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Summary
This summary is machine-generated.

We developed an analytic model for directed Watts-Strogatz small-world graphs, providing exact calculations for graph properties like asymmetry and clustering. This model enables precise analysis of finite-size small-world networks.

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Area of Science:

  • Graph Theory
  • Network Science
  • Mathematical Modeling

Background:

  • Watts-Strogatz graphs are fundamental models for complex networks.
  • Existing methods often rely on approximations or simulations for analysis.
  • Directed small-world networks require specialized analytical approaches.

Purpose of the Study:

  • To introduce a novel analytic model for directed Watts-Strogatz small-world graphs.
  • To derive an algebraic expression for the adjacency matrix of these graphs.
  • To enable exact, nonasymptotic analytical calculations of graph properties.

Main Methods:

  • Development of an analytic model for directed Watts-Strogatz graphs.
  • Algebraic derivation of the adjacency matrix.
  • Application of the matrix to calculate asymmetry index and clustering coefficient.

Main Results:

  • An algebraic expression for the adjacency matrix of directed Watts-Strogatz graphs was deduced.
  • Analytically exact formulas for the asymmetry index and clustering coefficient were derived.
  • The model is valid for all graph sizes and nonasymptotically.

Conclusions:

  • The proposed analytic model provides a powerful tool for studying directed small-world networks.
  • It allows for exact, finite-size analysis of graph-theoretical measures.
  • This approach generalizes to other algebraically defined network properties.