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Eigenvector dynamics under perturbation of modular networks.

Somwrita Sarkar1, Sanjay Chawla2, P A Robinson3

  • 1Design Lab, Faculty of Architecture, Design and Planning, University of Sydney, Australia NSW 2006 and ARC Centre of Excellence for Integrative Brain Function.

Physical Review. E
|July 15, 2016
PubMed
Summary
This summary is machine-generated.

Eigenvectors of modular network adjacency matrices reveal community structure. This study estimates the number of network modules algorithm-independently and defines detectability limits for sparse modular networks.

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Area of Science:

  • Network science
  • Graph theory
  • Linear algebra

Background:

  • Modular networks exhibit community structure, influencing their dynamics.
  • Eigenvector analysis of adjacency matrices is crucial for understanding network properties.

Purpose of the Study:

  • To analyze the rotation dynamics of eigenvectors in modular networks under perturbations.
  • To develop an algorithm-independent method for estimating the number of communities (modules) in a network.
  • To establish the theoretical detectability limit for modularity in sparse networks.

Main Methods:

  • Analysis of eigenvector rotation dynamics of modular network adjacency matrices.
  • Investigation of "community" eigenspaces versus "bulk" eigenspaces.
  • Derivation of theoretical detectability limits for sparse modular networks with q communities.

Main Results:

  • Eigenvectors corresponding to the largest eigenvalues form distinct "community" eigenspaces that rotate together.
  • The number of modules can be estimated algorithm-independently using eigenvector properties.
  • A "band" of difficulty exists for detecting clusters before the theoretical detectability limit is reached.

Conclusions:

  • Eigenvector rotation dynamics provide a robust method for community detection in modular networks.
  • The study establishes theoretical and practical bounds for detecting modularity in sparse networks.
  • Algorithm-independent estimation of network modules is achievable through spectral analysis.