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Geometric constraints influence spatial network formation. Our study reveals an algebraic scaling law for network extent, differing from unconstrained systems and aligning with protein structure observations.

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Area of Science:

  • Complex systems
  • Network science
  • Biophysics

Background:

  • Spatial network formation is crucial in diverse fields, from protein folding to particle aggregation.
  • The dynamic self-organization of interactions within these networks remains poorly understood.
  • Geometric constraints play a significant role in shaping these networks.

Purpose of the Study:

  • To analyze spatial network formation under geometric constraints.
  • To develop a framework mapping geometric constraints to graph-theoretic properties.
  • To elucidate the scaling laws governing network extent with system size.

Main Methods:

  • Mapping geometric constraints to graph-theoretic constraints.
  • Employing combined stochastic and mean-field analyses.
  • Deriving scaling laws for network diameter.

Main Results:

  • An algebraic scaling law for network extent (graph diameter) was identified.
  • This contrasts with the logarithmic scaling observed in unconstrained networks.
  • The derived exponent is intermediate between self-avoiding random walks and space-filling arrangements.

Conclusions:

  • Geometric constraints lead to distinct scaling behaviors in spatial networks.
  • The findings provide insights into the formation of complex structures like protein tertiary structures.
  • The study offers a novel analytical approach to understanding constrained network self-organization.