Growing Cayley trees described by a Fermi distribution

Area of Science:

• Complex Networks
• Statistical Physics
• Network Science

Background:

• Growing Cayley trees are a model for hierarchical network structures.
• Thermal noise introduces randomness and influences network properties.
• Understanding bond strength distributions is crucial for network behavior.

Purpose of the Study:

• To investigate the statistical properties of bond strengths in growing Cayley trees under thermal noise.
• To determine the applicable statistical mechanics models for network evolution.
• To relate findings to existing network models like scale-free networks.

Main Methods:

• Simulating growing Cayley trees with thermal noise.
• Analyzing the distribution of bond strengths (energies).
• Examining network evolution in limiting temperature cases (T→0 and T→∞).

Main Results:

• The distribution of bond strengths follows Fermi statistics.
• Network evolution converges to the Eden model as temperature approaches zero.
• Network evolution converges to the invasion percolation model as temperature approaches infinity.

Conclusions:

• Fermi statistics accurately describe bond strengths in these growing hierarchical networks.
• The study connects Cayley tree evolution to established models like Eden and invasion percolation.
• Results offer insights into the relationship between network structure and statistical mechanics, including scale-free networks described by Bose statistics.