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The principle that 'the brand effect is attractive' underlies preferentialattachment. Here we show that the brand effect is just one dimension ofattractiveness. Another dimension is competitiveness. We firstly develop ageneral framework that allows us to investigate the competitive aspect of realnetworks, instead of simply preferring popular nodes. Our model accuratelydescribes the evolution of social and technological networks. The phenomenonwhich more competitive nodes become richer links can help us to understand theevolution of many competitive systems in nature and society. In general, thepaper provides an explicit analytical expression of degree distributions of thenetwork. In particular, the model yields a nontrivial time evolution of nodes'properties and scale-free behavior with exponents depending on the microscopicparameters characterizing the competition rules. Secondly, through theoreticalanalysis and numerical simulations, it reveals that our model has not only theuniversality for the homogeneous weighted network, but also the character forthe heterogeneous weighted network. Thirdly, the paper also develops a modelbased on a profit-driven mechanism. It can better describe the observedphenomenon in enterprise cooperation networks. We show that standardpreferential attachment, the growing random graph, the initial attractivenessmodel, the fitness model and weighted networks, can all be seen as degeneratecases of our model.

The co-evolution of brand effect and competitiveness in evolving networks