Random growth scale-free networked models with an identical degree distribution and a tunable assortativity index

In this work, we propose two kinds of graphic operations by using triangle configuration, based on which we establish a family of random growth networked models G(t;p) where notations t and p represent time step and probability parameter, respectively. By studying some fundamental structural parameters both analytically and numerically, we show that (1) all the realizations G(t;p) follow the same power-law degree distribution with exponent γ=2+ln⁡3/ln⁡2 regardless of probability p and thus have scale-free feature; (2) each model G(t;p) has a relatively high clustering coefficient; and (3) while network G(t;1) has a small average path length, it is not a unique model possessing small-world property mainly because its diameter D(t;1) does not reach the theoretical lower bound. Next, we make use of assortativity index R to quantify the tendency of forming connection between vertices and observe that (1) model G(t;0) exhibits disassortative mixing because the corresponding index R(t;0) is non-positive, and (2) model G(t;1) is in the opposite direction. As a result, we demonstrate that random model G(t;p) has a tunable quantity R(t;p) controlled by probability p. In addition, we exactly determine the total number of spanning trees of deterministic models G(t;1) and G(t;0) and also calculate the entropy of spanning trees.