Growing random networks with fitness

Three models of growing random networks with fitness-dependent growth rates are analysed using the rate equations for the distribution of their connectivities. In the first model (A), a network is built by connecting incoming nodes to nodes of connectivity k and random additive fitness eta, with rate (k - 1) + eta. For eta > 0 we find the connectivity distribution is power law with exponent y = (eta) + 2. In the second model (B), the network is built by connecting nodes to nodes of connectivity k, random additive fitness eta and random multiplicative fitness zeta with rate zeta(k - 1) + eta. This model also has a power law connectivity distribution, but with an exponent which depends on the multiplicative fitness at each node. In the third model (C), a directed graph is considered and is built by the addition of nodes and the creation of links. A node with fitness (alpha, beta), i incoming links and j outgoing links gains a new incoming link with rate alpha(i + 1), and a new outgoing link with rate beta(j + 1). The distributions of the number of incoming and outgoing links both scale as power laws, with inverse logarithmic corrections. (C) 2002 Elsevier Science B.V. All rights reserved.