Reinventing the Triangles: Rule of Thumb for Assessing Detectability

TL;DR

This paper introduces a simple, fast rule of thumb based on the global clustering coefficient to assess the presence of detectable community structures in networks, addressing a longstanding challenge in network analysis.

Contribution

It proposes a novel criterion using the global clustering coefficient to determine detectability of communities without complex spectral analysis.

Findings

The criterion effectively distinguishes networks with detectable communities.

The method is computationally efficient and easier to implement than existing spectral methods.

It provides practical guidance for community detection in large networks.

Abstract

Statistical significance of network clustering has been an unresolved problem since it was observed that community detection algorithms produce false positives even in random graphs. After a phase transition between undetectable and detectable cluster structures was discovered, the connection between spectra of adjacency matrices and detectability limits were shown, and both were calculated for a wide range of networks with arbitrary degree distributions and community structure. In practice the full eigenspectrum is not known, and whether a given network has any communities within detectability regime cannot be easily established. Based on the global clustering coefficient we construct a criterion telling whether in an undirected, unweighted network there is some/no detectable community structure, or if the network is in a transient regime. The method is simple and faster than methods involving bootstrapping.