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Embedding the nodes of a large network into an Euclidean space is a common objective in modern machine learning, with a variety of tools available. These embeddings can then be used as features for tasks such as community detection/node clustering or link prediction, where they achieve state of the art performance. With the exception of spectral clustering methods, there is little theoretical understanding for commonly used approaches to learning embeddings. In this work we examine the theoretical properties of the embeddings learned by node2vec. Our main result shows that the use of $k$-means clustering on the embedding vectors produced by node2vec gives weakly consistent community recovery for the nodes in (degree corrected) stochastic block models. We also discuss the use of these embeddings for node and link prediction tasks. We demonstrate this result empirically, and examine how this relates to other embedding tools for network data., Comment: Camera ready version for Neurips 2024

The mixed membership stochastic blockmodel (MMSB) is a popular Bayesian network model for community detection. Fitting such large Bayesian network models quickly becomes computationally infeasible when the number of nodes grows into hundreds of thousands and millions. In this paper we propose a novel mini-batch strategy based on aggregated relational data that leverages nodal information to fit MMSB to massive networks. We describe a scalable inference method that can utilize nodal information that often accompanies real-world networks. Conditioning on this extra information leads to a model that admits a parallel stochastic variational inference algorithm, utilizing stochastic gradients of bipartite graph formed from aggregated network ties between node subpopulations. We apply our method to a citation network with over two million nodes and 25 million edges, capturing explainable structure in this network. Our method recovers parameters and achieves better convergence on simulated networks generated according to the MMSB.

Group-based social dominance hierarchies are of essential interest in animal behavior research. Studies often record aggressive interactions observed over time, and models that can capture such dynamic hierarchy are therefore crucial. Traditional ranking methods summarize interactions across time, using only aggregate counts. Instead, we take advantage of the interaction timestamps, proposing a series of network point process models with latent ranks. We carefully design these models to incorporate important characteristics of animal interaction data, including the winner effect, bursting and pair-flip phenomena. Through iteratively constructing and evaluating these models we arrive at the final cohort Markov-Modulated Hawkes process (C-MMHP), which best characterizes all aforementioned patterns observed in interaction data. We compare all models using simulated and real data. Using statistically developed diagnostic perspectives, we demonstrate that the C-MMHP model outperforms other methods, capturing relevant latent ranking structures that lead to meaningful predictions for real data., Comment: To appear in Journal of the Royal Statistical Society, Series C

A common goal in network modeling is to uncover the latent community structure present among nodes. For many real-world networks, the true connections consist of events arriving as streams, which are then aggregated to form edges, ignoring the dynamic temporal component. A natural way to take account of these temporal dynamics of interactions is to use point processes as the foundation of network models for community detection. Computational complexity hampers the scalability of such approaches to large sparse networks. To circumvent this challenge, we propose a fast online variational inference algorithm for estimating the latent structure underlying dynamic event arrivals on a network, using continuous-time point process latent network models. We describe this procedure for networks models capturing community structure. This structure can be learned as new events are observed on the network, updating the inferred community assignments. We investigate the theoretical properties of such an inference scheme, and provide regret bounds on the loss function of this procedure. The proposed inference procedure is then thoroughly compared, using both simulation studies and real data, to non-online variants. We demonstrate that online inference can obtain comparable performance, in terms of community recovery, to non-online variants, while realising computational gains. Our proposed inference framework can also be readily modified to incorporate other popular network structures., Comment: 45 pages

Embedding nodes of a large network into a metric (e.g., Euclidean) space has become an area of active research in statistical machine learning, which has found applications in natural and social sciences. Generally, a representation of a network object is learned in a Euclidean geometry and is then used for subsequent tasks regarding the nodes and/or edges of the network, such as community detection, node classification and link prediction. Network embedding algorithms have been proposed in multiple disciplines, often with domain-specific notations and details. In addition, different measures and tools have been adopted to evaluate and compare the methods proposed under different settings, often dependent of the downstream tasks. As a result, it is challenging to study these algorithms in the literature systematically. Motivated by the recently proposed Veridical Data Science (VDS) framework, we propose a framework for network embedding algorithms and discuss how the principles of predictability, computability and stability apply in this context. The utilization of this framework in network embedding holds the potential to motivate and point to new directions for future research.

Modeling event dynamics is central to many disciplines. Patterns in observed event arrival times are commonly modeled using point processes. Such event arrival data often exhibits self-exciting, heterogeneous and sporadic trends, which is challenging for conventional models. It is reasonable to assume that there exists a hidden state process that drives different event dynamics at different states. In this paper, we propose a Markov Modulated Hawkes Process (MMHP) model for learning such a mixture of event dynamics and develop corresponding inference algorithms. Numerical experiments using synthetic data demonstrate that MMHP with the proposed estimation algorithms consistently recover the true hidden state process in simulations, while email data from a large university and data from an animal behavior study show that the procedure captures distinct event dynamics that reveal interesting social structures in the real data.