Random walks and consensus problem on tree network with an identical degree distribution

It is of increasing interest to study various dynamics occurring on network models that reliably display some properties observed in real-world networks. In this work, we first introduce two graph operations and then propose a generative framework for creating an ensemble of new stochastic tree networks Tm,p(t) where p is probability parameter and m is a tunable parameter belonging to positive integer set. For given m, all the realizations in stochastic networks Tm,p(t) turn out to follow the same power-law degree distribution with exponent larger than 3, and thus have scale-free feature. In addition, fractal feature is found on our networks. Also, we study degree–degree correlation of stochastic networks Tm,p(t) by determining assortativity, and show that networks Tm,p(t) can have assortative structure by tuning parameters p and m. More importantly, networks Tm,1(t) are proved to be always disassortative independent of both the choice of seminal model and value for parameter m. Besides that, we obtain that there exists a structural transition in degree–degree correlation of tree network Tm,0(t) by properly selecting original model and parameter m. These degree–degree correlation phenomena have not been reported in previous scale-free tree networks with fractal feature. Then, we study random walks on stochastic networks Tm,p(t) in detail and obtain analytical solution to mean hitting time in a mathematically rigorous manner. The results mean that two fundamental structural parameters, namely, diameter and fractal dimension, of networks Tm,p(t) have a remarkable influence on the quantity. This further provides a guideline for optimizing the topological structure of tree networks. Next, we consider consensus problem on networks Tm,p(t) and analytically determine the solution of first-order noise coherence. Similarly, diameter and fractal dimension are proved to play a key role in determination of this parameter. Finally, we conduct extensive experiments, which demonstrates that computer simulations are in perfect agreement with the theoretical analysis.