Exact analytical solution of average path length for Apollonian networks

TL;DR

This paper derives an exact analytical formula for the average path length in Apollonian networks, revealing it grows logarithmically with network size, contrasting previous numerical estimates.

Contribution

It provides the first exact solution for the average path length in Apollonian networks using recursion relations based on their self-similar structure.

Findings

Average path length grows as ln N_t in large networks.

Analytical results agree with extensive numerical calculations.

Contrasts with previous numerical estimates of (ln N_t)^{3/4} growth.

Abstract

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 $\bar{d}_t \propto (\ln N_t)^{3/4}$ [Phys. Rev. Lett. \textbf{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.