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We identified a universal threshold for community detection using spectral modularity. Beyond this threshold, community identification becomes impossible, regardless of community size.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Data Mining

Background:

  • Community detection is crucial for understanding complex networks.
  • The spectral modularity method is a popular technique for identifying network communities.
  • The stochastic block model is a standard framework for generating networks with known community structures.

Purpose of the Study:

  • To investigate the existence and nature of a phase-transition threshold for community detectability using the spectral modularity method.
  • To determine the factors influencing this phase transition.
  • To derive a universal formula for the phase-transition threshold and propose a method for its estimation from real-world data.

Main Methods:

  • Theoretical analysis of the spectral modularity method under the stochastic block model.
  • Derivation of the asymptotic phase-transition threshold formula.
  • Computer simulations to validate the theoretical findings.

Main Results:

  • Existence of an asymptotic phase-transition threshold for community detectability.
  • The threshold is determined by the within-community edge probabilities (p1, p2) as p* = sqrt(p1*p2).
  • The threshold is universal, independent of community size ratios.

Conclusions:

  • The spectral modularity method has a fundamental limit in detecting communities, characterized by a universal phase transition.
  • The derived threshold provides a critical value for assessing the feasibility of community detection in networks.
  • The proposed empirical method allows for the estimation of this threshold from real-world network data.