SummaryWe consider multi-layer network data where the relationships between pairs of elements are reflected in multiple modalities, and may be described by multivariate or even high-dimensional vectors. Under the multi-layer stochastic block model framework we derive consistency results for a least squares estimation of memberships. Our theorems show that, as compared to single-layer community detection, a multi-layer network provides much richer information that allows for consistent community detection from a much sparser network, with required edge density reduced by a factor of the square root of the number of layers. Moreover, the multi-layer framework can detect cohesive community structure across layers, which might be hard to detect by any single-layer or simple aggregation. Simulations and a data example are provided to support the theoretical results.

中文翻译：

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多层网络数据中的一致社区检测

总结我们考虑多层网络数据，其中元素对之间的关​​系反映在多种模态中，并且可以用多变量甚至高维向量来描述。在多层随机块模型框架下，我们为成员资格的最小二乘估计得出一致性结果。我们的定理表明，与单层社区检测相比，多层网络提供了更丰富的信息，允许从更稀疏的网络进行一致的社区检测，所需的边缘密度减少了数字的平方根的一个因子层。此外，多层框架可以检测跨层的有凝聚力的社区结构，这可能很难被任何单层或简单的聚合检测到。