Stochastic growth tree networks with an identical fractal dimension: Construction and mean hitting time for random walks

Abstract

There is little attention paid to stochastic tree networks in comparison with the corresponding deterministic analogs in the current study of fractal trees. In this paper, we propose a principled framework for producing a family of stochastic growth tree networks [Formula: see text] possessing fractal characteristic, where [Formula: see text] represents the time step and parameter [Formula: see text] is the number of vertices newly created for each existing vertex at generation. To this end, we introduce two types of generative ways, i.e., Edge-Operation and Edge-Vertex-Operation. More interestingly, the resulting stochastic trees turn out to have an identical fractal dimension [Formula: see text] regardless of the introduction of randomness in the growth process. At the same time, we also study many other structural parameters including diameter and degree distribution. In both extreme cases, our tree networks are deterministic and follow multiple-point degree distribution and power-law degree distribution, respectively. Additionally, we consider random walks on stochastic growth tree networks [Formula: see text] and derive an expectation estimation for mean hitting time [Formula: see text] in an effective combinatorial manner instead of commonly used spectral methods. The result shows that on average, the scaling of mean hitting time [Formula: see text] obeys [Formula: see text], where [Formula: see text] represents vertex number and exponent [Formula: see text] is equivalent to [Formula: see text]. In the meantime, we conduct extensive experimental simulations and observe that empirical analysis is in strong agreement with theoretical results.