  The rate equations are used to study the scale-free behavior of the weight
distribution in evolving networks whose topology is determined only by degrees
of preexisting vertices. An analysis of these equations shows that the degree
distribution and thereby the weight distribution remain unchanged when the
probability rate of attaching new nodes is replaced with some unnormalized rate
determined by the ratio of the degree of a randomly selected old node to the
maximal node degree at the current stage of the network evolution. Such a
modification of the attachment rule is argued to accelerate considerably
numerical simulations of both unweighted and weighted networks belonging to the
class of investigated evolving systemes. It is also proved that the studied
rate equations have a solution corresponding to the total (concentrated at
individual vertices) distribution displaying the power-law behavior for
asymptotically large weights.
