Multiple probabilistic networks with scale-free property

It is well-known that the random scale-free networks are ubiquitous in the world and are applied in many areas of scientific research. Most previous networks are obtained from a single probability parameter, while our networks are produced by multiple probability parameters. This paper aims at generating a family of random scale-free networks by graphic operations based on probabilistic behaviors. These random scale-free networks can span a network space S(p, q, r, t) with three probabilistic parameters p, q, and r holding on p + q + r = 1 with 0 ≤ p, q, r ≤ 1 at each time step t. Each network N(p, q, r, t) of S(p, q, r, t) is a dynamic network that will be produced by N(p, q, r, t − 1) based on three types of operations, called the type-A operation, the type-B operation, and the type-C operation. We will show the topological structures of each network N(p, q, r, t) by its average degree, degree distribution, diameter, and clustering coefficient, and, furthermore, compute the degree exponent γ=1+ln(4−r)ln⁡2, as well as power-law distribution, in order to reveal the scale-free behavior of N(p, q, r, t), which induces the whole space S(p, q, r, t) to be scale-free. Our findings are able to enrich the fundamental structure properties of complex networks, in particular scale-free networks.