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.

翻译：我们研究了在网络中恢复植入的层次化划分结构的问题。单个植入划分的可检测性先前已被详细分析，并已识别出低于该界限则无法检测到划分的相变点。本文证明，在层次化场景下，存在额外的相变阶段，其中多个一致性划分的存在可能有助于或阻碍检测过程。因此，非层次化划分的可检测性界限通常无法为完整层次化结构的可检测性提供充分信息，我们通过若干构造性示例对此进行了重点说明。