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 qq, which are simultaneously scale-free, self-similar and disassortative. Our rigorous solution shows that the average path length of the HLs grows logarithmically as d̄t∼Ntlog(2q)2 in the infinite limit of network size of NtNt and that they are not small worlds but grow with a power-law relationship to the number of nodes.