Constructions and properties of a class of random scale-free networks

Complex networks have abundant and extensive applications in real life. Recently, researchers have proposed a number of complex networks, in which some are deterministic and others are random. Compared with deterministic networks, random network is not only interesting and typical but also practical to illustrate and study many real-world complex networks, especially for random scale-free networks. Here, we introduce three types of operations, i.e., type-A operation, type-B operation and type-C operation, for generating random scale-free networks $N(p,q,r,t)$. On the basis of our operations, we put forward the concrete process of producing networks, which constitute the network space $\mathcal{N}(p,q,r,t)$, and then discuss their topological properties. Firstly, we calculate the range of the average degree of each member in our network space and discover that each member is a sparse network. Secondly, we prove that each member in our space obeys the power-law distribution with degree exponent $γ=1+\frac{\ln(4-r)}{\ln2}$, which implies that each member is scale-free. Next, we analyze the diameter, and find that the diameter may abruptly transform from small to large due to type-B operation. Afterwards, we study the clustering coefficient of network and discover that its value is only determined by type-C operation. Ultimately, we make an elaborate conclusion. \\ \textbf{Keywords:} Random network; degree distribution; diameter; clustering coefficient.