Protean Graphs with a Variety of Ranking Schemes

We introduce a new class of random graph models for complex realworld networks, based on the protean graph model by à Luczak and PraÃlat. Our generalized protean graph models have two distinguishing features. First, they are not growth models, but instead are based on the assumption that a “steady state” of large but finite size has been reached. Second, the models assume that the vertices are ranked according to a given ranking scheme, and the rank of a vertex determines the probability that that vertex receives a link in a given time step. Precisely, the link probability is proportional to the rank raised to the power −α, where the attachment strength α is a tunable parameter. We show that the model leads to a power law degree distribution with exponent 1 + 1/α for ranking schemes based on a given prestige label, or on the degree of a vertex. We also study a scheme where each vertex receives an initial rank chosen randomly according to a biased distribution. In this case, the degree distribution depends on the distribution of the initial rank. For one particular choice of parameters we obtain a power law with an exponent that depends both on α and on a parameter determining the initial rank distribution.