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We found that maximizing modularity is equivalent to using a specific statistical model for network community detection. This clarifies the assumptions and provides an optimal resolution parameter for modularity maximization.

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Area of Science:

  • Network science
  • Statistical modeling
  • Data analysis

Background:

  • Community detection is crucial for understanding network structure.
  • Modularity maximization and stochastic block models are widely used methods.
  • Existing methods have different underlying assumptions and limitations.

Purpose of the Study:

  • To demonstrate an equivalence between modularity maximization and a specific stochastic block model.
  • To provide a principled derivation and clarify assumptions of modularity maximization.
  • To derive an explicit formula for the optimal resolution parameter.

Main Methods:

  • Equivalence demonstration between modularity maximization and the planted partition model (a special case of the degree-corrected stochastic block model).
  • Mathematical analysis to establish the exact equivalence.
  • Derivation of the generalized modularity function with a resolution parameter.

Main Results:

  • An exact mathematical equivalence is shown between generalized modularity maximization and the planted partition model.
  • This equivalence provides a principled derivation for the modularity function.
  • An explicit formula for the optimal resolution parameter in modularity maximization is derived.

Conclusions:

  • Modularity maximization is mathematically equivalent to a specific statistical model under certain assumptions.
  • The study clarifies the theoretical underpinnings and optimal usage of modularity maximization.
  • Findings offer a deeper understanding of network community detection methods.