Random walks on Fibonacci treelike models: emergence of power law.

The scale-free feature is prevalent in a large number of complex networks in nature and society. So, more and more researchers pay more attention to such kind of network models. In this paper, we propose a class of growth models, named Fibonacci trees $F(t)$, with respect to the intrinsic character of Fibonacci sequence $\{F_{t}\}$. First, we turn out model $F(t)$ to be scale-free because its degree distribution satisfies the power law with power-law exponent $\gamma$ greater than $3$. And then, we study analytically two significant indices correlated to random walks on networks, namely, both the optimal mean first-passage time ($OMFPT$) and the mean first-passage time ($MFPT$). We provide a deterministic algorithm for computing $OMFPT$ and then obtain a closed-form expression of $OMFPT$. Meanwhile, an approximation algorithm and a deterministic algorithm are introduced, respectively, to capture a valid solution to $MFPT$. We demonstrate that our algorithms are able to be widely applied to many network models with self-similar structure to derive desired solution to $OMFPT$ or $MFPT$. Especially, we capture a nontrivial result that the $MFPT$ reported by deterministic algorithm is no longer correlated linearly with the order of model $F(t)$. Extensive analytical results are in better agreement with our statements.