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 T possessing fractal characteristic, where t represents the time step and parameter m 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 d = ln ⁡ 2 ( m + 1 ) / ln ⁡ 2 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 T and derive an expectation estimation for mean hitting time ⟨ H ⟩ in an effective combinatorial manner instead of commonly used spectral methods. The result shows that on average, the scaling of mean hitting time ⟨ H ⟩ obeys ⟨ H ⟩ = | T |, where | T | represents vertex number and exponent λ is equivalent to 1 + ln ⁡ 2 / ln ⁡ 2 ( m + 1 ). In the meantime, we conduct extensive experimental simulations and observe that empirical analysis is in strong agreement with theoretical results.