perform model inference. Finally, we demonstrate the model's performance on both synthetic and real-world datasets compared to state-of-the-art methods.

Collective classification models attempt to improve classification performance by taking into account the class labels of related instances. However, they tend not to learn patterns of interactions between classes and/or make the assumption that instances of the same class link to each other (assortativity assumption). Blockmodels provide a solution to these issues, being capable of modelling assortative and disassortative interactions, and learning the pattern of interactions in the form of a summary network. The Supervised Blockmodel provides good classification performance using link structure alone, whilst simultaneously providing an interpretable summary of network interactions to allow a better understanding of the data. This work explores three variants of supervised blockmodels of varying complexity and tests them on four structurally different real world networks.

Discussion boards are a form of social media which allow users to discuss topics and exchange information in a complex manner, in a number of different settings. As the popularity of such message boards has increased, communities of users have emerged, and several prominent types of social role have been identified, such as Question Answerer, Celebrity, Discussion Person and Topic Initiator. Recent studies have noted the structural similarity of the egocentric network of users assigned the same role by qualitative criteria. In this paper a methodology is developed with which to cluster together users with similar ego-centric network structures. This is achieved using a mixed membership formulation which allows for the fact that different groups of users may have characteristics in common. The method is then applied to data taken from boards.ie, a medium sized message boards website. Prominent clusters of users are identified and discussed, and illustrative examples of user behaviour provided. The type of interaction, both locally and globally, taking place within forums is examined.