Structure Diversity and Mean Hitting Time for Random Walks on Stochastic Uniform Growth Tree Networks

In this article, we propose a principled framework using Vertex-based and Edge-based uniform generation mechanisms to build stochastic uniform growth tree networks that have a wide range of applications in various fields including physics, engineering, chemistry, ect., and then uncover the associated structural features analytically. When considering vertex-degree distribution, there exist three different classes of forms in the thermodynamic limit, i.e., exponential distribution, power-law distribution along with multiple-point distribution. At meantime, three distinct structural shapes are observed in the study of fractal phenomena, that is, fractal feature, critical phenomenon and non-fractal property. In addition, we obtain the analytical solution to fractal dimension for fractal structure from the probability point of view. More importantly, some well-known models, for instance, Vicsek fractal and T-graph, fall into our framework. Next, we precisely consider two families of stochastic uniform growth tree networks generated through the proposed framework. Specifically, we derive the analytic solution to mean hitting time <inline-formula><tex-math notation="LaTeX">$\langle \mathcal {H}\rangle$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>〈</mml:mo><mml:mi mathvariant="script">H</mml:mi><mml:mo>〉</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="ma-ieq1-3206210.gif"/></alternatives></inline-formula> for measuring efficiency of delivering information on networks in a random-walk-based manner, and find that the introduction of randomness certainly enriches the scaling exponent of quantity <inline-formula><tex-math notation="LaTeX">$\langle \mathcal {H}\rangle$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>〈</mml:mo><mml:mi mathvariant="script">H</mml:mi><mml:mo>〉</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="ma-ieq2-3206210.gif"/></alternatives></inline-formula>. Finally, we conduct extensive experiments, which suggests that computer simulations are in good agreement with theoretical analysis.