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In systems where values diminish by a constant proportion at each stage, the resulting sequence follows a geometric structure. Each new value in the sequence is obtained by applying a fixed multiplier to the preceding term. This regular, proportional decline type is often used to represent processes involving gradual loss, such as energy dissipation or reduction in amplitude over time.When analyzing the total effect of such a process across unlimited iterations, the series of values is referred...
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Scale-invariant geometric random graphs.

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This summary is machine-generated.

We introduce novel growing geometric random graphs with scale invariance, revealing unique degree distributions and network properties. These findings mirror characteristics of real-world web graphs, offering insights into complex network structures.

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Area of Science:

  • Complex networks
  • Network science
  • Statistical physics

Background:

  • Growing networks exhibit heterogeneity and specialization.
  • Understanding network evolution is crucial for various fields.
  • Existing models may not capture scale-invariant properties.

Purpose of the Study:

  • Introduce and analyze scale-invariant growing geometric random graphs.
  • Investigate the impact of scale invariance on network properties.
  • Compare model behavior to empirical web graph characteristics.

Main Methods:

  • Theoretical calculations.
  • Numerical simulations.
  • Analysis of degree distributions (in- and out-degree).

Main Results:

  • Dichotomy between scale-free and Poisson degree distributions observed.
  • Identification of a random number of hub nodes.
  • High clustering and unusual percolation behavior demonstrated.

Conclusions:

  • Scale invariance leads to unique properties in growing geometric random graphs.
  • The model replicates key features of empirical web graphs.
  • Provides a new framework for studying complex, evolving networks.