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)∼w-α with the exponent α∈(0,2]. The power-law dependence of the weight distribution is also proved to hold, for asymptotically large w, in the case of networks in which a link between nodes i and j carries a load wij, determined by node degrees ki and kj at the final stage of the network growth, according to the relation wij=(kikj)θ with θ∈(-1,0]. For this class of networks, the scaling exponent σ describing the weight distribution is found to satisfy the relationship σ=(λ+θ)/(1+θ), where λ is the scaling index characterizing the distribution of node degrees, n(k)∼k-λ.