The exact formula for the average path length of Apollonian networks is
found. With the help of recursion relations derived from the self-similar
structure, we obtain the exact solution of average path length, dΛtβ,
for Apollonian networks. In contrast to the well-known numerical result
dΛtββ(lnNtβ)3/4 [Phys. Rev. Lett. \textbf{94}, 018702
(2005)], our rigorous solution shows that the average path length grows
logarithmically as dΛtββlnNtβ in the infinite limit of network
size Ntβ. The extensive numerical calculations completely agree with our
closed-form solution.Comment: 8 pages, 4 figure