Exact analytical solution of average path length for Apollonian networks

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, ${\bar{d}}_t$, for Apollonian networks. In contrast to the well-known numerical result ${\bar{d}}_t\propto (\ln N_t{)}^{3\mathrm{/}4}$ [Phys. Rev. Lett. $\mathbf{\text{94}}$, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically as ${\bar{d}}_t\propto \ln N_t$ in the infinite limit of network size $N_t$. The extensive numerical calculations completely agree with our closed-form solution.