Analytical Solution Of Average Path Length For Apollonian Networks

With the help of recursion relations derived from the self-similar structure, we obtain the solution of average path length, (d) over bar (t), for Apollonian networks. In contrast to the well-known numerical result (d) over bar (t)proportional to(ln N-t)(3/4) [J. S. Andrade, Jr. , Phys. Rev. Lett. 94, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically as (d) over bar (t)proportional to ln N-t in the infinite limit of network size N-t. The extensive numerical calculations completely agree with our closed-form solution.