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Spatial network models reveal two distinct classes of geodesic wandering, characterized by different scaling exponents. These findings offer insights into Euclidean first-passage processes and their associated scaling laws.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Computational Geometry

Background:

  • First-passage percolation models analyze path lengths in networks.
  • Geodesic deviations and transversal fluctuations (wandering) are key metrics.
  • Spatial network models incorporate Euclidean distances and connectivity laws.

Purpose of the Study:

  • To investigate geodesic wandering and travel-time fluctuations in spatial network models.
  • To classify network models based on their wandering exponents.
  • To examine the validity of the Kardar-Parisi-Zhang (KPZ) relation in these models.

Main Methods:

  • Analysis of spatial network models with edges weighted by Euclidean span.
  • Monte Carlo simulations to determine wandering (ξ) and deviation (χ) exponents.
  • Numerical verification of the Kardar-Parisi-Zhang (KPZ) relation.

Main Results:

  • Two classes of spatial networks identified with distinct wandering exponents (ξ=3/5, χ=1/5 or ξ=7/10, χ=2/5).
  • Proximity graphs (e.g., random geometric graphs) exhibit minimal wandering (ξ=0.60±0.01, χ=0.20±0.01).
  • Excluded region graphs (e.g., β skeletons) show maximal wandering (ξ=0.70±0.01, χ=0.40±0.01).
  • Travel-time fluctuations are Gaussian, not Tracy-Widom.
  • The Kardar-Parisi-Zhang (KPZ) relation (χ=2ξ-1) holds for all models.

Conclusions:

  • Spatial network connectivity laws dictate geodesic wandering behavior.
  • The study provides empirical evidence for theoretical scaling laws in Euclidean first-passage processes.
  • Open questions remain regarding maximal wandering and non-Gaussian travel fluctuations in embedded models.