  The principle that 'the brand effect is attractive' underlies preferential
attachment. Here we show that the brand effect is just one dimension of
attractiveness. Another dimension is competitiveness. We firstly develop a
general framework that allows us to investigate the competitive aspect of real
networks, instead of simply preferring popular nodes. Our model accurately
describes the evolution of social and technological networks. The phenomenon
which more competitive nodes become richer links can help us to understand the
evolution of many competitive systems in nature and society. In general, the
paper provides an explicit analytical expression of degree distributions of the
network. In particular, the model yields a nontrivial time evolution of nodes'
properties and scale-free behavior with exponents depending on the microscopic
parameters characterizing the competition rules. Secondly, through theoretical
analysis and numerical simulations, it reveals that our model has not only the
universality for the homogeneous weighted network, but also the character for
the heterogeneous weighted network. Thirdly, the paper also develops a model
based on a profit-driven mechanism. It can better describe the observed
phenomenon in enterprise cooperation networks. We show that standard
preferential attachment, the growing random graph, the initial attractiveness
model, the fitness model and weighted networks, can all be seen as degenerate
cases of our model.
