Full eigenvalues of the Markov matrix for scale-free polymer networks

Area of Science:

• Complex systems analysis
• Network science
• Statistical mechanics

Background:

• Spectral methods using Markov matrix eigenvalues are crucial for understanding complex system dynamics.
• Random walks on networks provide insights into structural and dynamical properties.

Purpose of the Study:

• To derive exact analytical expressions for eigenvalues and their multiplicities of Markov matrices for scale-free polymer networks.
• To obtain an explicit formula for random target access time on these networks.
• To validate the derived eigenvalues and degeneracies using their connection to spanning trees.

Main Methods:

• Analysis of Markov matrix properties for a specific class of scale-free polymer networks.
• Derivation of exact analytical expressions for eigenvalues and their multiplicities.
• Formulation of a random target access time formula based on spectral properties.
• Utilizing the relationship between eigenvalues and the number of spanning trees for validation.

Main Results:

• Exact analytical expressions for all eigenvalues and their multiplicities of the Markov matrix for scale-free polymer networks.
• An explicit formula for the random target access time on these networks.
• Confirmation of eigenvalue validity and degeneracies through their link to spanning trees.

Conclusions:

• The study provides a complete spectral characterization of scale-free polymer networks.
• The derived random target access time formula offers a new tool for network analysis.
• The findings validate the utility of spectral methods in complex network research.