Multilayer networks are used to represent the interdependence between the relational data of individuals interacting with each other via different types of relationships. To study the information-theoretic phase transitions in detecting the presence of planted partition among the nodes of a multi-layer network with additional nodewise covariate information and diverging average degree, Ma and Nandy (2023) introduced Multi-Layer Contextual Stochastic Block Model. In this paper, we consider the problem of detecting planted partitions in the Multi-Layer Contextual Stochastic Block Model, when the average node degrees for each network is greater than $1$. We establish the sharp phase transition threshold for detecting the planted bi-partition. Above the phase-transition threshold testing the presence of a bi-partition is possible, whereas below the threshold no procedure to identify the planted bi-partition can perform better than random guessing. We further establish that the derived detection threshold coincides with the threshold for weak recovery of the partition and provide a quasi-polynomial time algorithm to estimate it.

翻译：多层网络用于表示个体通过不同类型关系互动时，其关联数据间的相互依赖关系。为研究在节点附带协变量信息且平均度发散的多层网络中检测植入分区存在性的信息论相变，Ma与Nandy（2023）提出了多层上下文随机区块模型。本文探讨当各网络平均节点度大于$1$时，在该模型中检测植入分区的问题。我们建立了检测植入二分区的尖锐相变阈值：高于该阈值时，检验二分区的存在性成为可能；低于该阈值时，任何识别植入二分区的算法性能均不优于随机猜测。进一步证明该检测阈值与分区弱恢复的阈值重合，并提出一种拟多项式时间算法对其进行估计。