Detectability of hierarchical communities in networks

TL;DR

This paper investigates the detectability of hierarchical community structures in networks, revealing complex phase transitions and showing that traditional detectability limits for single partitions are insufficient for hierarchies.

Contribution

It introduces a detailed analysis of hierarchical community detection, identifying new phases where multiple partitions influence detectability, extending beyond single-partition results.

Findings

Multiple phases affect hierarchical detectability

Detectability limits for single partitions are insufficient for hierarchies

Constructive examples illustrate complex phase behavior

Abstract

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.