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1. Optimal Bayesian estimation for random dot product graphs

   https://doi.org/10.1093/biomet/asaa031  

   Xie, Fangzheng; Xu, Yanxun (July 2020, Biometrika)

   Summary We propose and prove the optimality of a Bayesian approach for estimating the latent positions in random dot product graphs, which we call posterior spectral embedding. Unlike classical spectral-based adjacency, or Laplacian spectral embedding, posterior spectral embedding is a fully likelihood-based graph estimation method that takes advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax lower bound for estimating the latent positions, and show that posterior spectral embedding achieves this lower bound in the following two senses: it both results in a minimax-optimal posterior contraction rate and yields a point estimator achieving the minimax risk asymptotically. The convergence results are subsequently applied to clustering in stochastic block models with positive semidefinite block probability matrices, strengthening an existing result concerning the number of misclustered vertices. We also study a spectral-based Gaussian spectral embedding as a natural Bayesian analogue of adjacency spectral embedding, but the resulting posterior contraction rate is suboptimal by an extra logarithmic factor. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of Wikipedia graph data. 

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2. Joint latent space models for network data with high-dimensional node variables

   https://doi.org/10.1093/biomet/asab063  

   Zhang, Xuefei; Xu, Gongjun; Zhu, Ji (December 2021, Biometrika)

   Summary Network latent space models assume that each node is associated with an unobserved latent position in a Euclidean space, and such latent variables determine the probability of two nodes connecting with each other. In many applications, nodes in the network are often observed along with high-dimensional node variables, and these node variables provide important information for understanding the network structure. However, classical network latent space models have several limitations in incorporating node variables. In this paper, we propose a joint latent space model where we assume that the latent variables not only explain the network structure, but are also informative for the multivariate node variables. We develop a projected gradient descent algorithm that estimates the latent positions using a criterion incorporating both network structure and node variables. We establish theoretical properties of the estimators and provide insights into how incorporating high-dimensional node variables could improve the estimation accuracy of the latent positions. We demonstrate the improvement in latent variable estimation and the improvements in associated downstream tasks, such as missing value imputation for node variables, by simulation studies and an application to a Facebook data example. 

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3. Task-Guided Pair Embedding in Heterogeneous Network

   https://doi.org/10.1145/3357384.3357982  

   Park, Chanyoung; Kim, Donghyun; Zhu, Qi; Han, Jiawei; Yu, Hwanjo (November 2019, Proceedings of the 28th {ACM} International Conference on Information and Knowledge Management, {CIKM} 2019)

   Many real-world tasks solved by heterogeneous network embedding methods can be cast as modeling the likelihood of a pairwise relationship between two nodes. For example, the goal of author identification task is to model the likelihood of a paper being written by an author (paper–author pairwise relationship). Existing taskguided embedding methods are node-centric in that they simply measure the similarity between the node embeddings to compute the likelihood of a pairwise relationship between two nodes. However, we claim that for task-guided embeddings, it is crucial to focus on directly modeling the pairwise relationship. In this paper, we propose a novel task-guided pair embedding framework in heterogeneous network, called TaPEm, that directly models the relationship between a pair of nodes that are related to a specific task (e.g., paper-author relationship in author identification). To this end, we 1) propose to learn a pair embedding under the guidance of its associated context path, i.e., a sequence of nodes between the pair, and 2) devise the pair validity classifier to distinguish whether the pair is valid with respect to the specific task at hand. By introducing pair embeddings that capture the semantics behind the pairwise relationships, we are able to learn the fine-grained pairwise relationship between two nodes, which is paramount for task-guided embedding methods. Extensive experiments on author identification task demonstrate that TaPEm outperforms the state-of-the-art methods, especially for authors with few publication records. 

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4. Multi-Class Imbalanced Graph Convolutional Network Learning

   https://doi.org/10.24963/ijcai.2020/398  

   Shi, Min; Tang, Yufei; Zhu, Xingquan; Wilson, David; Liu, Jianxun (July 2020, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence (IJCAI))

   null (Ed.)

   Networked data often demonstrate the Pareto principle (i.e., 80/20 rule) with skewed class distributions, where most vertices belong to a few majority classes and minority classes only contain a handful of instances. When presented with imbalanced class distributions, existing graph embedding learning tends to bias to nodes from majority classes, leaving nodes from minority classes under-trained. In this paper, we propose Dual-Regularized Graph Convolutional Networks (DR-GCN) to handle multi-class imbalanced graphs, where two types of regularization are imposed to tackle class imbalanced representation learning. To ensure that all classes are equally represented, we propose a class-conditioned adversarial training process to facilitate the separation of labeled nodes. Meanwhile, to maintain training equilibrium (i.e., retaining quality of fit across all classes), we force unlabeled nodes to follow a similar latent distribution to the labeled nodes by minimizing their difference in the embedding space. Experiments on real-world imbalanced graphs demonstrate that DR-GCN outperforms the state-of-the-art methods in node classification, graph clustering, and visualization. 

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5. Multi-Class Imbalanced Graph Convolutional Network Learning

   Shi, Min; Tang, Yufei; Zhu, Xingquan; Wilson, David; Liu, Jianxun (January 2020, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence (IJCAI-20))

   null (Ed.)

   Networked data often demonstrate the Pareto principle (i.e., 80/20 rule) with skewed class distributions, where most vertices belong to a few majority classes and minority classes only contain a handful of instances. When presented with imbalanced class distributions, existing graph embedding learning tends to bias to nodes from majority classes, leaving nodes from minority classes under-trained. In this paper, we propose Dual-Regularized Graph Convolutional Networks (DRGCN) to handle multi-class imbalanced graphs, where two types of regularization are imposed to tackle class imbalanced representation learning. To ensure that all classes are equally represented, we propose a class-conditioned adversarial training process to facilitate the separation of labeled nodes. Meanwhile, to maintain training equilibrium (i.e., retaining quality of fit across all classes), we force unlabeled nodes to follow a similar latent distribution to the labeled nodes by minimizing their difference in the embedding space. Experiments on real-world imbalanced graphs demonstrate that DR-GCN outperforms the state-of-the-art methods in node classification, graph clustering, and visualization. 

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