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 ]_(t) , for Apollonian networks. In contrast to the well-known numerical result d[over ]_{t} proportional, variant(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 ]_(t) proportional, variantln N_(t) in the infinite limit of network size N_(t) . The extensive numerical calculations completely agree with our closed-form solution.