Complex growing networks with intrinsic vertex fitness

One of the major questions in complex network research is to identify the range of mechanisms by which a complex network can self organize into a scale-free state. In this paper we investigate the interplay between a fitness linking mechanism and both random and preferential attachment. In our models, each vertex is assigned a fitness x, drawn from a probability distribution rho(x). In Model A, at each time step a vertex is added and joined to an existing vertex, selected at random, with probability p and an edge is introduced between vertices with fitnesses x and y, with a rate f(x,y), with probability 1-p. Model B differs from Model A in that, with probability p, edges are added with preferential attachment rather than randomly. The analysis of Model A shows that, for every fixed fitness x, the network's degree distribution decays exponentially. In Model B we recover instead a power-law degree distribution whose exponent depends only on p, and we show how this result can be generalized. The properties of a number of particular networks are examined.