The average path length for a class of scale-free fractal hierarchical lattices: Rigorous results

With the help of the recurrence relations derived from the self-similar structure, we obtain the closed-form solution for the average path length of a class of scale-free fractal hierarchical lattices (HLs) with a general parameter q, which are simultaneously scale-free, self-similar and disassortative. Our rigorous solution shows that the average path length of the HLs grows logarithmically as in the infinite limit of network size of Nt and that they are not small worlds but grow with a power-law relationship to the number of nodes.