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Analytical solution of average path length for Apollonian networks.
Zhongzhi Zhang1, Lichao Chen, Shuigeng Zhou
1Department of Computer Science and Engineering, Fudan University, Shanghai 200433, China.
Summary
This study provides a rigorous solution for the average path length in Apollonian networks. The findings reveal that average path length grows logarithmically with network size, differing from previous numerical estimations.
Area of Science:
- Network Science
- Complex Systems Analysis
- Mathematical Physics
Background:
- Apollonian networks are a class of fractal networks with a self-similar structure.
- Previous studies suggested a numerical relationship for average path length growth (d[over ]_{t} proportional, variant(ln N_(t));(3/4)).
Purpose of the Study:
- To derive a rigorous analytical solution for the average path length of Apollonian networks.
- To investigate the asymptotic behavior of average path length as network size approaches infinity.
Main Methods:
- Utilized recursion relations derived from the self-similar properties of Apollonian networks.
- Developed a closed-form solution for the average path length.
Main Results:
- The rigorous solution demonstrates that the average path length (d[over ]_(t)) grows logarithmically with the network size (N_(t)), specifically d[over ]_(t) proportional, variantln N_(t).
- This analytical result contrasts with prior numerical findings.
Conclusions:
- The derived closed-form solution accurately describes the average path length in Apollonian networks.
- The study establishes a logarithmic growth of average path length in the infinite limit of network size.
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