On the emergence of scaling in weighted networks

General conditions for the appearance of the power-law distribution of total weights concentrated in vertices of complex network systems are established. By use of the rate equation approach for networks evolving by connectivity-governed attachment of every new node to p >= 1 exiting nodes and by ascription to every new link a weight taken from algebraic distributions, independent of network topologies, it is shown that the distribution of the total weight w asymptotically follows the power law, P(w)similar to w(-alpha) with the exponent alpha epsilon (0, 2]. The power-law dependence of the weight distribution is also proved to hold, for asymptotically large iv, in the case of networks in which a link between nodes i and j carries a load w(ij), determined by node degrees k(i) and k(j) at the final stage of the network growth, according to the relation w(ij) = (k(i)k(j))(0) with theta epsilon (-1,0]. For this class of networks, the scaling exponent sigma describing the weight distribution is found to satisfy the relationship sigma = (lambda + theta)/(l + theta), where lambda is the scaling index characterizing the distribution of node degrees, n(k)similar to k(-alpha). (C) 2007 Elsevier B.V. All rights reserved.