Self-avoiding walks and connective constants in clustered scale-free networks

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Random walks are common for modeling network navigation.
• Self-avoiding walks (SAWs) offer better insights into long-distance network characteristics.
• Scale-free networks exhibit a power-law degree distribution P(k)∼k^{-γ}.

Purpose of the Study:

• Investigate self-avoiding walks (SAWs) in clustered scale-free networks.
• Analyze the impact of clustering (triangle insertions) on network properties.
• Determine the connective constant (μ) and its dependence on network parameters.

Main Methods:

• Direct enumeration of self-avoiding walks (a_n) to calculate the connective constant (μ).
• Development of an analytical approach to complement enumeration results.
• Comparison of results from both direct enumeration and analytical methods.

Main Results:

• Clustered networks generally exhibit a larger number of SAWs (a_n) compared to unclustered networks with the same degree distribution.
• The asymptotic limit of the connective constant (μ) is dependent on the exponent (γ) of the degree distribution.
• For γ>3, μ converges to a finite value; for γ=3, μ_N diverges as lnN; for γ<3, μ_N∼N^{(3-γ)/2}.

Conclusions:

• Clustering significantly influences the long-distance behavior of self-avoiding walks in scale-free networks.
• The connective constant's asymptotic behavior is precisely characterized by the degree distribution exponent γ.
• The study provides a robust framework for understanding navigation and information spread in complex, clustered network structures.