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 large variety of complex networks, in which some are deterministic and others are random. The goal of this paper is to generate a class of random scale-free networks. To achieve this, we introduce three types of operations, i.e., rectangle operation, diamond operation, and triangle operation, and provide the concrete process for generating random scale-free networks N(p,q,r,t), where probability parameters p,q,r hold on p+q+r=1 with 0≤p,q,r≤1. We then discuss their topological properties, such as average degree, degree distribution, diameter, and clustering coefficient. First, we calculate the average degree of each member and discover that each member is a sparse graph. Second, by computing the degree distribution of our network N(p,q,r,t), we find that degree distribution obeys the power-law distribution, which implies that each member is scale-free. Next, according to our analysis of the diameter of our network N(p,q,r,t), we reveal the fact that the diameter may abruptly transform from small to large. Afterward, we give the calculation process of the clustering coefficient and discover that its value is mainly determined by r.