Analytical solution of average path length for Apollonian networks
Abstract
With the help of recursion relations derived from the self-similar structure, we obtain the solution of average path length, dmacr t , for Apollonian networks. In contrast to the well-known numerical result dmacr t∝(lnNt)3/4 [J. S. Andrade, Jr. , Phys. Rev. Lett. 94, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically as dmacr t∝lnNt in the infinite limit of network size Nt . The extensive numerical calculations completely agree with our closed-form solution.
- Publication:
-
Physical Review E
- Pub Date:
- January 2008
- DOI:
- 10.1103/PhysRevE.77.017102
- arXiv:
- arXiv:0706.3491
- Bibcode:
- 2008PhRvE..77a7102Z
- Keywords:
-
- 89.75.Hc;
- 89.75.Da;
- 02.10.Ox;
- 05.10.-a;
- Networks and genealogical trees;
- Systems obeying scaling laws;
- Combinatorics;
- graph theory;
- Computational methods in statistical physics and nonlinear dynamics;
- Condensed Matter - Statistical Mechanics;
- Physics - Physics and Society
- E-Print:
- 8 pages, 4 figures