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Distinct types of eigenvector localization in networks.

Romualdo Pastor-Satorras1, Claudio Castellano2,3

  • 1Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain.

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|January 13, 2016
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This summary is machine-generated.

Principal eigenvector localization in complex networks reveals two distinct modes. Localization occurs on hubs for high heterogeneity (γ > 5/2) and on mesoscopic subgraphs for lower heterogeneity (γ < 5/2).

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Area of Science:

  • Network Science
  • Graph Theory
  • Statistical Physics

Background:

  • The spectral properties of adjacency matrices are key to understanding complex network structure and dynamics.
  • The principal eigenvector is vital for node centrality and analyzing dynamical processes.

Purpose of the Study:

  • To investigate and identify distinct localization phenomena of the principal eigenvector in heterogeneous networks.
  • To explore the relationship between network heterogeneity and eigenvector localization patterns.

Main Methods:

  • Analysis of synthetic networks with power-law degree distributions (P(q) ~ q(-γ)).
  • Application of K-core decomposition to identify network structures.
  • Examination of real-world network data to validate findings.

Main Results:

  • Two distinct localization modes of the principal eigenvector were identified.
  • Localization occurs on the largest hub when γ > 5/2.
  • A novel localization on mesoscopic subgraphs (K-core shells) emerges when γ < 5/2.
  • Empirical evidence for these distinct modes was observed in real-world networks.

Conclusions:

  • The study reveals fundamental differences in principal eigenvector localization based on network heterogeneity.
  • Findings offer new insights into network dynamics and centrality measures.
  • Results suggest implications for understanding information diffusion and network robustness.