  We consider a fitness model assumed to generate simple graphs with power-law
heavy-tailed degree sequence: P(k) \propto k^{-1-\alpha} with 0 < \alpha < 1,
in which the corresponding distributions do not posses a mean. We discuss the
situations in which the model is used to produce a multigraph and examine what
happens if the multiple edges are merged to a single one and thus a simple
graph is built. We give the relation between the (normalized) fitness parameter
r and the expected degree \nu of a node and show analytically that it possesses
non-trivial intermediate and final asymptotic behaviors. We show that the model
produces P(k) \propto k^{-2} for large values of k independent of \alpha. Our
analytical findings are confirmed by numerical simulations.
