An ensemble of random graphs with identical degree distribution.

Degree distribution, or equivalently called degree sequence, has been commonly used to study a large number of complex networks in the past few years. This reveals some intriguing results, for instance, the popularity of power-law distribution in most of these networks under consideration. Along such a research line, in this paper, we generate an ensemble of random graphs with an identical degree distribution P(k)∼k (γ=3) as proved shortly, denoted as graph space N(p,q,t), where probability parameters p and q hold on p+q=1. Next, we study some topological structure properties of great interest on each member in the graph space N(p,q,t) using both precisely analytical calculations and extensively numerical simulations, as follows. From the theoretical point of view, given an ultrasmall constant p, perhaps only the graph model N(1,0,t) is small-world and the others are not in terms of diameter. Then, we obtain exact solutions for a spanning tree number on two deterministic graph models in the graph space N(p,q,t), which gives both upper bound and lower bound for that of other members. Meanwhile, for an arbitrary p(≠1), we prove using the Pearson correlation coefficient that the graph model N(p,q,t) does go through two phase transitions over time, i.e., starting by a nonassortative pattern, then suddenly going into a disassortative region, and gradually converging to an initial position (nonassortative point). Therefore, to some extent, the three topological parameters above can serve as the complementary measures for degree distribution to help us clearly distinguish all members in the graph space N(p,q,t). In addition, one "null" graph model is built.