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Area of Science:

  • Complex networks analysis
  • Geometric network theory
  • Mathematical physics

Background:

  • The latent space approach reveals network principles and symmetries, enabling geometric methods.
  • Conditions linking network topology to geometricity remain unclear.
  • Understanding this link is crucial for network mapping and understanding graph-continuous space equivalences.

Purpose of the Study:

  • To mathematically prove and empirically validate the conditions under which complex network topology implies latent geometry.
  • To identify the critical role of multiscale self-similarity in establishing network geometricity.
  • To investigate the relationship between local clustering, self-similarity, and inherent latent geometry.

Main Methods:

  • Mathematical proof using degree-thresholding renormalization.
  • Analysis of random scale-free networks in Riemannian manifolds.
  • Empirical validation of theoretical findings.

Main Results:

  • Multiscale self-similarity is proven to be a crucial factor for latent geometry in complex networks.
  • A general class of random scale-free networks in constant curvature Riemannian manifolds are shown to be self-similar.
  • Both nonvanishing local clustering and self-similarity are demonstrated as necessary conditions for network geometricity.

Conclusions:

  • Network geometricity requires both local clustering and multiscale self-similarity.
  • Correlated links can yield clustering without self-similarity, thus lacking inherent latent geometry.
  • Findings have significant implications for network mapping and ensemble equivalence between discrete graphs and continuous spaces.