  We consider a class of simple, non-trivial models of evolving weighted
scale-free networks. The network evolution in these models is determined by
attachment of new vertices to ends of preferentially chosen weighted edges.
Resulting networks have scale-free distributions of the edge weight, of the
vertex degree, and of the vertex strength. We discuss situations where this
mechanism operates. Apart of stochastic models of weighted networks, we
introduce a wide class of deterministic, scale-free, weighted graphs with the
small-world effect. We show also how one can easily construct an equilibrium
weighted network by using a generalization of the configuration model.

Statistical Mechanics