We study the problem of recovering a planted hierarchy of partitions in a network. The detectability of a single planted partition has previously been analysed in detail and a phase transition has been identified below which the partition cannot be detected. Here we show that, in the hierarchical setting, there exist additional phases in which the presence of multiple consistent partitions can either help or hinder detection. Accordingly, the detectability limit for non-hierarchical partitions typically provides insufficient information about the detectability of the complete hierarchical structure, as we highlight with several constructive examples.

中文翻译：

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网络中分层社区的可检测性

我们研究了在网络中恢复种植的分区层次结构的问题。先前已详细分析了单个种植分区的可检测性，并且已识别出低于该分区无法检测到的相变。在这里，我们表明，在分层设置中，存在其他阶段，其中多个一致分区的存在可以帮助或阻碍检测。因此，正如我们用几个建设性的例子强调的那样，非分层分区的可检测性限制通常提供关于完整分层结构的可检测性的不足信息。