Detectability of minority communities in networks

TL;DR

This paper analyzes the theoretical limits of detecting small minority communities within networks, revealing phase transitions and comparing spectral clustering with belief propagation.

Contribution

It introduces a detailed phase diagram for minority community detectability in the Stochastic Block Model and evaluates the performance of spectral methods versus belief propagation.

Findings

Three phases of community detectability are identified.

Spectral clustering performs worse than belief propagation for minority communities.

Detection thresholds are explicitly derived from eigenvalue structures.

Abstract

Community structure is prevalent in real-world networks, with empirical studies revealing heterogeneous distributions where a few dominant majority communities coexist with many smaller groups. These small-scale groups, which we term minority communities, are critical for understanding network organization but pose significant challenges for detection. Here, we investigate the detectability of minority communities from a theoretical perspective using the Stochastic Block Model. We identify three distinct phases of community detection: the detectable phase, where overall community structure is recoverable but minority communities are merged into majority groups; the distinguishable phase, where minority communities form a coherent group separate from the majority but remain unresolved internally; and the resolvable phase, where each minority community is fully distinguishable. These phases correspond to phase transitions at the Kesten-Stigum threshold and two additional thresholds determined by the eigenvalue structure of the signal matrix, which we derive explicitly. Furthermore, we demonstrate that spectral clustering with the Bethe Hessian exhibits significantly weaker detection performance for minority communities compared to belief propagation, revealing a specific limitation of spectral methods in identifying fine-grained community structure despite their capability to detect macroscopic structures down to the theoretical limit.