Community detection is an important problem when processing network data. Traditionally, this is done by exploiting the connections between nodes, but connections can be too sparse to detect communities in many real datasets. Node covariates can be used to assist community detection; see Binkiewicz et al. (2017); Weng and Feng (2022); Yan and Sarkar (2021); Yang et al. (2013). However, how to combine covariates with network connections is challenging, because covariates may be high-dimensional and inconsistent with community labels. To study the relationship between covariates and communities, we propose the degree corrected stochastic block model with node covariates (DCSBM-NC). It allows degree heterogeneity among communities and inconsistent labels between communities and covariates. Based on DCSBM-NC, we design the adjusted neighbor-covariate (ANC) data matrix, which leverages covariate information to assist community detection. We then propose the covariate-assisted spectral clustering on ratios of singular vectors (CA-SCORE) method on the ANC matrix. We prove that CA-SCORE successfully recovers community labels when 1) the network is relatively dense; 2) the covariate class labels match the community labels; 3) the data is a mixture of 1) and 2). CA-SCORE has good performance on synthetic and real datasets. The algorithm is implemented in the R(R Core Team (2021)) package CASCORE.

翻译：社区检测是处理网络数据时的重要问题。传统方法通过利用节点间的连接关系进行检测，但在许多真实数据集中，连接过于稀疏而难以识别社区。节点协变量可用于辅助社区检测（参见Binkiewicz等（2017）；Weng和Feng（2022）；Yan和Sarkar（2021）；Yang等（2013））。然而，如何将协变量与网络连接相结合具有挑战性，因为协变量可能是高维的，且与社区标签不一致。为研究协变量与社区之间的关系，我们提出了带节点协变量的度修正随机块模型（DCSBM-NC）。该模型允许社区间的度异质性以及社区与协变量之间的标签不一致。基于DCSBM-NC，我们设计了调整后的邻居-协变量（ANC）数据矩阵，利用协变量信息辅助社区检测。进而提出了基于奇异向量比值的协变量辅助谱聚类（CA-SCORE）方法，应用于ANC矩阵。我们证明：当（1）网络相对稠密；（2）协变量类别标签与社区标签一致；（3）数据是（1）和（2）的混合时，CA-SCORE能成功恢复社区标签。CA-SCORE在合成数据集和真实数据集上均表现良好。该算法已实现于R语言（R Core Team，2021）的CASCORE包中。