Effective dimensions and percolation in hierarchically structured scale-free networks.

Area of Science:

• Network Science
• Complex Systems Analysis
• Fractal Geometry

Background:

• Complex networks exhibit diverse topological and dynamical properties.
• Characterizing the intrinsic dimensionality of these networks is crucial for understanding their behavior.
• Hierarchically structured networks present unique challenges in dimensional analysis.

Purpose of the Study:

• To introduce precise definitions for characterizing the fractal dimensions of complex networks.
• To compute these dimensions for a specific hierarchically structured network.
• To investigate the dimensional properties of complex networks with scale-free connectivity.

Main Methods:

• Development of appropriate mathematical definitions for fractal dimensions.
• Computation of these dimensions on a hierarchically structured network.
• Comparison of dimensional characterization with site percolation statistics and dynamical processes.

Main Results:

• The proposed definitions effectively characterize fractal properties.
• The analyzed hierarchically structured network, despite its complexity and scale-free connectivity, exhibits an approximate one-dimensional nature.
• Dimensional characterization is consistent with findings from percolation theory and dynamical simulations.

Conclusions:

• Complex networks can possess surprisingly low dimensionality.
• The introduced dimensional characterization provides a valuable tool for analyzing network structures and dynamics.
• The one-dimensional nature impacts various network processes, including percolation and site dynamics.