A box-covering algorithm for fractal scaling in scale-free networks

Area of Science:

• Network science
• Complex systems analysis
• Fractal geometry

Background:

• Scale-free (SF) networks exhibit complex structures.
• Measuring fractal dimension in SF networks is challenging.
• Existing algorithms may not capture SF network properties accurately.

Purpose of the Study:

• Investigate a random sequential box-covering algorithm for fractal dimension measurement in SF networks.
• Determine the necessity of box-splitting allowance for fractal scaling.
• Analyze vertex distribution from a cluster-growing perspective.

Main Methods:

• Utilized a random sequential box-covering algorithm with Monte Carlo steps.
• Introduced box-split allowance where divided boxes are counted once.
• Examined algorithm behavior with and without box-split allowance.
• Applied cluster-growing perspective, allowing box overlap.

Main Results:

• Box-split allowance is crucial for fractal scaling in SF networks.
• Box-split allowance is inessential for regular lattices and Euclidean fractals.
• Vertex box membership distribution is heterogeneous for SF networks.
• Vertex box membership distribution is Poisson-type for conventional fractals.

Conclusions:

• The box-splitting mechanism is key to accurately measuring fractal dimensions in SF networks.
• The algorithm's behavior differs significantly between SF networks and conventional fractal objects.
• Understanding vertex distribution provides insights into network topology.