Gaussian mixture block models are distributions over graphs that strive to model modern networks: to generate a graph from such a model, we associate each vertex $i$ with a latent feature vector $u_i \in \mathbb{R}^d$ sampled from a mixture of Gaussians, and we add edge $(i,j)$ if and only if the feature vectors are sufficiently similar, in that $\langle u_i,u_j \rangle \ge \tau$ for a pre-specified threshold $\tau$. The different components of the Gaussian mixture represent the fact that there may be different types of nodes with different distributions over features -- for example, in a social network each component represents the different attributes of a distinct community. Natural algorithmic tasks associated with these networks are embedding (recovering the latent feature vectors) and clustering (grouping nodes by their mixture component). In this paper we initiate the study of clustering and embedding graphs sampled from high-dimensional Gaussian mixture block models, where the dimension of the latent feature vectors $d\to \infty$ as the size of the network $n \to \infty$. This high-dimensional setting is most appropriate in the context of modern networks, in which we think of the latent feature space as being high-dimensional. We analyze the performance of canonical spectral clustering and embedding algorithms for such graphs in the case of 2-component spherical Gaussian mixtures, and begin to sketch out the information-computation landscape for clustering and embedding in these models.

翻译：高斯混合块模型是旨在模拟现代网络的图分布：要从此类模型生成图，我们为每个顶点$i$关联一个从高斯混合中采样的潜在特征向量$u_i \in \mathbb{R}^d$，并且当且仅当特征向量足够相似（即$\langle u_i,u_j \rangle \ge \tau$，其中$\tau$为预设阈值）时添加边$(i,j)$。高斯混合的不同分量体现了存在具有不同特征分布的不同类型节点的可能性——例如，在社交网络中，每个分量代表不同社区的不同属性。与这些网络相关的自然算法任务包括嵌入（恢复潜在特征向量）和聚类（按混合分量对节点分组）。本文首次研究了从高维高斯混合块模型中采样的图的聚类与嵌入问题，其中潜在特征向量的维度$d\to \infty$与网络规模$n \to \infty$同步增长。这种高维设置最适合现代网络场景，因为我们认为潜在特征空间具有高维特性。我们分析了2分量球面高斯混合情况下此类图的经典谱聚类与嵌入算法性能，并初步勾勒出这些模型中聚类与嵌入问题的信息-计算图谱。