Scale-free networks with invariable diameter and density feature: Counterexamples.

Area of Science:

• Network science
• Statistical physics

Background:

• Existing scale-free network models often assume sparsity and constant average degrees.
• Theoretical models with power-law degree distributions have limitations in simultaneously satisfying key network properties.

Purpose of the Study:

• To propose a new class of scale-free networks G(t;m) with unique properties.
• To analyze the trapping problem and derive a closed-form solution for mean hitting time on these networks.

Main Methods:

• Development of a novel network model G(t;m) with specific properties.
• Mathematical analysis of network characteristics including average degree, power-law parameter, and diameter.
• Investigation of the trapping problem and calculation of mean hitting time.

Main Results:

• The proposed networks G(t;m) exhibit dense properties, a fixed power-law parameter (γ=2), and invariant diameter during growth.
• A closed-form solution for the mean hitting time (〈H〉_{t}) was derived.
• The mean hitting time was found to be logarithmically related to the number of vertices, an unexpected finding.

Conclusions:

• The novel scale-free networks G(t;m) serve as counterexamples to many existing power-law network models.
• These models provide new insights into network structure and dynamics, particularly for the trapping problem.